Calculus: Early Transcendentals

James Stewart

Chapter 3

Differentiation Rules - all with Video Answers

Educators

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Section 1

Derivatives of Polynomials and Exponential Functions

Problem 1

(a) How is the number $ e $ defined?
(b) Use a calculator to estimate the values of the limits

$ \displaystyle \lim_{h\to 0}\frac {2.7^h - 1}{h} $ and $ \displaystyle \lim_{h\to 0}\frac {2.8^h - 1}{h} $

correct to two decimal places. What can you conclude about the value of $ e $?

Clarissa Noh

Clarissa Noh

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Problem 2

(a) Sketch, by hand, the graph of the function $ f(x) = e^x, $ paying particular attention to the graph crosses the y-axes.
What fact allows you to do this?
(b) What types of functions are $ f(x) = e^x $ and $ g(x) = x^e? $
Compare the differentiation formulas for $ f $ and $ g. $
(c) Which of the two functions in part (b) grows more rapidly when $ x $ is large?

Clarissa Noh

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Problem 3

Differentiate the function.
$ f(x) = 2^{40} $

Clarissa Noh

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Problem 4

Differentiate the function.
$ f(x) = e^5 $

Clarissa Noh

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Problem 5

Differentiate the function.
$ f(x) = 5.2x + 2.3 $

CP

Cortney Powers

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Problem 6

Differentiate the function.
$ g(x) = \frac{7}{4}x^2 - 3x + 12 $

Clarissa Noh

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Problem 7

Differentiate the function.
$ f(t) = 2t^3 - 3t^2 - 4t $

Clarissa Noh

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Problem 8

Differentiate the function.
$ f(t) = 1.4t^5 - 2.5t^2 + 6.7 $

Clarissa Noh

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Problem 9

Differentiate the function.
$ g(x) = x^2(1-2x) $

Clarissa Noh

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Problem 10

Differentiate the function.
$ H(u) = (3u - 1)(u + 2) $

Clarissa Noh

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Problem 11

Differentiate the function.
$ g(t) = 2t^{-3/4} $

Clarissa Noh

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Problem 12

Differentiate the function.
$ B(y) = cy^{-6} $

Clarissa Noh

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Problem 13

Differentiate the function.
$ F(r) = \frac{5}{r^3} $

Clarissa Noh

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Problem 14

Differentiate the function.
$ y = x^{5/3} - x^{2/3} $

Mary Wakumoto

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Problem 15

Differentiate the function.
$ R(a) = (3a + 1)^2 $

Clarissa Noh

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Problem 16

Differentiate the function.
$ h(t) = \sqrt[4]{t} - 4e^1 $

Clarissa Noh

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Problem 17

Differentiate the function.
$ S(p) = \sqrt{p} - p $

Clarissa Noh

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Problem 18

Differentiate the function.
$ y = \sqrt[3]{x}(2 + x ) $

Clarissa Noh

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Problem 19

Differentiate the function.
$ y = 3e^x + \frac{4}{\sqrt[3]{x}} $

Clarissa Noh

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Problem 20

Differentiate the function.
$ S(R) = 4\pi R^2 $

Clarissa Noh

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Problem 21

Differentiate the function.
$ h(u) = Au^3 + Bu^2 + Cu $

Clarissa Noh

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Problem 22

Differentiate the function.
$ y = \frac {\sqrt{x + x}}{x^2} $

Clarissa Noh

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Problem 23

Differentiate the function.
$ y = \frac {x^2+4x+3}{\sqrt{x}} $

Clarissa Noh

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Problem 24

Differentiate the function.
$ G(t) = \sqrt{5t} + \frac {\sqrt{7}}{t} $

Clarissa Noh

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Problem 25

Differentiate the function.
$ j(x) = x^{2.4} + e^{2.4} $

Clarissa Noh

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Problem 26

Differentiate the function.
$ k(r) = e^r + r^e $

Clarissa Noh

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Problem 27

Differentiate the function.
$ G(q) = (1 + q^{-1})^2 $

Clarissa Noh

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Problem 28

Differentiate the function.
$ F(z) = \frac{A + Bz + Cz^2}{z^2} $

Clarissa Noh

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Problem 29

Differentiate the function.
$ f(v) = \frac{\sqrt[3]{v - 2ve^v}}{v} $

Clarissa Noh

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Problem 30

Differentiate the function.
$ D(t) = \frac {1 + 16t^2}{(4t)^3} $

Clarissa Noh

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Problem 31

Differentiate the function.
$ z = \frac {A}{y^{10}} + Be^y $

Carson Merrill

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Problem 32

Differentiate the function.
$ y = e^{x+1} + 1 $

Clarissa Noh

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Problem 33

Find an equation of the tangent line to the curve at the given point.
$ y = 2x^3 - x^2 + 2, (1,3) $

