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Population Growth: The Standard & Logistic Equations

Slide Duration:

Section 1: Limits and Derivatives
Overview & Slopes of Curves

42m 8s

Intro
0:00
Overview & Slopes of Curves
0:21
Differential and Integral
0:22
Fundamental Theorem of Calculus
6:36
Differentiation or Taking the Derivative
14:24
What Does the Derivative Mean and How do We Find it?
15:18
Example: f'(x)
19:24
Example: f(x) = sin (x)
29:16
General Procedure for Finding the Derivative of f(x)
37:33
More on Slopes of Curves

50m 53s

Intro
0:00
Slope of the Secant Line along a Curve
0:12
Slope of the Tangent Line to f(x) at a Particlar Point
0:13
Slope of the Secant Line along a Curve
2:59
Instantaneous Slope
6:51
Instantaneous Slope
6:52
Example: Distance, Time, Velocity
13:32
Instantaneous Slope and Average Slope
25:42
Slope & Rate of Change
29:55
Slope & Rate of Change
29:56
Example: Slope = 2
33:16
Example: Slope = 4/3
34:32
Example: Slope = 4 (m/s)
39:12
Example: Density = Mass / Volume
40:33
Average Slope, Average Rate of Change, Instantaneous Slope, and Instantaneous Rate of Change
47:46
Example Problems for Slopes of Curves

59m 12s

Intro
0:00
Example I: Water Tank
0:13
Part A: Which is the Independent Variable and Which is the Dependent?
2:00
Part B: Average Slope
3:18
Part C: Express These Slopes as Rates-of-Change
9:28
Part D: Instantaneous Slope
14:54
Example II: y = √(x-3)
28:26
Part A: Calculate the Slope of the Secant Line
30:39
Part B: Instantaneous Slope
41:26
Part C: Equation for the Tangent Line
43:59
Example III: Object in the Air
49:37
Part A: Average Velocity
50:37
Part B: Instantaneous Velocity
55:30
Desmos Tutorial

18m 43s

Intro
0:00
Desmos Tutorial
1:42
Desmos Tutorial
1:43
Things You Must Learn To Do on Your Particular Calculator
2:39
Things You Must Learn To Do on Your Particular Calculator
2:40
Example I: y=sin x
4:54
Example II: y=x³ and y = d/(dx) (x³)
9:22
Example III: y = x² {-5 <= x <= 0} and y = cos x {0 < x < 6}
13:15
The Limit of a Function

51m 53s

Intro
0:00
The Limit of a Function
0:14
The Limit of a Function
0:15
Graph: Limit of a Function
12:24
Table of Values
16:02
lim x→a f(x) Does not Say What Happens When x = a
20:05
Example I: f(x) = x²
24:34
Example II: f(x) = 7
27:05
Example III: f(x) = 4.5
30:33
Example IV: f(x) = 1/x
34:03
Example V: f(x) = 1/x²
36:43
The Limit of a Function, Cont.
38:16
Infinity and Negative Infinity
38:17
Does Not Exist
42:45
Summary
46:48
Example Problems for the Limit of a Function

24m 43s

Intro
0:00
Example I: Explain in Words What the Following Symbols Mean
0:10
Example II: Find the Following Limit
5:21
Example III: Use the Graph to Find the Following Limits
7:35
Example IV: Use the Graph to Find the Following Limits
11:48
Example V: Sketch the Graph of a Function that Satisfies the Following Properties
15:25
Example VI: Find the Following Limit
18:44
Example VII: Find the Following Limit
20:06
Calculating Limits Mathematically

53m 48s

Intro
0:00
Plug-in Procedure
0:09
Plug-in Procedure
0:10
Limit Laws
9:14
Limit Law 1
10:05
Limit Law 2
10:54
Limit Law 3
11:28
Limit Law 4
11:54
Limit Law 5
12:24
Limit Law 6
13:14
Limit Law 7
14:38
Plug-in Procedure, Cont.
16:35
Plug-in Procedure, Cont.
16:36
Example I: Calculating Limits Mathematically
20:50
Example II: Calculating Limits Mathematically
27:37
Example III: Calculating Limits Mathematically
31:42
Example IV: Calculating Limits Mathematically
35:36
Example V: Calculating Limits Mathematically
40:58
Limits Theorem
44:45
Limits Theorem 1
44:46
Limits Theorem 2: Squeeze Theorem
46:34
Example VI: Calculating Limits Mathematically
49:26
Example Problems for Calculating Limits Mathematically

21m 22s

Intro
0:00
Example I: Evaluate the Following Limit by Showing Each Application of a Limit Law
0:16
Example II: Evaluate the Following Limit
1:51
Example III: Evaluate the Following Limit
3:36
Example IV: Evaluate the Following Limit
8:56
Example V: Evaluate the Following Limit
11:19
Example VI: Calculating Limits Mathematically
13:19
Example VII: Calculating Limits Mathematically
14:59
Calculating Limits as x Goes to Infinity

50m 1s

Intro
0:00
Limit as x Goes to Infinity
0:14
Limit as x Goes to Infinity
0:15
Let's Look at f(x) = 1 / (x-3)
1:04
Summary
9:34
Example I: Calculating Limits as x Goes to Infinity
12:16
Example II: Calculating Limits as x Goes to Infinity
21:22
Example III: Calculating Limits as x Goes to Infinity
24:10
Example IV: Calculating Limits as x Goes to Infinity
36:00
Example Problems for Limits at Infinity

36m 31s

Intro
0:00
Example I: Calculating Limits as x Goes to Infinity
0:14
Example II: Calculating Limits as x Goes to Infinity
3:27
Example III: Calculating Limits as x Goes to Infinity
8:11
Example IV: Calculating Limits as x Goes to Infinity
14:20
Example V: Calculating Limits as x Goes to Infinity
20:07
Example VI: Calculating Limits as x Goes to Infinity
23:36
Continuity

53m

Intro
0:00
Definition of Continuity
0:08
Definition of Continuity
0:09
Example: Not Continuous
3:52
Example: Continuous
4:58
Example: Not Continuous
5:52
Procedure for Finding Continuity
9:45
Law of Continuity
13:44
Law of Continuity
13:45
Example I: Determining Continuity on a Graph
15:55
Example II: Show Continuity & Determine the Interval Over Which the Function is Continuous
17:57
Example III: Is the Following Function Continuous at the Given Point?
22:42
Theorem for Composite Functions
25:28
Theorem for Composite Functions
25:29
Example IV: Is cos(x³ + ln x) Continuous at x=π/2?
27:00
Example V: What Value of A Will make the Following Function Continuous at Every Point of Its Domain?
34:04
Types of Discontinuity
39:18
Removable Discontinuity
39:33
Jump Discontinuity
40:06
Infinite Discontinuity
40:32
Intermediate Value Theorem
40:58
Intermediate Value Theorem: Hypothesis & Conclusion
40:59
Intermediate Value Theorem: Graphically
43:40
Example VI: Prove That the Following Function Has at Least One Real Root in the Interval [4,6]
47:46
Derivative I

40m 2s

Intro
0:00
Derivative
0:09
Derivative
0:10
Example I: Find the Derivative of f(x)=x³
2:20
Notations for the Derivative
7:32
Notations for the Derivative
7:33
Derivative & Rate of Change
11:14
Recall the Rate of Change
11:15
Instantaneous Rate of Change
17:04
Graphing f(x) and f'(x)
19:10
Example II: Find the Derivative of x⁴ - x²
24:00
Example III: Find the Derivative of f(x)=√x
30:51
Derivatives II

53m 45s

Intro
0:00
Example I: Find the Derivative of (2+x)/(3-x)
0:18
Derivatives II
9:02
f(x) is Differentiable if f'(x) Exists
9:03
Recall: For a Limit to Exist, Both Left Hand and Right Hand Limits Must Equal to Each Other
17:19
Geometrically: Differentiability Means the Graph is Smooth
18:44
Example II: Show Analytically that f(x) = |x| is Nor Differentiable at x=0
20:53
Example II: For x > 0
23:53
Example II: For x < 0
25:36
Example II: What is f(0) and What is the lim |x| as x→0?
30:46
Differentiability & Continuity
34:22
Differentiability & Continuity
34:23
How Can a Function Not be Differentiable at a Point?
39:38
How Can a Function Not be Differentiable at a Point?
39:39
Higher Derivatives
41:58
Higher Derivatives
41:59
Derivative Operator
45:12
Example III: Find (dy)/(dx) & (d²y)/(dx²) for y = x³
49:29
More Example Problems for The Derivative

