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Workbook for Stewart’s "Calculus"

Cory Wilson

Section 3.9 Antiderivatives

Subsection 3.9.1 Before Class

https://mymedia.ou.edu/media/3.9-1/1_47ve5csu
Figure 29. Pre-Class Video 1

Subsubsection 3.9.1.1 Derivatives

Function Derivative Function Derivative
\(c\) \(0\) \(x^n\) \(nx^{n-1}\)
\(\sin x\) \(\cos x\) \(\cos x\) \(-\sin x\)
\(\tan x\) \(\sec^2 x\) \(\cot x\) \(-\csc^2 x\)
\(\sec x\) \(\sec x\tan x\) \(\csc x\) \(-\csc x\cot x\)
\(c\cdot f(x)\) \(c\cdot f'(x)\) \(f(x)\pm g(x)\) \(f'(x)\pm g'(x)\)
\(f(x)\cdot g(x)\) \(f'(x)g(x) + f(x)g'(x)\) \(\dfrac{f(x)}{g(x)}\) \(\dfrac{f'(x)g(x)-f(x)g'(x)}{[g(x)]^2}\)
\(f(g(x))\) \(f'(g(x))\cdot g'(x)\)

Subsubsection 3.9.1.2 The Antiderivative

Definition 3.9.1. Antiderivative.
A function \(F\) is called an antiderivative of \(f\) on an interval \(I\) if \(F'(x) = f(x)\) for all \(x\in I\text{.}\)
Example 3.9.2.
Find five antiderivatives for the function \(f(x) = 2x\text{.}\)
Solution.
Answers vary
Question 3.9.3.
How might we write the general antiderivative \(F(x)\) for \(f(x) = 2x\text{?}\)
Solution.
\(F(x) = x^2 + C\)
Example 3.9.4.
Find the most general antiderivative for the functions \(f(x) = \cos x\text{,}\) \(g(x) = x^n\) (\(n\geq 0\)), and \(h(x) = x^{-2}\text{.}\)
Solution.
\(F(x) = \sin x + C, G(x) = \dfrac{x^{n+1}}{n+1}+C, H(x) = -x^{-1} + C\)
A table of useful antiderivatives is given below:
Function Antiderivative Function Antiderivative
\(0\) \(C\) \(k\) \(kx + C\)
\(x^n\) \(\dfrac{x^{n+1}}{n+1} + C\) \(\sin x\) \(-\cos x + C\)
\(\cos x\) \(\sin x + C\) \(\sec^2 x\) \(\tan x + C\)
\(\csc^2 x\) \(-\cot x + C\) \(\sec x\tan x\) \(\sec x + C\)
\(\cot x \csc x\) \(-\csc x\) \(c\cdot f(x)\) \(c\cdot F(x)\)
\(f(x)\pm g(x)\) \(F(x) \pm G(x)\)
Example 3.9.5.
\(g'(x) = 3\sin x - \dfrac{9x^4-\sqrt[3]{x}}{x^2}\text{.}\) Find \(g(x)\text{,}\) the most general antiderivative of \(g\text{.}\)
Solution.
\(g(x) = -3\cos x - 3x^3 - \dfrac{3}{2}x^{-2/3} + C\)
Example 3.9.6.
Find \(f\) if \(f'(x) = x\sqrt{x}\) and \(f(1) = 5\text{.}\)
Solution.
\(f(x) = \dfrac{2}{5}x^{5/2} + \dfrac{23}{5}\)

Subsection 3.9.2 Pre-Class Activities

Example 3.9.7.