Clarissa Noh

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Problem 34

Find an equation of the tangent line to the curve at the given point.
$ y = 2e^x + x, (0,2) $

Clarissa Noh

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Problem 35

Find an equation of the tangent line to the curve at the given point.
$ y = x + \frac{2}{x} , (2,3) $

Clarissa Noh

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Problem 36

Find an equation of the tangent line to the curve at the given point.
$ y = \sqrt[4]{x} - x, (1,0) $

MI

Mahasin Irfan

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Problem 37

Find equations of the tangent line and normal line to the curve at the given point.
$ y = x^4 + 2e^x, (0,2) $

Clarissa Noh

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Problem 38

Find equations of the tangent line and normal line to the curve at the given point.
$ y^2 = x^3, (1,1) $

Clarissa Noh

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Problem 39

Fidn an equation of the tangent line to the curve at the given point. Illustrate by graphing the curve and the tangent line on the same scree.
$ y = 3x^2 - x^3, (1,2) $

Clarissa Noh

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Problem 40

Fidn an equation of the tangent line to the curve at the given point. Illustrate by graphing the curve and the tangent line on the same scree.
$ y = x - \sqrt{x}, (1,0) $

Clarissa Noh

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Problem 41

Find $ f'(x) $. Compare the graphs of $ f $ and $ f' $ and use them to explain why your answer is reasonable.
$ f(x) = x^4 - 2x^3 + x^2 $

Clarissa Noh

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Problem 42

Find $ f'(x) $. Compare the graphs of $ f $ and $ f' $ and use them to explain why your answer is reasonable.
$ f(x) = x^5 - 2x^3 + x - 1 $

Clarissa Noh

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Problem 43

(a) Graph the function
$ f(x) = x^4 - 3x^3 - 6x^2 + 7x + 30 $
in the viewing rectangle [-3,5] by [-10,50].
(b) Using the graph in part (a) to estimate slopes, make a rough sketch, by hand, of the graph of $ f' $.
(c) Calculate $ f'(x) $ and use this expression, with graphing device, to graph $ f' $. Compare with your sketch in part (b).

Clarissa Noh

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Problem 44

(a) Graph the function $ g(x) = e^x - 3x^2 $ in the viewing rectangle [-1,4] by [-8,8].
(b) Using the graph in part (a) to estimate slopes, make a rough sketch, by hand, of the graph of $ g' $.
(c) Calculate $ g'(x) $ and use this expression, with a graphing device, to graph $ g' $. Compare with your sketch in part (b).

Clarissa Noh

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Problem 45

Find the first and second derivatives of the function.
$ f(x) = 0.001x^5 - 0.02x^3 $

Clarissa Noh

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Problem 46

Find the first and second derivatives of the function.
$ G(r) = \sqrt{r} + \sqrt[3]{r} $

Clarissa Noh

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Problem 47

Find the first and second derivatives of the function. Check to see that your answers are reasonable by comparing the graphs of $ f, f', $ and $ f". $
$ f(x) = 2x - 5x^{3/4} $

Clarissa Noh

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Problem 48

Find the first and second derivatives of the function. Check to see that your answers are reasonable by comparing the graphs of $ f, f', $ and $ f". $
$ f(x) = e^x - x^3 $

Clarissa Noh

Numerade Educator

Problem 49

The equation of motion of a particle is $ = t^3 - 3t, $ where is in meters and is in seconds. Find
(a) the velocity and acceleration as functions of $ t, $
(b) the acceleration after $ 2 s, $ and
(c) the acceleration when the velocity is 0.

Clarissa Noh

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Problem 50

The equation of motion of a particle is, $ s = 2t^4 - 2t^3 + t^2 - t $ , where is in meters and is in
seconds.
(a) Find the velocity and acceleration as functions of .
(b) Find the acceleration after 1 s.
(c) Graph the position, velocity, and acceleration functions on the same screen.