31m 38s

Intro
0:00
Example I: Sketch f'(x)
0:10
Example II: Sketch f'(x)
2:14
Example III: Find the Derivative of the Following Function sing the Definition
3:49
Example IV: Determine f, f', and f'' on a Graph
12:43
Example V: Find an Equation for the Tangent Line to the Graph of the Following Function at the Given x-value
13:40
Example VI: Distance vs. Time
20:15
Example VII: Displacement, Velocity, and Acceleration
23:56
Example VIII: Graph the Displacement Function
28:20
Section 2: Differentiation
Differentiation of Polynomials & Exponential Functions

47m 35s

Intro
0:00
Differentiation of Polynomials & Exponential Functions
0:15
Derivative of a Function
0:16
Derivative of a Constant
2:35
Power Rule
3:08
If C is a Constant
4:19
Sum Rule
5:22
Exponential Functions
6:26
Example I: Differentiate
7:45
Example II: Differentiate
12:38
Example III: Differentiate
15:13
Example IV: Differentiate
16:20
Example V: Differentiate
19:19
Example VI: Find the Equation of the Tangent Line to a Function at a Given Point
12:18
Example VII: Find the First & Second Derivatives
25:59
Example VIII
27:47
Part A: Find the Velocity & Acceleration Functions as Functions of t
27:48
Part B: Find the Acceleration after 3 Seconds
30:12
Part C: Find the Acceleration when the Velocity is 0
30:53
Part D: Graph the Position, Velocity, & Acceleration Graphs
32:50
Example IX: Find a Cubic Function Whose Graph has Horizontal Tangents
34:53
Example X: Find a Point on a Graph
42:31
The Product, Power & Quotient Rules

47m 25s

Intro
0:00
The Product, Power and Quotient Rules
0:19
Differentiate Functions
0:20
Product Rule
5:30
Quotient Rule
9:15
Power Rule
10:00
Example I: Product Rule
13:48
Example II: Quotient Rule
16:13
Example III: Power Rule
18:28
Example IV: Find dy/dx
19:57
Example V: Find dy/dx
24:53
Example VI: Find dy/dx
28:38
Example VII: Find an Equation for the Tangent to the Curve
34:54
Example VIII: Find d²y/dx²
38:08
Derivatives of the Trigonometric Functions

41m 8s

Intro
0:00
Derivatives of the Trigonometric Functions
0:09
Let's Find the Derivative of f(x) = sin x
0:10
Important Limits to Know
4:59
d/dx (sin x)
6:06
d/dx (cos x)
6:38
d/dx (tan x)
6:50
d/dx (csc x)
7:02
d/dx (sec x)
7:15
d/dx (cot x)
7:27
Example I: Differentiate f(x) = x² - 4 cos x
7:56
Example II: Differentiate f(x) = x⁵ tan x
9:04
Example III: Differentiate f(x) = (cos x) / (3 + sin x)
10:56
Example IV: Differentiate f(x) = e^x / (tan x - sec x)
14:06
Example V: Differentiate f(x) = (csc x - 4) / (cot x)
15:37
Example VI: Find an Equation of the Tangent Line
21:48
Example VII: For What Values of x Does the Graph of the Function x + 3 cos x Have a Horizontal Tangent?
25:17
28:23
Example IX: Evaluate
33:22
Example X: Evaluate
36:38
The Chain Rule

24m 56s

Intro
0:00
The Chain Rule
0:13
Recall the Composite Functions
0:14
Derivatives of Composite Functions
1:34
Example I: Identify f(x) and g(x) and Differentiate
6:41
Example II: Identify f(x) and g(x) and Differentiate
9:47
Example III: Differentiate
11:03
Example IV: Differentiate f(x) = -5 / (x² + 3)³
12:15
Example V: Differentiate f(x) = cos(x² + c²)
14:35
Example VI: Differentiate f(x) = cos⁴x +c²
15:41
Example VII: Differentiate
17:03
Example VIII: Differentiate f(x) = sin(tan x²)
19:01
Example IX: Differentiate f(x) = sin(tan² x)
21:02
More Chain Rule Example Problems

25m 32s

Intro
0:00
Example I: Differentiate f(x) = sin(cos(tanx))
0:38
Example II: Find an Equation for the Line Tangent to the Given Curve at the Given Point
2:25
Example III: F(x) = f(g(x)), Find F' (6)
4:22
Example IV: Differentiate & Graph both the Function & the Derivative in the Same Window
5:35
Example V: Differentiate f(x) = ( (x-8)/(x+3) )⁴
10:18
Example VI: Differentiate f(x) = sec²(12x)
12:28
Example VII: Differentiate
14:41
Example VIII: Differentiate
19:25
Example IX: Find an Expression for the Rate of Change of the Volume of the Balloon with Respect to Time
21:13
Implicit Differentiation

52m 31s

Intro
0:00
Implicit Differentiation
0:09
Implicit Differentiation
0:10
Example I: Find (dy)/(dx) by both Implicit Differentiation and Solving Explicitly for y
12:15
Example II: Find (dy)/(dx) of x³ + x²y + 7y² = 14
19:18
Example III: Find (dy)/(dx) of x³y² + y³x² = 4x
21:43
Example IV: Find (dy)/(dx) of the Following Equation
24:13
Example V: Find (dy)/(dx) of 6sin x cos y = 1
29:00
Example VI: Find (dy)/(dx) of x² cos² y + y sin x = 2sin x cos y
31:02
Example VII: Find (dy)/(dx) of √(xy) = 7 + y²e^x
37:36
Example VIII: Find (dy)/(dx) of 4(x²+y²)² = 35(x²-y²)
41:03
Example IX: Find (d²y)/(dx²) of x² + y² = 25
44:05
Example X: Find (d²y)/(dx²) of sin x + cos y = sin(2x)
47:48
Section 3: Applications of the Derivative
Linear Approximations & Differentials

47m 34s

Intro
0:00
Linear Approximations & Differentials
0:09
Linear Approximations & Differentials
0:10
Example I: Linear Approximations & Differentials
11:27
Example II: Linear Approximations & Differentials
20:19
Differentials
30:32
Differentials
30:33
Example III: Linear Approximations & Differentials
34:09
Example IV: Linear Approximations & Differentials
35:57
Example V: Relative Error
38:46
Related Rates

45m 33s

Intro
0:00
Related Rates
0:08
Strategy for Solving Related Rates Problems #1
0:09
Strategy for Solving Related Rates Problems #2
1:46
Strategy for Solving Related Rates Problems #3
2:06
Strategy for Solving Related Rates Problems #4
2:50
Strategy for Solving Related Rates Problems #5
3:38
Example I: Radius of a Balloon
5:15
12:52
Example III: Water Tank
19:08
Example IV: Distance between Two Cars
29:27
Example V: Line-of-Sight
36:20
More Related Rates Examples

37m 17s

Intro
0:00
0:14
Example II: Particle
4:45
Example III: Water Level
10:28
Example IV: Clock
20:47
Example V: Distance between a House and a Plane
29:11
Maximum & Minimum Values of a Function

40m 44s

Intro
0:00
Maximum & Minimum Values of a Function, Part 1
0:23
Absolute Maximum
2:20
Absolute Minimum
2:52
Local Maximum
3:38
Local Minimum
4:26
Maximum & Minimum Values of a Function, Part 2
6:11
Function with Absolute Minimum but No Absolute Max, Local Max, and Local Min
7:18
Function with Local Max & Min but No Absolute Max & Min
8:48
Formal Definitions
10:43
Absolute Maximum
11:18
Absolute Minimum
12:57
Local Maximum
14:37
Local Minimum
16:25
Extreme Value Theorem
18:08
Theorem: f'(c) = 0
24:40
Critical Number (Critical Value)
26:14
Procedure for Finding the Critical Values of f(x)
28:32
Example I: Find the Critical Values of f(x) x + sinx
29:51
Example II: What are the Absolute Max & Absolute Minimum of f(x) = x + 4 sinx on [0,2π]
35:31
Example Problems for Max & Min

