# College Calculus: Level I Applied Optimization

Section 5: Application of Derivatives: Lecture 6 | 25:37 min

In this video we are going to talk about Applied Optimization problems. There are many math problems where, based on a given set of constraints, we must minimize something, like the cost of producing a container, or maximize something, like an area that will be fenced in. These are called optimization problems, since we will find an optimum value for a given parameter. Essentially, these problems involve finding the absolute maximum or minimum value of a function over a given interval. We will see in the examples we cover that optimization problems have many applications in real-life.

Professor Switkes

Applied Optimization

Slide Duration:Table of Contents

26m 29s

- Intro0:00
- What is a Function0:10
- Domain and Range0:21
- Vertical Line Test0:31
- Example: Vertical Line Test0:47
- Function Examples1:57
- Example: Squared2:10
- Example: Natural Log2:41
- Example: Exponential3:21
- Example: Not Function3:54
- Odd and Even Functions4:39
- Example: Even Function5:10
- Example: Odd Function5:53
- Odd and Even Examples6:48
- Odd Function6:55
- Even Function8:43
- Increasing and Decreasing Functions10:15
- Example: Increasing10:42
- Example: Decreasing10:55
- Increasing and Decreasing Examples11:41
- Example: Increasing11:48
- Example: Decreasing12:33
- Types of Functions13:32
- Polynomials13:45
- Powers14:06
- Trigonometric14:34
- Rational14:50
- Exponential15:13
- Logarithmic15:29
- Lecture Example 115:55
- Lecture Example 217:51
- Additional Example 3-1
- Additional Example 4-2

12m 29s

- Intro0:00
- Compositions0:09
- Alternative Notation0:32
- Three Functions0:47
- Lecture Example 11:19
- Lecture Example 23:25
- Lecture Example 36:45
- Additional Example 4-1
- Additional Example 5-2

20m 59s

- Intro0:00
- Rates of Change0:11
- Average Rate of Change0:21
- Instantaneous Rate of Change0:33
- Slope of the Secant Line0:46
- Slope of the Tangent Line1:00
- Lecture Example 11:14
- Lecture Example 26:36
- Lecture Example 311:30
- Additional Example 4-1
- Additional Example 5-2

22m 37s

- Intro0:00
- What is a Limit?0:10
- Lecture Example 10:56
- Lecture Example 25:28
- Lecture Example 39:27
- Additional Example 4-1
- Additional Example 5-2

28m 19s

- Intro0:00
- Evaluating Limits0:09
- Lecture Example 11:06
- Lecture Example 25:16
- Lecture Example 38:15
- Lecture Example 412:58
- Additional Example 5-1
- Additional Example 6-2

23m 39s

- Intro0:00
- Formal Definition0:13
- Template0:55
- Epsilon and Delta1:24
- Lecture Example 11:40
- Lecture Example 29:20
- Additional Example 3-1
- Additional Example 4-2

19m 9s

- Intro0:00
- Continuity0:13
- Continuous0:16
- Discontinuous0:37
- Intermediate Value Theorem0:52
- Example1:22
- Lecture Example 12:58
- Lecture Example 29:02
- Additional Example 3-1
- Additional Example 4-2

22m 52s

- Intro0:00
- Limit Definition of the Derivative0:11
- Three Versions0:13
- Lecture Example 11:02
- Lecture Example 24:33
- Lecture Example 36:49
- Lecture Example 410:11
- Additional Example 5-1
- Additional Example 6-2

26m 1s

- Intro0:00
- Power Rule of Differentiation0:14
- Power Rule with Constant0:41
- Sum/Difference1:15
- Lecture Example 11:59
- Lecture Example 26:48
- Lecture Example 311:22
- Additional Example 4-1
- Additional Example 5-2

14m 54s

- Statement of the Product Rule0:08
- Lecture Example 10:41
- Lecture Example 22:27
- Lecture Example 35:03
- Additional Example 4-1
- Additional Example 5-2

19m 17s

- Intro0:00
- Statement of the Quotient Rule0:07
- Carrying out the Differentiation0:23
- Quotient Rule in Words1:00
- Lecture Example 11:19
- Lecture Example 24:23
- Lecture Example 38:00
- Additional Example 4-1
- Additional Example 5-2

17m 43s

- Intro0:00
- Rates of Change0:11
- Lecture Example 10:44
- Lecture Example 25:16
- Lecture Example 37:38
- Additional Example 4-1
- Additional Example 5-2

26m 58s

- Intro0:00
- Six Basic Trigonometric Functions0:11
- Patterns0:47
- Lecture Example 11:18
- Lecture Example 27:38
- Lecture Example 312:15
- Lecture Example 414:25
- Additional Example 5-1
- Additional Example 6-2