Clarissa Noh

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Problem 51

Biologists have proposed a cubic polynomial to model the length $ L $ of Alaskan rockfish at age $ A: $
$ L = 0.0155A^3 - 0.372A^2 + 3.95A + 1.21 $
where $ L $ is measured in inches and $ A $ in years. Calculate
$ \frac{dL}{dA}\mid A=12 $

Clarissa Noh

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Problem 52

The number of tree species $ S $ in a given area $ A $ in the Pasoh Forest Reserve in Malaysia has been modeled by the power function

$ S(A) = 0.882A^0.842 $

where A is measured in square meters. Find $ S' (100) $ and interpret your answer.

Clarissa Noh

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Problem 53

Boyle's Law states that when a sample of gas is compressed at a constant temperature, the pressure $ P $ of the gas is inversely proportional to the volume $ V $ of the gas.
(a) Suppose that the pressure of a sample of air that occupies $ 0.106 m^3 at 25^{\circ} C $ is 50 kPa. Write $ V $ as a function of $ P. $
(b) Calculate $ dV\mid dP $ when $ P $ = 50 kPA. What is the meaning of the derivative? What are its units?

Clarissa Noh

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Problem 54

Car tires need to be inflated properly because overinflation or underinflation can cause premature tread wear. The data in tge tabe shoe tire life $ L $ (in thousands of miles) for a certain type of tire at various pressures $ P (in lb/in^2). $

(a) Use a calculator to model tire life with a quadratic function of the pressure.
(b) Use the model to estimate $ dL/dP $ when $ P = 30 $ and when $ P = 40. $ What is the meaning of the derivative? What are the units? What is the significance of the signs of the derivatives?

Clarissa Noh

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Problem 55

Find the points on the curve $ y = 2x^3 + 3x^2 - 12x + 1 $ where the tangent is horizontal.

Clarissa Noh

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Problem 56

For what value of $ x $ does the graph of $ f(x) = e^x - 2x $ have a horizontal tangent?

Clarissa Noh

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Problem 57

Show that the curve $ y = 2e^x + 3x + 5x^3 $ has no tangent line with slope 2.

Clarissa Noh

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Problem 58

Find an equation of the tangent line to the curve $ y = x^4 + 1 $ that is parallel to the line $ 32^x - y = 15. $

Clarissa Noh

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Problem 59

Find equation of both lines that are tangent to the curve $ y = x^3 - 3x^2 + 3x - 3 $ and are parallel to the line $ 3x - y = 15. $

Clarissa Noh

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Problem 60

At what point on the curve $ y = 1 + 2e^x - 3x $ is the tangent line parallel to the line $ 3x - y = 5? $ Illustrate with a sketch.

Clarissa Noh

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Problem 61

Find an equation of the normal line to the curve $ y = \sqrt{x} $ that is parallel to the line $ 2x + y = 1. $

Clarissa Noh

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Problem 62

Where does the normal line to the parabola $ y = x^2 - 1 $ at the point (-1, 0) intersect the parabola a second time? Illustrate with a sketch.

Clarissa Noh

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Problem 63

Draw a diagram to shoe that there are two tangent lines to the parabola $ y = x^2 $ that pass through the point (0, -4). Find the coordinates of the point where these tangent lines intersect the parabola.

Clarissa Noh

Numerade Educator

Problem 64

(a) Find equations of both lines through the point (2, -3) that are tangent to the parabola $ y = x^2 + x. $
(b) Show that there is no line through the point (2,7) that is tangent to the parabola. Then draw a diagram to see why.

Michael Crabtree

Michael Crabtree

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Problem 65

Use the definition of a derivative to show that if $ f(x) = 1/x, $ then $ f'(x) = -1/x^2. $ (This proves the Power Rule for the case $ n = -1.$ )

Clarissa Noh

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Problem 66

Find the $ n $ th derivative of each function by calculating the first few derivatives and observing the pattern that occurs.
(a) $ f(x) = x" $ (b) $ f(x) = 1/x $

Clarissa Noh

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Problem 67

Find a second-degree polynomial & P & such that $ P(2) = 5, P'(2) = 3, $ and $ P"(2) = 2. $

Clarissa Noh

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Problem 68

The equation $ y" + y' - 2y = x^2 $ is called a differential equation because it involves an unknown function $ y $ and its derivatives $ y' $ and $ y". $ Find constant $ A, B, $ and $ C $ such that the function $ y = Ax^2 + Bx + C $ satisfies this equation. (Differential equations will be studied in detail in Chapter 9.)

Clarissa Noh

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Problem 69

Find a cubic function $ y = ax^3 + bx^2 + cx + d $ whose graph has horizontal tangent at the points (-2, 6) and (2, 0).