40m 44s

Intro
0:00
Example I: Identify Absolute and Local Max & Min on the Following Graph
0:11
Example II: Sketch the Graph of a Continuous Function
3:11
Example III: Sketch the Following Graphs
4:40
Example IV: Find the Critical Values of f (x) = 3x⁴ - 7x³ + 4x²
6:13
Example V: Find the Critical Values of f(x) = |2x - 5|
8:42
Example VI: Find the Critical Values
11:42
Example VII: Find the Critical Values f(x) = cos²(2x) on [0,2π]
16:57
Example VIII: Find the Absolute Max & Min f(x) = 2sinx + 2cos x on [0,(π/3)]
20:08
Example IX: Find the Absolute Max & Min f(x) = (ln(2x)) / x on [1,3]
24:39
The Mean Value Theorem

25m 54s

Intro
0:00
Rolle's Theorem
0:08
Rolle's Theorem: If & Then
0:09
Rolle's Theorem: Geometrically
2:06
There May Be More than 1 c Such That f'( c ) = 0
3:30
Example I: Rolle's Theorem
4:58
The Mean Value Theorem
9:12
The Mean Value Theorem: If & Then
9:13
The Mean Value Theorem: Geometrically
11:07
Example II: Mean Value Theorem
13:43
Example III: Mean Value Theorem
21:19
Using Derivatives to Graph Functions, Part I

25m 54s

Intro
0:00
Using Derivatives to Graph Functions, Part I
0:12
Increasing/ Decreasing Test
0:13
Example I: Find the Intervals Over Which the Function is Increasing & Decreasing
3:26
Example II: Find the Local Maxima & Minima of the Function
19:18
Example III: Find the Local Maxima & Minima of the Function
31:39
Using Derivatives to Graph Functions, Part II

44m 58s

Intro
0:00
Using Derivatives to Graph Functions, Part II
0:13
Concave Up & Concave Down
0:14
What Does This Mean in Terms of the Derivative?
6:14
Point of Inflection
8:52
Example I: Graph the Function
13:18
Example II: Function x⁴ - 5x²
19:03
Intervals of Increase & Decrease
19:04
Local Maxes and Mins
25:01
Intervals of Concavity & X-Values for the Points of Inflection
29:18
Intervals of Concavity & Y-Values for the Points of Inflection
34:18
Graphing the Function
40:52
Example Problems I

49m 19s

Intro
0:00
Example I: Intervals, Local Maxes & Mins
0:26
Example II: Intervals, Local Maxes & Mins
5:05
Example III: Intervals, Local Maxes & Mins, and Inflection Points
13:40
Example IV: Intervals, Local Maxes & Mins, Inflection Points, and Intervals of Concavity
23:02
Example V: Intervals, Local Maxes & Mins, Inflection Points, and Intervals of Concavity
34:36
Example Problems III

59m 1s

Intro
0:00
Example I: Intervals, Local Maxes & Mins, Inflection Points, Intervals of Concavity, and Asymptotes
0:11
Example II: Intervals, Local Maxes & Mins, Inflection Points, Intervals of Concavity, and Asymptotes
21:24
Example III: Cubic Equation f(x) = Ax³ + Bx² + Cx + D
37:56
Example IV: Intervals, Local Maxes & Mins, Inflection Points, Intervals of Concavity, and Asymptotes
46:19
L'Hospital's Rule

30m 9s

Intro
0:00
L'Hospital's Rule
0:19
Indeterminate Forms
0:20
L'Hospital's Rule
3:38
Example I: Evaluate the Following Limit Using L'Hospital's Rule
8:50
Example II: Evaluate the Following Limit Using L'Hospital's Rule
10:30
Indeterminate Products
11:54
Indeterminate Products
11:55
Example III: L'Hospital's Rule & Indeterminate Products
13:57
Indeterminate Differences
17:00
Indeterminate Differences
17:01
Example IV: L'Hospital's Rule & Indeterminate Differences
18:57
Indeterminate Powers
22:20
Indeterminate Powers
22:21
Example V: L'Hospital's Rule & Indeterminate Powers
25:13
Example Problems for L'Hospital's Rule

38m 14s

Intro
0:00
Example I: Evaluate the Following Limit
0:17
Example II: Evaluate the Following Limit
2:45
Example III: Evaluate the Following Limit
6:54
Example IV: Evaluate the Following Limit
8:43
Example V: Evaluate the Following Limit
11:01
Example VI: Evaluate the Following Limit
14:48
Example VII: Evaluate the Following Limit
17:49
Example VIII: Evaluate the Following Limit
20:37
Example IX: Evaluate the Following Limit
25:16
Example X: Evaluate the Following Limit
32:44
Optimization Problems I

49m 59s

Intro
0:00
Example I: Find the Dimensions of the Box that Gives the Greatest Volume
1:23
Fundamentals of Optimization Problems
18:08
Fundamental #1
18:33
Fundamental #2
19:09
Fundamental #3
19:19
Fundamental #4
20:59
Fundamental #5
21:55
Fundamental #6
23:44
Example II: Demonstrate that of All Rectangles with a Given Perimeter, the One with the Largest Area is a Square
24:36
Example III: Find the Points on the Ellipse 9x² + y² = 9 Farthest Away from the Point (1,0)
35:13
Example IV: Find the Dimensions of the Rectangle of Largest Area that can be Inscribed in a Circle of Given Radius R
43:10
Optimization Problems II

55m 10s

Intro
0:00
Example I: Optimization Problem
0:13
Example II: Optimization Problem
17:34
Example III: Optimization Problem
35:06
Example IV: Revenue, Cost, and Profit
43:22
Newton's Method

30m 22s

Intro
0:00
Newton's Method
0:45
Newton's Method
0:46
Example I: Find x2 and x3
13:18
Example II: Use Newton's Method to Approximate
15:48
Example III: Find the Root of the Following Equation to 6 Decimal Places
19:57
Example IV: Use Newton's Method to Find the Coordinates of the Inflection Point
23:11
Section 4: Integrals
Antiderivatives

55m 26s

Intro
0:00
Antiderivatives
0:23
Definition of an Antiderivative
0:24
Antiderivative Theorem
7:58
Function & Antiderivative
12:10
x^n
12:30
1/x
13:00
e^x
13:08
cos x
13:18
sin x
14:01
sec² x
14:11
secxtanx
14:18
1/√(1-x²)
14:26
1/(1+x²)
14:36
-1/√(1-x²)
14:45
Example I: Find the Most General Antiderivative for the Following Functions
15:07
Function 1: f(x) = x³ -6x² + 11x - 9
15:42
Function 2: f(x) = 14√(x) - 27 4√x
19:12
Function 3: (fx) = cos x - 14 sinx
20:53
Function 4: f(x) = (x⁵+2√x )/( x^(4/3) )
22:10
Function 5: f(x) = (3e^x) - 2/(1+x²)
25:42
Example II: Given the Following, Find the Original Function f(x)
26:37
Function 1: f'(x) = 5x³ - 14x + 24, f(2) = 40
27:55
Function 2: f'(x) 3 sinx + sec²x, f(π/6) = 5
30:34
Function 3: f''(x) = 8x - cos x, f(1.5) = 12.7, f'(1.5) = 4.2
32:54
Function 4: f''(x) = 5/(√x), f(2) 15, f'(2) = 7
37:54
Example III: Falling Object
41:58
Problem 1: Find an Equation for the Height of the Ball after t Seconds
42:48
Problem 2: How Long Will It Take for the Ball to Strike the Ground?
48:30
Problem 3: What is the Velocity of the Ball as it Hits the Ground?
49:52
Problem 4: Initial Velocity of 6 m/s, How Long Does It Take to Reach the Ground?
50:46
The Area Under a Curve

51m 3s

Intro
0:00
The Area Under a Curve
0:13
Approximate Using Rectangles
0:14
Let's Do This Again, Using 4 Different Rectangles
9:40
Approximate with Rectangles
16:10
Left Endpoint
18:08
Right Endpoint
25:34
Left Endpoint vs. Right Endpoint
30:58
Number of Rectangles
34:08
True Area
37:36
True Area
37:37
Sigma Notation & Limits
43:32
When You Have to Explicitly Solve Something
47:56
Example Problems for Area Under a Curve