23m 47s

- Intro0:00
- Statement of the Chain Rule0:09
- Chain Rule for Three Functions0:27
- Lecture Example 11:00
- Lecture Example 24:34
- Lecture Example 37:23
- Additional Example 4-1
- Additional Example 5-2

27m 5s

- Intro0:00
- Six Basic Inverse Trigonometric Functions0:10
- Lecture Example 11:11
- Lecture Example 28:53
- Lecture Example 312:37
- Additional Example 4-1
- Additional Example 5-2

15m 52s

- Intro0:00
- Point Slope Form0:10
- Lecture Example 10:47
- Lecture Example 23:15
- Lecture Example 36:10
- Additional Example 4-1
- Additional Example 5-2

30m 5s

- Intro0:00
- Purpose0:09
- Implicit Function0:20
- Lecture Example 10:32
- Lecture Example 27:14
- Lecture Example 311:22
- Lecture Example 416:43
- Additional Example 5-1
- Additional Example 6-2

13m 16s

- Intro0:00
- Notation0:08
- First Type0:19
- Second Type0:54
- Lecture Example 11:41
- Lecture Example 23:15
- Lecture Example 34:57
- Additional Example 4-1
- Additional Example 5-2

17m 42s

- Intro0:00
- Essential Equations0:12
- Lecture Example 11:34
- Lecture Example 22:48
- Lecture Example 35:54
- Additional Example 4-1
- Additional Example 5-2

14m 30s

- Intro0:00
- Essential Equations0:15
- Six Basic Hyperbolic Trigc Functions0:32
- Six Basic Inverse Hyperbolic Trig Functions1:21
- Lecture Example 11:48
- Lecture Example 23:45
- Lecture Example 37:09
- Additional Example 4-1
- Additional Example 5-2

29m 5s

- Intro0:00
- What Are Related Rates?0:08
- Lecture Example 10:35
- Lecture Example 25:25
- Lecture Example 311:54
- Additional Example 4-1
- Additional Example 5-2

23m 52s

- Intro0:00
- Essential Equations0:09
- Linear Approximation (Tangent Line)0:18
- Example: Graph1:18
- Differential (df)2:06
- Delta F5:10
- Lecture Example 16:38
- Lecture Example 211:53
- Lecture Example 315:54
- Additional Example 4-1
- Additional Example 5-2

18m 57s

- Intro0:00
- Minimums and Maximums0:09
- Absolute Minima and Maxima (Extrema)0:53
- Critical Points1:25
- Lecture Example 12:58
- Lecture Example 26:57
- Lecture Example 310:02
- Additional Example 4-1
- Additional Example 5-2

20m

- Intro0:00
- Theorems0:09
- Mean Value Theorem0:13
- Graphical Explanation0:36
- Rolle's Theorem2:06
- Graphical Explanation2:28
- Lecture Example 13:36
- Lecture Example 26:33
- Lecture Example 39:32
- Additional Example 4-1
- Additional Example 5-2

27m 11s

- Intro0:00
- Local Minimum and Local Maximum0:14
- Example1:01
- First and Second Derivative Test1:26
- First Derivative Test1:36
- Example2:00
- Second Derivative Test (Concavity)2:58
- Example: Concave Down3:15
- Example: Concave Up3:54
- Inconclusive4:19
- Lecture Example 15:23
- Lecture Example 212:03
- Lecture Example 315:54
- Additional Example 4-1
- Additional Example 5-2

23m 9s

- Intro0:00
- Using L'Hopital's Rule0:09
- Informal Definition0:34
- Lecture Example 11:27
- Lecture Example 24:00
- Lecture Example 35:40
- Lecture Example 49:38
- Additional Example 5-1
- Additional Example 6-2

40m 16s

- Intro0:00
- Collecting Information0:15
- Domain and Range0:17
- Intercepts0:21
- Symmetry Properties (Even/Odd/Periodic)0:33
- Asymptotes (Vertical/Horizontal/Slant)0:45
- Critical Points1:15
- Increasing/Decreasing Intervals1:24
- Inflection Points1:38
- Concave Up/Down1:52
- Maxima/Minima2:03
- Lecture Example 12:58
- Lecture Example 210:52
- Lecture Example 317:55
- Additional Example 4-1
- Additional Example 5-2

25m 37s

- Intro0:00
- Real World Problems0:08
- Sketch0:11
- Interval0:20
- Rewrite in One Variable0:26
- Maximum or Minimum0:34
- Critical Points0:42
- Optimal Result0:52
- Lecture Example 11:05
- Lecture Example 26:12
- Lecture Example 313:31
- Additional Example 4-1
- Additional Example 5-2