Clarissa Noh

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Problem 70

Find a parabola with equation $ y = ax^2 + bx + c $ that has slope 4 at $ x = 1, $ slope -8 at $ x = -1, $ and passes through the point (2, 15).

Clarissa Noh

Numerade Educator

Problem 71

Let.
$ f(x) = \left\{
\begin{array}{ll}
x^2 + 1 & \mbox{if} x < 1\\
x + 1 & \mbox{if} x \ge 1 \\
\end{array} \right.$
Is $ f $ differentiable at 1? Sketch the graphs of $ f $ and $ f '. $

Frank Lin

Numerade Educator

Problem 72

At what numbers is the following function $ g $ differentiable?
$ g(x) = \left\{
\begin{array}{ll}
2x & \mbox{if}x \le 0\\
2x - x^2 & \mbox{if}0 < x < 2\\
2 - x & \mbox{if}x \ge 2
\end{array} \right. $
Give a formula for $ g' $ and sketch the graphs of $ g $ and $ g'. $

Frank Lin

Numerade Educator

Problem 73

(a) For what values of $ x $ is the function $ f(x) = \mid x^2 - 9 \mid $ differentiable? Find a formula for $ f'. $
(b) Sketch the graph of $ f $ and $ f'. $

Clarissa Noh

Numerade Educator

Problem 74

Where is the function $ h(x) = \mid x - 1 \mid + \mid x + 2 \mid $ differentiable? Give a formula for $ h' $ and sketch the graph of $ h $ and $ h'. $

Clarissa Noh

Numerade Educator

Problem 75

Find the parabola with equation $ y = ax^2 + bx $ whose tangent line at (1, 1) has equation $ y = 3x - 2. $

Clarissa Noh

Numerade Educator

Problem 76

Suppose the curve $ y = x^4 + ax^3 + bx^2 + cx + d $ has a tangent line when $ x = 0 $ with equation $ y = 2x + 1 $ and tangent line when $ x = 1 $ with equation $ y = 2 - 3x. $ Find the values of $ a,b,c, $ and $ d. $

Clarissa Noh

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Problem 77

For what values of $ a $ and $ b $ is the line $ 2x + y = b $ tangent to the parabola $ y = ax^2 $ when $ x = 2? $

Clarissa Noh

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Problem 78

Find the value of $ c $ such that the line $ y = \frac{3}{2}x + 6 $ is tangent to the curve $ y = c \sqrt{x}.

Clarissa Noh

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Problem 79

What is the value of $ c $ such that the line $ y = 2x + 3 $ is tangent to the parabola $ y = cx^2? $

Clarissa Noh

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Problem 80

The graph of any quadratic function $ f(x) = ax^2 + bx + c $ is a parabola. Prove that the average of the slopes of the slopes of the tangent lines to the parabola at the endpoints of any interval $ [p, q] $ equals the slope of the tangent line at the midpoint of the interval.

Clarissa Noh

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Problem 81

Let
$ f(x) = \left\{
\begin{array}{ll}
x^2 + & \mbox{if} x \le 2\\
mx + b & \mbox{if} x > 2\\
\end{array} \right. $
Find the values of $ m $ and $ b $ that make $ f $ differentiable everywhere.

Clarissa Noh

Numerade Educator

Problem 82

A tangent line is drawn to the hyperbola $ xy = c $ at a point $ P. $
(a) Show that the midpoint of the line segment cut from this tangent line by the coordinate axes is $ P. $
(b) Show that the triangle formed by the tangent line and the coordinate axes always has the same area, no matter where $ P $ is located on the hyperbola.

Clarissa Noh

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Problem 83

Evaluate $ \lim_{x\to1} \frac {x^{1000} - 1}{x - 1}. $

Clarissa Noh

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Problem 84

Draw a diagram showing two perpendicular lines that intersect on the y-axis and are both tangent to the parabola $ y = x^2. $ Where do these lines intersect?

Clarissa Noh

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Problem 85

If $ c > \frac {1}{2}, $ how many lines through the point (0, c) are normal lines to the parabola $ y = x^2? $ What if $ c \le \frac {1}{2}? $

Clarissa Noh

Numerade Educator

Problem 86

Sketch the parabolas $ y = x^2 and $ y = x^2 - 2x + 2. $ Do you think there is a line that is tangent to both curves? If so. find its equation. If not, why not?

Clarissa Noh

Numerade Educator