33m 7s

Intro
0:00
Example I: Using Left Endpoint & Right Endpoint to Approximate Area Under a Curve
0:10
Example II: Using 5 Rectangles, Approximate the Area Under the Curve
11:32
Example III: Find the True Area by Evaluating the Limit Expression
16:07
Example IV: Find the True Area by Evaluating the Limit Expression
24:52
The Definite Integral

43m 19s

Intro
0:00
The Definite Integral
0:08
Definition to Find the Area of a Curve
0:09
Definition of the Definite Integral
4:08
Symbol for Definite Integral
8:45
Regions Below the x-axis
15:18
Associating Definite Integral to a Function
19:38
Integrable Function
27:20
Evaluating the Definite Integral
29:26
Evaluating the Definite Integral
29:27
Properties of the Definite Integral
35:24
Properties of the Definite Integral
35:25
Example Problems for The Definite Integral

32m 14s

Intro
0:00
Example I: Approximate the Following Definite Integral Using Midpoints & Sub-intervals
0:11
Example II: Express the Following Limit as a Definite Integral
5:28
Example III: Evaluate the Following Definite Integral Using the Definition
6:28
Example IV: Evaluate the Following Integral Using the Definition
17:06
Example V: Evaluate the Following Definite Integral by Using Areas
25:41
Example VI: Definite Integral
30:36
The Fundamental Theorem of Calculus

24m 17s

Intro
0:00
The Fundamental Theorem of Calculus
0:17
Evaluating an Integral
0:18
Lim as x → ∞
12:19
Taking the Derivative
14:06
Differentiation & Integration are Inverse Processes
15:04
1st Fundamental Theorem of Calculus
20:08
1st Fundamental Theorem of Calculus
20:09
2nd Fundamental Theorem of Calculus
22:30
2nd Fundamental Theorem of Calculus
22:31
Example Problems for the Fundamental Theorem

25m 21s

Intro
0:00
Example I: Find the Derivative of the Following Function
0:17
Example II: Find the Derivative of the Following Function
1:40
Example III: Find the Derivative of the Following Function
2:32
Example IV: Find the Derivative of the Following Function
5:55
Example V: Evaluate the Following Integral
7:13
Example VI: Evaluate the Following Integral
9:46
Example VII: Evaluate the Following Integral
12:49
Example VIII: Evaluate the Following Integral
13:53
Example IX: Evaluate the Following Graph
15:24
Local Maxs and Mins for g(x)
15:25
Where Does g(x) Achieve Its Absolute Max on [0,8]
20:54
On What Intervals is g(x) Concave Up/Down?
22:20
Sketch a Graph of g(x)
24:34
More Example Problems, Including Net Change Applications

34m 22s

Intro
0:00
Example I: Evaluate the Following Indefinite Integral
0:10
Example II: Evaluate the Following Definite Integral
0:59
Example III: Evaluate the Following Integral
2:59
Example IV: Velocity Function
7:46
Part A: Net Displacement
7:47
Part B: Total Distance Travelled
13:15
Example V: Linear Density Function
20:56
Example VI: Acceleration Function
25:10
Part A: Velocity Function at Time t
25:11
Part B: Total Distance Travelled During the Time Interval
28:38
Solving Integrals by Substitution

27m 20s

Intro
0:00
Table of Integrals
0:35
Example I: Evaluate the Following Indefinite Integral
2:02
Example II: Evaluate the Following Indefinite Integral
7:27
Example IIII: Evaluate the Following Indefinite Integral
10:57
Example IV: Evaluate the Following Indefinite Integral
12:33
Example V: Evaluate the Following
14:28
Example VI: Evaluate the Following
16:00
Example VII: Evaluate the Following
19:01
Example VIII: Evaluate the Following
21:49
Example IX: Evaluate the Following
24:34
Section 5: Applications of Integration
Areas Between Curves

34m 56s

Intro
0:00
Areas Between Two Curves: Function of x
0:08
Graph 1: Area Between f(x) & g(x)
0:09
Graph 2: Area Between f(x) & g(x)
4:07
Is It Possible to Write as a Single Integral?
8:20
Area Between the Curves on [a,b]
9:24
Absolute Value
10:32
Formula for Areas Between Two Curves: Top Function - Bottom Function
17:03
Areas Between Curves: Function of y
17:49
What if We are Given Functions of y?
17:50
Formula for Areas Between Two Curves: Right Function - Left Function
21:48
Finding a & b
22:32
Example Problems for Areas Between Curves

42m 55s

Intro
0:00
Instructions for the Example Problems
0:10
Example I: y = 7x - x² and y=x
0:37
Example II: x=y²-3, x=e^((1/2)y), y=-1, and y=2
6:25
Example III: y=(1/x), y=(1/x³), and x=4
12:25
Example IV: 15-2x² and y=x²-5
15:52
Example V: x=(1/8)y³ and x=6-y²
20:20
Example VI: y=cos x, y=sin(2x), [0,π/2]
24:34
Example VII: y=2x², y=10x², 7x+2y=10
29:51
Example VIII: Velocity vs. Time
33:23
Part A: At 2.187 Minutes, Which care is Further Ahead?
33:24
Part B: If We Shaded the Region between the Graphs from t=0 to t=2.187, What Would This Shaded Area Represent?
36:32
Part C: At 4 Minutes Which Car is Ahead?
37:11
Part D: At What Time Will the Cars be Side by Side?
37:50
Volumes I: Slices

34m 15s

Intro
0:00
Volumes I: Slices
0:18
Rotate the Graph of y=√x about the x-axis
0:19
How can I use Integration to Find the Volume?
3:16
Slice the Solid Like a Loaf of Bread
5:06
Volumes Definition
8:56
Example I: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Given Functions about the Given Line of Rotation
12:18
Example II: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Given Functions about the Given Line of Rotation
19:05
Example III: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Given Functions about the Given Line of Rotation
25:28
Volumes II: Volumes by Washers

51m 43s

Intro
0:00
Volumes II: Volumes by Washers
0:11
Rotating Region Bounded by y=x³ & y=x around the x-axis
0:12
Equation for Volumes by Washer
11:14
Process for Solving Volumes by Washer
13:40
Example I: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Following Functions around the Given Axis
15:58
Example II: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Following Functions around the Given Axis
25:07
Example III: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Following Functions around the Given Axis
34:20
Example IV: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Following Functions around the Given Axis
44:05
Volumes III: Solids That Are Not Solids-of-Revolution

49m 36s

Intro
0:00
Solids That Are Not Solids-of-Revolution
0:11
Cross-Section Area Review
0:12
Cross-Sections That Are Not Solids-of-Revolution
7:36
Example I: Find the Volume of a Pyramid Whose Base is a Square of Side-length S, and Whose Height is H
10:54
Example II: Find the Volume of a Solid Whose Cross-sectional Areas Perpendicular to the Base are Equilateral Triangles
20:39
Example III: Find the Volume of a Pyramid Whose Base is an Equilateral Triangle of Side-Length A, and Whose Height is H
29:27
Example IV: Find the Volume of a Solid Whose Base is Given by the Equation 16x² + 4y² = 64
36:47
Example V: Find the Volume of a Solid Whose Base is the Region Bounded by the Functions y=3-x² and the x-axis
46:13
Volumes IV: Volumes By Cylindrical Shells

50m 2s

Intro
0:00
Volumes by Cylindrical Shells
0:11
Find the Volume of the Following Region
0:12
Volumes by Cylindrical Shells: Integrating Along x
14:12
Volumes by Cylindrical Shells: Integrating Along y
14:40
Volumes by Cylindrical Shells Formulas
16:22
Example I: Using the Method of Cylindrical Shells, Find the Volume of the Solid
18:33
Example II: Using the Method of Cylindrical Shells, Find the Volume of the Solid
25:57
Example III: Using the Method of Cylindrical Shells, Find the Volume of the Solid
31:38
Example IV: Using the Method of Cylindrical Shells, Find the Volume of the Solid
38:44
Example V: Using the Method of Cylindrical Shells, Find the Volume of the Solid
44:03
The Average Value of a Function