Find the most general antiderivative of the following functions; check your answer by differentiation.
  1. \(\displaystyle f(x) = 5x + 3\)
  2. \(\displaystyle f(x) = x(12x + 8)\)
  3. \(\displaystyle g(x) = \sqrt{5}\)
  4. \(\displaystyle h(x) = -3\sqrt{x} - 8\sqrt[3]{x}\)
  5. \(\displaystyle k(x) = \dfrac{6}{x^7}\)
  6. \(\displaystyle f(t) = \dfrac{5-4t^3 + 2t^6}{t^6}\)
Solution.
  1. \(\displaystyle F(x) = \dfrac{5}{2}x^2 + 3x + C\)
  2. \(\displaystyle F(x) = 4x^3 + 4x^2 + C\)
  3. \(\displaystyle G(x) = x\sqrt{5}+ C\)
  4. \(\displaystyle H(x) = -2x^{3/2} - 6x^{4/3} + C\)
  5. \(\displaystyle k(x) = -x^{-6} + C\)
  6. \(\displaystyle F(t) = -t^{-5} + 2t^{-2} + 2t + C\)

Subsection 3.9.3 In Class

General Antiderivative vs. Specific Antiderivative.

A general antiderivative takes the form \(F(x) + C\text{,}\) while a specific antiderivative takes the form \(F(x)\text{.}\) For a specific antiderivative, we must have information about the value(s) of the original function.

Example 3.9.8.

Find \(f\) if \(f''(x) = 12x^2+6x-4\text{,}\) \(f(0) = 4\text{,}\) and \(f(1) = 2\text{.}\)
Solution.
\(f(x) = x^4 + x^3-2x^2-2x+4\)

Example 3.9.9.

A particle moves in a straight line, and its acceleration is given by \(a(t) = 3t+2\text{.}\) Its initial velocity is \(v(0) = -3\) cm/s, and its initial displacement is \(s(0) = 5\) cm. Find the position function \(s(t)\text{.}\)
Solution.
\(s(t) = \dfrac{1}{2}t^3 + t^2-3t+5\)

Example 3.9.10.

A ball is thrown upward with speed 24 ft/s from the edge of a cliff which is 432 ft above the ground. Find its height above the ground \(t\) seconds later. When does it reach its maximum height? When does it hit the ground?
Solution.
The position function is \(s(t) = -16t^2 + 24t + 432\text{.}\) The ball reaches maximum height \(\dfrac{3}{4}\) seconds after being thrown, and hits the ground 6 seconds after being thrown.

Example 3.9.11.

Find the most general antiderivative of \(f(x) = (x-5)^2\text{.}\)
Solution.
\(F(x) = \dfrac{1}{3}x^3-5x^2+25x+C\)

Example 3.9.12.

Find the most general antiderivative of \(f(x) = \pi^2\text{.}\)
Solution.
\(F(x) = \pi^2x + C\)

Example 3.9.13.

Find the most general antiderivative of \(f(t) = 3\cos t - 4\sin t\text{.}\)
Solution.
\(F(t) = 3\sin t + 4\cos t + C\)

Example 3.9.14.

Find the most general antiderivative of \(f(x) = 8\sqrt{x}-\sec x\tan x\text{.}\)
Solution.
\(F(x) = \dfrac{16}{3}x^{3/2} - \sec x + C\)

Example 3.9.15.

Find \(f\text{,}\) if \(f''(x) = \sin x + \cos x\text{,}\) \(f(0) = 3\text{,}\) and \(f'(0)= 4\text{.}\)
Solution.
\(f(x) = -\sin x - \cos x + 5x + 4\)

Example 3.9.16.

The graph of \(f\) passes through the point \((2,5)\text{,}\) and the slope of its tangent line at \((x,f(x))\) is \(3-4x\text{.}\) Find \(f(1)\text{.}\)
Solution.
\(f(1) = 8\)

Subsection 3.9.4 After Class Activities

Example 3.9.17.

Find \(g\text{,}\) if \(g'''(t) = 12+\cos t\text{.}\)
Solution.
\(g(t) = 2t^3-\sin t + Ct^2 + Dt + E\)

Example 3.9.18.

A car is traveling at 60 mi/h when the brakes are fully applied, producing a constant deceleration of 20 ft/s\(^2\text{.}\) What is the distance traveled before the car comes to a stop?
Solution.
90 feet

Example 3.9.19.

The graph of a function is show below. Make a rough sketch of an antiderivative \(F\text{,}\) if \(F(0) = 1\text{.}\)
Solution.
Answers vary
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