25m 13s

- Intro0:00
- Approximating Using Newton's Method0:10
- Good Guesses for Convergence0:32
- Lecture Example 10:49
- Lecture Example 24:21
- Lecture Example 37:59
- Additional Example 4-1
- Additional Example 5-2

36m 50s

- Intro0:00
- Three Approximations0:12
- Right Endpoint, Left Endpoint, Midpoint0:22
- Formulas1:05
- Velocity and Distance1:35
- Lecture Example 12:28
- Lecture Example 212:10
- Lecture Example 319:43
- Additional Example 4-1
- Additional Example 5-2

22m 2s

- Intro0:00
- Important Equations0:22
- Riemann Sum0:28
- Integral1:58
- Integrand2:35
- Limits of Integration (Upper Limit, Lower Limit)2:43
- Other Equations3:05
- Fundamental Theorem of Calculus4:00
- Lecture Example 15:04
- Lecture Example 210:43
- Lecture Example 313:52
- Additional Example 4-1
- Additional Example 5-2

23m 19s

- Intro0:00
- U-Substitution0:13
- Important Equations0:30
- Purpose0:36
- Lecture Example 11:30
- Lecture Example 26:17
- Lecture Example 39:00
- Lecture Example 411:24
- Additional Example 5-1
- Additional Example 6-2

19m 59s

- Intro0:00
- Area Between Two Curves0:12
- Graphic Description0:34
- Lecture Example 11:44
- Lecture Example 25:39
- Lecture Example 38:45
- Additional Example 4-1
- Additional Example 5-2

24m 22s

- Intro0:00
- Important Equations0:16
- Equation 1: Rotation about x-axis (disks)0:27
- Equation 2: Two curves about x-axis (washers)3:38
- Equation 3: Rotation about y-axis5:31
- Lecture Example 16:05
- Lecture Example 28:28
- Lecture Example 311:55
- Additional Example 4-1
- Additional Example 5-2

30m 29s

- Intro0:00
- Important Equations0:50
- Equation 1: Rotation about y-axis1:04
- Equation 2: Rotation about y-axis (2 curves)7:34
- Equation 3: Rotation about x-axis8:15
- Lecture Example 18:57
- Lecture Example 214:26
- Lecture Example 318:15
- Additional Example 4-1
- Additional Example 5-2

16m 31s

- Intro0:00
- Important Equations0:11
- Origin of Formula0:34
- Lecture Example 12:51
- Lecture Example 25:30
- Lecture Example 38:13
- Additional Example 4-1
- Additional Example 5-2

23m 58s

- Intro0:00
- Slope Function of f(x)0:41
- Slope is Zero0:53
- Slope is Positive1:03
- Slope is Negative1:13
- Slope Function of f'(x)1:31
- Slope is Zero1:42
- Slope is Positive1:48
- Slope is Negative1:54
- Lecture Example 12:23
- Lecture Example 28:06
- Lecture Example 312:36
- Additional Example 4-1
- Additional Example 5-2

18m 32s

- Intro0:00
- Things to Remember0:13
- Graphic Description0:42
- Lecture Example 11:44
- Lecture Example 26:59
- Lecture Example 39:46
- Additional Example 4-1
- Additional Example 5-2

17m 4s

- Intro0:00
- Differential Equations0:10
- Focus on Exponential Growth/Decay0:27
- Separating Variables0:47
- Lecture Example 11:35
- Lecture Example 26:41
- Lecture Example 39:36
- Additional Example 4-1
- Additional Example 5-2

For more information, please see full course syllabus of College Calculus: Level I

0 answers

Post by Mohsin Alibrahim on April 6, 2015

In Ex 1

Where did get 4.25 when you evaluated v(4.25)

0 answers

Post by edder villegas on October 26, 2014

on example 2, can you explain please how did we get 0 and 4 as the values for the other two second travel options, thank you

0 answers

Post by Roy Yakobovich on April 23, 2014

I am really confused on how you moved the -1/5 to the other side to become 5/3. I understand how you got 5/3, by moving -1/5 to the other side, multiplying it by 5 and then dividing by 3. But don't you also need to multiply by 5 to both sides of the equation? Because I have been taught what you do one side to the equation you must do to the other.

0 answers

Post by Fadel Hanoun on November 22, 2013

I really enjoy your lectures! Thank you!

Are you using Stewart Calculus? because the order of your lectures matches the order of the chapter.

1 answer

Last reply by: Angela Patrick

Fri Dec 6, 2013 5:11 PM

Post by Sitora Muhamedova on July 26, 2013

I think there is a typo, when u divided 36/16, I got an answer of 9/4

1 answer

Last reply by: sonia Orjuela Orjuela

Thu Nov 11, 2010 11:14 AM

Post by sonia Orjuela Orjuela on November 11, 2010

why did you solve for y and not x?