32m 13s

Intro
0:00
The Average Value of a Function
0:07
Average Value of f(x)
0:08
What if The Domain of f(x) is Not Finite?
2:23
Let's Calculate Average Value for f(x) = x² [2,5]
4:46
Mean Value Theorem for Integrate
9:25
Example I: Find the Average Value of the Given Function Over the Given Interval
14:06
Example II: Find the Average Value of the Given Function Over the Given Interval
18:25
Example III: Find the Number A Such that the Average Value of the Function f(x) = -4x² + 8x + 4 Equals 2 Over the Interval [-1,A]
24:04
Example IV: Find the Average Density of a Rod
27:47
Section 6: Techniques of Integration
Integration by Parts

50m 32s

Intro
0:00
Integration by Parts
0:08
The Product Rule for Differentiation
0:09
Integrating Both Sides Retains the Equality
0:52
Differential Notation
2:24
Example I: ∫ x cos x dx
5:41
Example II: ∫ x² sin(2x)dx
12:01
Example III: ∫ (e^x) cos x dx
18:19
Example IV: ∫ (sin^-1) (x) dx
23:42
Example V: ∫₁⁵ (lnx)² dx
28:25
Summary
32:31
Tabular Integration
35:08
Case 1
35:52
Example: ∫x³sinx dx
36:39
Case 2
40:28
Example: ∫e^(2x) sin 3x
41:14
Trigonometric Integrals I

24m 50s

Intro
0:00
Example I: ∫ sin³ (x) dx
1:36
Example II: ∫ cos⁵(x)sin²(x)dx
4:36
Example III: ∫ sin⁴(x)dx
9:23
Summary for Evaluating Trigonometric Integrals of the Following Type: ∫ (sin^m) (x) (cos^p) (x) dx
15:59
#1: Power of sin is Odd
16:00
#2: Power of cos is Odd
16:41
#3: Powers of Both sin and cos are Odd
16:55
#4: Powers of Both sin and cos are Even
17:10
Example IV: ∫ tan⁴ (x) sec⁴ (x) dx
17:34
Example V: ∫ sec⁹(x) tan³(x) dx
20:55
Summary for Evaluating Trigonometric Integrals of the Following Type: ∫ (sec^m) (x) (tan^p) (x) dx
23:31
#1: Power of sec is Odd
23:32
#2: Power of tan is Odd
24:04
#3: Powers of sec is Odd and/or Power of tan is Even
24:18
Trigonometric Integrals II

22m 12s

Intro
0:00
Trigonometric Integrals II
0:09
Recall: ∫tanx dx
0:10
Let's Find ∫secx dx
3:23
Example I: ∫ tan⁵ (x) dx
6:23
Example II: ∫ sec⁵ (x) dx
11:41
Summary: How to Deal with Integrals of Different Types
19:04
Identities to Deal with Integrals of Different Types
19:05
Example III: ∫cos(5x)sin(9x)dx
19:57
More Example Problems for Trigonometric Integrals

17m 22s

Intro
0:00
Example I: ∫sin²(x)cos⁷(x)dx
0:14
Example II: ∫x sin²(x) dx
3:56
Example III: ∫csc⁴ (x/5)dx
8:39
Example IV: ∫( (1-tan²x)/(sec²x) ) dx
11:17
Example V: ∫ 1 / (sinx-1) dx
13:19
Integration by Partial Fractions I

55m 12s

Intro
0:00
Integration by Partial Fractions I
0:11
Recall the Idea of Finding a Common Denominator
0:12
Decomposing a Rational Function to Its Partial Fractions
4:10
2 Types of Rational Function: Improper & Proper
5:16
Improper Rational Function
7:26
Improper Rational Function
7:27
Proper Rational Function
11:16
Proper Rational Function & Partial Fractions
11:17
Linear Factors
14:04
15:02
Case 1: G(x) is a Product of Distinct Linear Factors
17:10
Example I: Integration by Partial Fractions
20:33
Case 2: D(x) is a Product of Linear Factors
40:58
Example II: Integration by Partial Fractions
44:41
Integration by Partial Fractions II

42m 57s

Intro
0:00
Case 3: D(x) Contains Irreducible Factors
0:09
Example I: Integration by Partial Fractions
5:19
Example II: Integration by Partial Fractions
16:22
Case 4: D(x) has Repeated Irreducible Quadratic Factors
27:30
Example III: Integration by Partial Fractions
30:19
Section 7: Differential Equations
Introduction to Differential Equations

46m 37s

Intro
0:00
Introduction to Differential Equations
0:09
Overview
0:10
Differential Equations Involving Derivatives of y(x)
2:08
Differential Equations Involving Derivatives of y(x) and Function of y(x)
3:23
Equations for an Unknown Number
6:28
What are These Differential Equations Saying?
10:30
Verifying that a Function is a Solution of the Differential Equation
13:00
Verifying that a Function is a Solution of the Differential Equation
13:01
Verify that y(x) = 4e^x + 3x² + 6x + e^π is a Solution of this Differential Equation
17:20
General Solution
22:00
Particular Solution
24:36
Initial Value Problem
27:42
Example I: Verify that a Family of Functions is a Solution of the Differential Equation
32:24
Example II: For What Values of K Does the Function Satisfy the Differential Equation
36:07
Example III: Verify the Solution and Solve the Initial Value Problem
39:47
Separation of Variables

28m 8s

Intro
0:00
Separation of Variables
0:28
Separation of Variables
0:29
Example I: Solve the Following g Initial Value Problem
8:29
Example II: Solve the Following g Initial Value Problem
13:46
Example III: Find an Equation of the Curve
18:48
Population Growth: The Standard & Logistic Equations

51m 7s

Intro
0:00
Standard Growth Model
0:30
Definition of the Standard/Natural Growth Model
0:31
Initial Conditions
8:00
The General Solution
9:16
Example I: Standard Growth Model
10:45
Logistic Growth Model
18:33
Logistic Growth Model
18:34
Solving the Initial Value Problem
25:21
What Happens When t → ∞
36:42
Example II: Solve the Following g Initial Value Problem
41:50
Relative Growth Rate
46:56
Relative Growth Rate
46:57
Relative Growth Rate Version for the Standard model
49:04
Slope Fields

24m 37s

Intro
0:00
Slope Fields
0:35
Slope Fields
0:36
Graphing the Slope Fields, Part 1
11:12
Graphing the Slope Fields, Part 2
15:37
Graphing the Slope Fields, Part 3
17:25
Steps to Solving Slope Field Problems
20:24
Example I: Draw or Generate the Slope Field of the Differential Equation y'=x cos y
22:38
Section 8: AP Practic Exam
AP Practice Exam: Section 1, Part A No Calculator

45m 29s

Intro
0:00
0:10
Problem #1
1:26
Problem #2
2:52
Problem #3
4:42
Problem #4
7:03
Problem #5
10:01
Problem #6
13:49
Problem #7
15:16
Problem #8
19:06
Problem #9
23:10
Problem #10
28:10
Problem #11
31:30
Problem #12
33:53
Problem #13
37:45
Problem #14
41:17
AP Practice Exam: Section 1, Part A No Calculator, cont.

41m 55s

Intro
0:00
Problem #15
0:22
Problem #16
3:10
Problem #17
5:30
Problem #18
8:03
Problem #19
9:53
Problem #20
14:51
Problem #21
17:30
Problem #22
22:12
Problem #23
25:48
Problem #24
29:57
Problem #25
33:35
Problem #26
35:57
Problem #27
37:57
Problem #28
40:04
AP Practice Exam: Section I, Part B Calculator Allowed

58m 47s

Intro
0:00
Problem #1
1:22
Problem #2
4:55
Problem #3
10:49
Problem #4
13:05
Problem #5
14:54
Problem #6
17:25
Problem #7
18:39
Problem #8
20:27
Problem #9
26:48
Problem #10
28:23
Problem #11
34:03
Problem #12
36:25
Problem #13
39:52
Problem #14
43:12
Problem #15
47:18
Problem #16
50:41
Problem #17
56:38
AP Practice Exam: Section II, Part A Calculator Allowed

25m 40s

Intro
0:00
Problem #1: Part A
1:14
Problem #1: Part B
4:46
Problem #1: Part C
8:00
Problem #2: Part A
12:24
Problem #2: Part B
16:51
Problem #2: Part C
17:17
Problem #3: Part A
18:16
Problem #3: Part B
19:54
Problem #3: Part C
21:44
Problem #3: Part D
22:57
AP Practice Exam: Section II, Part B No Calculator

