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INSTRUCTORS Raffi Hovasapian John Zhu
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For more information, please see full course syllabus of Calculus AB
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Exponential Functions

  • Useful identities:
  • Sometimes helpful to think of numbers as exponents: , , etc.

Exponential Functions

Simplify (x2)3
  • This is equivalent to x2x2x2, which would give an exponent of 2 + 2 + 2, or 2 * 3
x6
Rewrite (ex)−2
(ex)−2 = e−2x = [1/(e2x)]
Rewrite (ex)x
(ex)x = ex * x = ex2
Combine the exponents of ([(ex)/(ey)])z
([(ex)/(ey)])z = (ex e−y)z = (ex − y)z = ez(x − y)
Combine the exponents of [(x−5)/(x4)]
[(x−5)/(x4)] = [1/(x4 x5)] = [1/(x4 + 5)] = [1/(x9)]
Simplify [(cos3(x)sin2(x))/(cos5(x)sin3(x) + sin4(x))]
  • [(cos3(x)sin2(x))/(cos5(x)sin3(x) + sin4(x))] =
  • [(cos3(x)sin2(x))/(sin3(x)(cos5(x) + sin(x)))]=
  • [(sin2(x))/(sin3(x))] [(cos3(x))/(cos5(x) + sin(x))] =
  • [1/sin(x)] [(cos3(x))/(cos5(x) + sin(x))]=
[(csc(x)cos3(x))/(cos5(x) + sin(x))]
Simplify e−iθe
e−iθe = e−iθ+ iθ = e0 = 1
Solve for x, 2x = 16
  • 2x = 16
  • 2x = 42
  • 2x = (22)2
  • 2x = 24
x = 4
Solve for x, 3x + 1 = 81
  • 3x + 1 = 81
  • 3x + 1 = 92
  • 3x + 1 = (33)2
  • 3x + 1 = 36
  • x + 1 = 6
x = 5
Prove [(2(e − e−iθ)(e + e−iθ))/4i] = [(e2iθ − e−2iθ)/2i]
  • [(2(e − e−iθ)(e + e−iθ))/4i]
  • =[2/4i] (e − e−iθ)(e + e−iθ)
  • = [1/2i] (ee − e−iθe + ee−iθ − e−iθe−iθ)
  • = [1/2i] (e2iθ − e0 + e0 − e−2iθ)
  • = [(e2iθ − e−2iθ)/2i]
  • ADVANCED NOTE: This is the exponential expression of the trig identity 2sin(θ)cos(θ) = sin(2θ)
[(2(e − e−iθ)(e + e−iθ))/4i] =[2/4i] (e − e−iθ)(e + e−iθ)
= [1/2i] (ee − e−iθe + ee−iθ − e−iθe−iθ)
= [1/2i] (e2iθ − e0 + e0 − e−2iθ)
= [(e2iθ − e−2iθ)/2i]

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

Exponential Functions

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
  • Types of Functions: Exponentials 0:07
    • General Form
    • Special Exponential Function
  • Example 1: Using Exponential Properties 0:46
  • Example 2: Using Exponential Properties 1:58
  • Example 3: Using Trig Identities & Exponential Properties 3:16
  • Example 4: Using Exponential Properties 4:37