31m 20s

Intro
0:00
Problem #4: Part A
1:35
Problem #4: Part B
5:54
Problem #4: Part C
8:50
Problem #4: Part D
9:40
Problem #5: Part A
11:26
Problem #5: Part B
13:11
Problem #5: Part C
15:07
Problem #5: Part D
19:57
Problem #6: Part A
22:01
Problem #6: Part B
25:34
Problem #6: Part C
28:54

• ## Transcription

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### Population Growth: The Standard & Logistic Equations

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

• Intro 0:00
• Standard Growth Model 0:30
• Definition of the Standard/Natural Growth Model
• Initial Conditions
• The General Solution
• Example I: Standard Growth Model 10:45
• Logistic Growth Model 18:33
• Logistic Growth Model
• Solving the Initial Value Problem
• What Happens When t → ∞
• Example II: Solve the Following g Initial Value Problem 41:50
• Relative Growth Rate 46:56
• Relative Growth Rate
• Relative Growth Rate Version for the Standard model

### Transcription: Population Growth: The Standard & Logistic Equations

Hello, welcome back to www.educator.com, and welcome back to AP Calculus.0000

Today, we are going to be talking about a particular differential equation.0005

There are several that we could have chosen from but we decided to go with population growth.0010

We are going to be discussing the standard and the logistic equations for population growth.0014

Two differential equations, the standard we are just going to go through to mention it.0019

It is the logistic equation the one that we really want to concentrate on.0024

Let us jump right on in.0028

There are many differential equations that exist that try to model how populations grow and decay.0032

Let us see, I think I will go ahead and work in blue.0041

There are many differential equations that try to describe population growth and decay.0043

I will just put growth/decay.0069

The simplest of these is something called the standard model or the natural model standard equation, the natural equation.0074

Whenever you hear the word model, essentially, you are talking about a different equation0082

or series of differential equations that try to describe the situation.0086

Model is just a fancy word for that.0089

The simplest of these equations, the simplest of these is the standard model or natural model.0092

It says the following, it says that the rate of change of a population0116

is directly proportional to the size of the population at the moment.0135

The size is of the population at the moment.0151

Symbols is this, dp dt, the rate of change of the population.0164

The rate at which the population changes, dp, per unit change in time dt.0178

The derivative is a rate of change or rate of change is just a derivative.0183

The rate of change of a population is directly proportional to,0188

that means some constant × the size of the population at the moment p.0191

That is it, this is a model.0195

The differential equation is this.0200

We seek a function p(t), notice we did not use dy dx, dp dt, change in population versus the time.0203

Here, the independent variable is the time, the dependent variable is the population.0219

We seek a function p(t), that will predict what the population will be at any future time.0222

We want to solve this differential equation.0250

Once we have the solution, this p(t), we want to compare it to data0254

that we have collected on how populations grow and decay.0259

If the equation matches the empirical data, the real life data, your model is a good model.0264

If it is not, then we have to go back and modify the model or come up with a different model altogether.0271

That is essentially all that we do in science.0276

As we go into the lab and we collect a certain amount of data, we try to describe it mathematically with some model.0279

In other words, a set of differential equations, 1 equation, 2 equations, 3 equations, whatever it is that we need.0286

We try to see if it matches, once we solve the differential equation, we try to see if it matches the actual data.0292

That is it, that is really all science comes down to.0298

Sometimes the data itself will, oftentimes, the data itself will tell us, gives us clues on how the model should look.0301

It is sort of back and fourth between the data, the model,0309

until we have refined the model so much that it actually describes what we observe in the real world.0312

Once that happens consistently, we call that a theory.0322

That is all that is going on in science.0325

Let us go ahead and see what we can do.0328

Let us to solve this particular equation.0330

We have that the rate of change of population with respect to time is proportional to the size of the population of the moment.0333

It looks like this is a separable equation.0342

Let us go ahead and separate variables here.0344

We are going to end up with dp/p = k × dt.0347

Sure enough, it is separable, that t is on one side and the p is on the other.0351

Since it is separable, all we have to do is integrate both sides.0354

The integral of dp/p is the natlog of p.0358

I do not need the absolute value sign here because a population is always going to be positive.0363

P is always positive, it is just the natlog of p is equal to kt + c.0367

This is our general solution of this differential equation.0374

Let us go ahead and change the form and make it look like an exponential.0378

Let us exponentiate both sides.0382

We end up with p is equal to e ⁺kt + c.0384

Kt + c, the exponent, this is just e ⁺kt × e ⁺c.0391

Same base at the exponents.0398

E ⁺c is just some number is just some number based on c.0400

I will call the whole thing, this whole thing is just some constant.0415

We will call it a.0421

What we have is the population at time t is equal to some a × e ⁺kt.0426

The standard model predicts an exponential growth of a population.0433

Another reason why it is actually called the natural model because it is based on the natural logarithm.0441

This is the natural model.0445

Basic, first, approximation model, we predict exponential growth.0447

This equation is usually accompanied by some initial values, initial conditions.0457

In other words, we know what a population is at any given moment, when we start to collect the data.0462

In the year 2000, the population of New Haven, Connecticut was 400,000, whatever it was.0469

We have some initial data.0476

That is our initial value problem.0477

This equation is usually accompanied by a set of initial conditions, usually, just one initial condition.0481

In other words, the population at time 0, whatever our 0 time happens to be, is going to equal some number.0510

I will call it p sub i, it is the initial population of our group.0518

This is just the initial population at time 0.0524

The initial value problem is nothing more than the differential equation dp dt = kp.0540

We have p at time 0 = p sub i.0549

We have the general solution.0555

We just solved it, we have the general solution.0560

That is just p(t) is equal to ae ⁺kt.0566

Let us see if we can use this initial value to find what a is going to be.0573

p(0) which means put 0 in for x, or in this case t, is equal to a × e ⁺k × 0.0579

They are telling me that p(0) which is equal to this, when I put it in the equation is equal to p sub i.0594

e ⁺k0 is e⁰ it is equal to 1.0602

a × 1 = the initial population, a is equal to the initial population.0610

Our solution is, the population at time t is equal to the initial population that I start with at time 0 × e ⁺kt.0614

K is the constant, in this case.0627

For different values of k, there is going to be different degrees of growth.0631

Let us do an example, this is our general equation.0639

At 1:00 pm on June 15, a biologist counts a bacterial population of 100 in a culture.0647

At 11:00 pm on the same day, the culture has grown to 187 bacteria.0653

Assuming a standard growth model, how much bacteria will there be at 1:00 pm on June 18, 3 days later?0659

Three days later, 1 pm, 1 pm, let us see what we have got.0669

P(t), we are assuming a standard model so our solution is p(t) = π e ⁺kt.0677

We know what the initial population is, it is 100 bacteria in the culture, at = 0 which is 1:00 pm on June 15.0689

We have to choose a unit of time for t.0702

Are we going to work in days, minutes, seconds, years, months.0704

June 15 and June 18, that is three days.0707

1 pm, 1 pm, that is 11:00 pm.0711

Let us work in hours, actually, it looks like.0714

We choose to express time in hours.0720

I think it will be the best.0736

We know what the initial population is, we need to find what k is.0745

Once we find what k is, then we can go ahead and plug in a value of t to find what p(t) is.0755

The π we know, in this case, we have one constant π, we have another constant k.0764

We need to find both of these.0768

The π, we know that is just 100.0769

We have p(t) is equal to 100 × e ⁺kt.0774

Let us go ahead and solve for k.0794

Let us see what I have got.0798

P(t) is equal to, I already got that, I do not have to repeat that.0800

At 11:00 pm, the population is 187 bacteria which means,0811

What that means is that, p(10), 1:00 pm is t = 0.0834

11:00 pm is 10 hours later, at t = 10.0842

P(10) = 187, I can put these values into this equation to see what I get for k.0847

P(10) which is equal to 100 e ⁺k × 10 is equal to 187.0862

I solve here, e ⁺10k is equal to 1.87, 10k is equal to the natlog of 1.87.0877

I get k is equal to the natlog of 1.87 divided by 10.0895

The natlog of 1.87 divided by 10 which is equal to 0.06259.0905

Our equation becomes, population of t is equal to 100, the initial population, × e raised the power of 0.06259 t.0922

1:00 pm on June 18, three days later exactly, three days is 72 hours, three days later exactly, t = 72 hours.0948

Therefore, what we are looking for is p(72).0976

P(72) is equal to 100 × e⁰.06259 × 72.0979

When I do that, I get 9,060 bacteria.0991

Let us look at this equation.1001

Let me go back to blue here.1007

I have population = an initial population × e ⁺kt.1020

There are four parameters here.1029

There is final population, the initial population, the growth constant, k and t.1032

There are four parameters in this equation.1050

If you have any three of them, you can find the fourth by just rearranging, solving, that is the whole idea.1061

If you have any three of them, you can solve for the fourth.1068

The nature of the problems that you are given are going to be such that they allow you to find three of them.1086

You do not know which three, that is why no two problems actually look the same.1095

You are going to have to reason out which three you are going to find.1101

Once you found those three, you will find the fourth.1104

That is it, that is all that is going on here.1106

Let us go back to blue.1113

A much improved model for population growth.1116

The standard model, it is not bad, it works to a certain degree.1120

But just from your experience, you know that an exponential growth, something cannot just keep growing exponentially.1125

A population is not just going to keep growing and growing and growing.1131

People leave, people die, people are born, people move in.1136

Not to mention the fact that there is only a certain number of amount of resources that are available.1144

A particular environment can only support a maximum number of people.1150

You know just intuitively that a population will tend to grow but that eventually it will start to level off,1156

once you actually reached what we call the carrying capacity of that particular environment.1162

Whether it is a petri dish or whether it is a city or a country or whatever it is.1167

A little bit better, a lot better, a much improved version and much improved model1172

for population growth and decay is the logistic model or the logistic equation.1185

This particular differential equation looks like this.1204

It says the rate of change of a population is not just directly proportional to the population.1207

It is directly proportional to the population × 1 – what the population is/ c, where c is the carrying capacity of the environment.1213

In other words, carrying capacity of an environment is the maximum number of people that the environment can support.1230

Like a room, when you go to a room and it says maximum occupancy 250, only 250 people can be in that room.1236

If there is more than 250, people have to leave.1246

If there is less than 250, people will keep coming in and the population will increase until you reach 250.1248

That is it, it is just exactly what it sounds like, the carrying capacity.1254

Where c is the carrying capacity which is the number of individuals,1257

whether that would be people, bacteria, whatever, the number of individuals an environment can carry and support.1265

Now this is our differential equation, the logistic model.1284

Let us take a look at what is going to happen here.1289

Those of you who would actually go on in your scientific careers, you are going to run through the differential equations.1294

Oftentimes, you actually do not need to solve them.1300

All you need to do is analyze them and look for what we call qualitative behavior.1302

You are going to look at a different equation, you may not be able to solve the equation1309

but you can extract a lot of information from just looking at the equation1312

and seeing if you can get this qualitative information from it.1317

Usually that will be enough for you to answer whatever question you are trying to answer.1322

In this particular case, let us see what happens,1325

if the population at any given moment is less than the carrying capacity.1328

If p is less than c, this p/c is less than 1.1333

1 - a number less than 1 that means this 1 - p/c term is positive.1345

This implies that dp dt, if this is positive, this is going to be positive which means the population is going to grow.1358

Dp dt will be positive.1366

A positive rate of change means something is growing, which means that p is actually growing.1376

The population will grow.1382

It just confirms our intuition, population will grow until it levels off at c.1385

In which case, when the population reaches c, this becomes 1 – 1, it is 0.1401

There is no more population growth, it just levels off.1410

The rate of change of population is 0.1414

It is not growing, it is not decaying, it just stops right there.1415

This is qualitative information, if the population at any given moment is actually greater than the carrying capacity,1420

then this term 1 - p/c is negative.1431

This implies that dp dt is negative.1438

A negative rate of change means that the dependent variable, the population in this case, is declining.1446

It is negative which means the population will decrease until it levels off at the carrying capacity.1458

We already have some idea of what the solutions are supposed to look like graphically.1483

You are going to see a population increase and eventually it is going to level off at the caring capacity.1488

If the population is greater than the carrying capacity, the population is going to decline until it levels off at the carrying capacity.1494

We already have some idea what the graph should look like, what the solution should look like.1501

That is why this qualitative information is very important.1505

Let us go ahead and solve this equation.1513

Let us solve this initial value problem.1521

The initial value problem is dp dt is equal to k × p × 1 - p/c.1532

P(0) is equal to p sub 0 or p sub i, initial population, some number.1542

This is separable, this differential equation is separable.1550

May not look like it, but it actually is.1559

Let us go ahead and do, here is what I'm going to do.1564

I’m going to write this equation and I’m going to multiply this out.1568

I’m going to write this as dp dt is equal to k × pc - p²/ c.1570

That is just this, multiplied out, common denominator multiplied out.1585

Now what I’m going to do is I'm going to move the dt up here.1590

I’m going to move c, it is just a constant.1594

I’m just going to bring everything here this way.1595

What I end up with is, once I separated out and hopefully you can take care of this.1598

I’m not going to go through the entire process pc - p² dp is equal to k dt.1605

Now the p’s are all on one side, constant does not matter, it is just a constant.1616

The t’s are on one side.1619

I can go ahead and I can integrate.1622

Now integrating a rational function.1625

This c, I’m going to decompose this by partial fraction, left side, when I integrate this.1630

The difficulty here is not the separation, the difficulty here is the actual integration.1639

pc - p², I'm going to factor the denominator c/p × c - p1645

which is equal to some a/p, partial fraction decomposition + b/ c – p.1655

Now I have got, let me go to the next page here.1667

Let me write this again.1679

I have got c/ p × c - p is equal to a/p + b/ c – p.1681

Multiply, multiply, partial fraction decomposition, I get c is equal to a × c - p + bp.1695

c is equal to ac - ap + bp.1706

c is equal to ac + -a + b × p.1715

Therefore, ac 1c, this coefficient is equal to the coefficient from here.1728

a is equal to 1, that takes care of the a.1737

I have got -a + p is equal to 0 because there is no p term here on the left,1741

which implies that a is equal to b which implies that b is also equal to 1.1750

Our c/ p × c - p is equal to actually 1/ p + 1/ c – p.1761

This is our a and this is our b.1771

Therefore, the integral of our c/ p × c - p dp is equal to the integral of 1/ p dp + the integral of 1/ c – p dp,1774

which is equal to the natlog of p - the natlog of c – p.1796

Therefore, what we have is c – p.1809

This is the left side of the integral.1820

Let us go ahead and rewrite what is it that we actually did.1834

We had the integral of c/ p × c - p = the integral of k dt.1843

We just did that integral, that is ln of p - ln of c - p = the integral of this is just kt + some constant.1853

I will go ahead write out constant, I do not want to use the word c1870

because I do not want you to confuse it with this c right here.1873

Over here, we have the natlog of p/ c - p is equal to kt + a constant.1878

I’m going to flip this and I’m going to write this as the natlog of c - p/ p = -kt - the constant.1895

I hope that make sense.1908

Log of a/b is log(a) – log(b).1910

If I flip this, it just takes the negative sign out here.1914

I move the negative sign to this side, now I have got that.1917

And now I exponentiate both sides.1922

I end up with c - p/ p is equal to e ⁻kt × e ⁻some constant.1926

This constant, I'm just going to call a.1942

I’m going to separate this out, this is going to be c/p - p/p which is 1 = a × e ⁻kt.1952

I have got, this is going to end up giving me c/p = a × e ⁻kt + 1.1968

When I solve this for p, I’m going to move the p up here, move this down here.1985

I'm left with p is equal to c which is the carrying capacity not the constant, = a which is some constant e ⁻kt + 1.1989

This is my solution to the logistic equation.2002

My population is equal to the carrying capacity divided by this thing,2007

some constant a × the exponential, some growth constant t + 1.2012

Let us go ahead and deal with the initial value.2019

Initial value, it is said that the population at time 0 is equal to some initial population.2023

Therefore, our p(0) is equal to c.2032

We use our equation ae ⁻kt × 0 + 1.2041

They tell me that it is equal to p(0), the initial population.2050

e⁰ is just 1, what I have here is c / a + 1 = p(0).2055

I rearrange this, c/ p(0) = a + 1.2068

I get a is equal to c/ p(0) – 1.2084

This is one version of it or when I do a common denominator, I get a = c – p(0)/ p0.2091

Our equation that we solved was the carrying capacity/ a × e ⁻kt + 1.2104

This value of a, I get it by taking the carrying capacity - the initial population divided by the initial population.2112

I have a way of finding a.2120

Let us go back to blue.2129

Our initial value problem dp dt = kp × 1 - the population/ c, where c is the carrying capacity.2133

p(0) = p sub 0, some number, which is the initial population.2165

It gives the following.2172

Our solution is, the carrying capacity divided by a × e ⁻kt + 1,2174

where a is equal to the carrying capacity - the initial population/ the initial population.2186

This is the solution to our problem.2193

What happens as t goes to infinity, as time just grows?2198

Let us analyze this solution, what happens when t goes to infinity?2204

When t goes to infinity, e ⁻kt goes to 0.2219

a × e ⁻kt goes to 0.2228

As this goes to 0, the population goes to c.2238

As time increases, the population goes to c, the carrying capacity, exactly what we said.2246

A population will grow and then it will level off at the carrying capacity.2251

Let us actually see what the solutions look like.2260

I have a couple of cases.2266

Case 1, what if the initial population is less than c?2270

Let us just pick some numbers, let us say our initial population of 100 and a carrying capacity of 2000.2278

2000 is the maximum population in this environment.2286

We have a formula for a, a is the carrying capacity - the initial population/ the initial population2289

which is 2000 - 100/ 100, that is going to equal 19.2295

Case 2, it is where the initial population is actually bigger than the carrying capacity.2304

Let us pick some numbers.2312

Let us do an initial population of 3000, where the carrying capacity is 2000.2314

We have a, a is equal to the carrying capacity - the initial population2321

divided by the initial population which is equal to 2000 - 3000/ 3000.2333

It is going to be equal to -1/3.2343

Our case 1, our equation, p is going to equal the carrying capacity 2000 divided by a which is 19 × e ⁻kt + 1.2348

In our case 2, our population is going to be the carrying capacity 2000 divided by a which is -1/3 e ⁻kt + 1.2367

What I'm going to do is I’m going to graph that equation and I'm going to graph that equation for different values of k.2382

k is the nature of the growth.2390

The value of k is something that we deduce from data that we have collected.2393

Let us take a look at what these look like.2398

I have taken a particular case, a particular initial population, a particular carrying capacity.2399

A particular initial population and a particular carrying capacity.2404

Here is what they look like.2410

Initial population of 100 for different values of k, k(0.5), k(0.7), k(0.9).2420

They all start at an initial population, they start to grow, and then they level off at the carrying capacity.2431

The carrying capacity is 2000.2439

For different value of k, same thing.2442

The behavior is the same, it is the nature of the growth.2444

This definitely is a much better model for how populations actually behave.2447

They will eventually start to grow, that is resources start to deplete,2452

the environment can only support so much so they achieve a maximum carrying capacity.2456

This is for when the initial population is less than the carrying capacity.2462

The initial population is 100, carrying capacity was 2000.2469

There is going to be initial growth and it is going to the exponential growth, but then it is going to level off.2474

That is what is happening here.2479

For the case where the initial population is greater than the carrying capacity,2481

the initial population of 3000 with a carrying capacity of 2000, it is going to just decay.2485

For different values of k, that is all it is.2492

These different graphs are just different values of k, that has to do with the nature of the environment.2494

It is going to decline until it reaches the carrying capacity and it is going to level off.2500

That is what is happening here.2506

Let us go ahead and do an example.2508

Solve the following initial value problem then use the solution to find the time,2512

when the population reaches 75% of its carrying capacity.2518

In this particular case, dp dt = 0.065.2524

This is our k, they gave us the k, in this case.2530

P1 p/c, this is our carrying capacity.2533

This is our c, our carrying capacity.2539

The initial population is 120.2540

We have a lot of information here.2544

We have our k which is equal to 0.065.2546

We have an initial population equal to 120.2551

We have a carrying capacity of 1500.2556

We know what our solution looks like.2561

First of all, let us find what a is.2563

We know what the solution is already, we solve this differential equation.2568

It is equal to c/a × e ⁻kt + 1.2571

Let us go ahead and find what a is.2578

I know what a is, a is equal to carrying capacity - initial population/ initial population.2581

It is equal to 1500 which is the carrying capacity - 120 which is the initial population/ 120.2590

I get an a value of equal to 11.5.2598

This is the value that I put in there.2602

I have a k value of 0.065, I put that value in there.2605

My equation is p is equal to the carrying capacity which is 1500, and that goes here.2610

I have already found most of the parameters here.2618

1500 divided by 11.5 × e⁰.065 t + 1, that is my solution.2621

I want to find the time when the population reaches 75% of its carrying capacity.2640

75% of c is just equal to 0.75 × 1500 which is equal to 1125.2651

The equation that I’m going to solve is, when the population is 1125.2663

I want to solve for t.2668

It is 1500/ 11.5 × e⁰.065 t + 1.2671

I want to solve for t.2684

I’m going to do 1125 × 11.5 e⁰.065 t + 1 = 1500.2691

I get 12937.5 e⁰.065 t + 1125 = 15002707

e⁰.065 t is equal to 0.028986 subtract 1125 divided by the 012937.5.2721

I end up with this, take the natlog of both sides.2737

The natlog of this, the natlog of that, I end up with a final value of t = 54.5.2742

I do not know the units.2751

The units could be seconds, hours, minutes, days, years, whatever.2751

It is just 54.5 time units.2755

Let us see what this looks like.2759

This is what this looks like.2762

An initial population of 120, the carrying capacity of 1500, it is right there.2763

It starts to grow, at the t value of 54.5 which is here, I labeled it as 54.48,2774

I hit a population of 1125 which is 75% of my carrying capacity, which happens to be up here.2784

That is all that is going on here, I hope that made sense.2794

Let us see here, what else can I do?2802

One last thing to round things out.2807

There is one final concept that comes up in your problems, may or may not come up, but just in case it does.2811

One final concept that can come up.2820

We speak about the rate of change of population dp dt.2832

Sometimes we speak of the relative rate of change of population.2836

There is a difference, here is what they are.2840

Relative rate, you need to know how to turn that word problem into some symbolic form.2843

Relative rate, this is the topic that can come up now for population.2857

Growth rate is what we already know, that is just dp dt.2868

Growth rate, it is the rate of change of the population per unit change in time.2876

It is the derivative, dp dt.2880

Relative growth rate or relative rate of growth or relative rate, you might see it that way too.2882

Relative growth rate, it is equal to dp dt, the growth rate divided by the population at the time.2893

Or you can do it this way, 1/p dp dt.2907

The growth rate is dp dt, the normal derivative.2913

The relative growth rate is the normal derivative divided by whatever the population happens to be.2916

In other words, it is how fast the population is changing relative to how many there are in the population.2923

Relative just means divide by that number.2932

It is a ratio that you are actually coming up with.2936

In terms of equations, it is going to look like this.2942

For the standard model, we said that dp dt is equal kp.2945

The rate of growth is directly proportional to the population at the time.2962

Relative growth version is this.2968

Relative growth version is 1/p × dp dt is equal to k.2975

In words, this one is going to say, the relative growth rate of the population is a constant.2987

They are the same equation.2993

The only difference between this equation and this equation is I actually just divided by p and I brought it over here.2994

In words, it is a same equation, there is no difference.2999

The solution that we got for the standard model is going to be the same solution here.3005

We are still going to separate variables and integrate.3012

The only difference is it is going to be worded differently.3014

Here they are going to say the rate of change in the population is directly proportional to the population.3016

Here they are going to say the relative rate of change in the population is constant.3023

You are going to say the relative rate of change divided by the population is equal to a constant.3028

That is all that is happening here.3034

Growth rate, relative growth rate, just divided by whatever the dependent variable is.3037

It is the same equation just worded differently.3047

You have to be aware of that, just worded differently.3055

Thank you so much for joining us here at www.educator.com.3064

We will see you next time, bye.3066

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