Chapter 3

Differentiation Rules

Educators

FL

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 N.
Numerade Educator

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 N.
Numerade Educator

Problem 3

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

Clarissa N.
Numerade Educator

Problem 4

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

Clarissa N.
Numerade Educator

Problem 5

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

Clarissa N.
Numerade Educator

Problem 6

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

Clarissa N.
Numerade Educator

Problem 7

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

Clarissa N.
Numerade Educator

Problem 8

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

Clarissa N.
Numerade Educator

Problem 9

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

Clarissa N.
Numerade Educator

Problem 10

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

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Problem 11

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

Clarissa N.
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Problem 12

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

Clarissa N.
Numerade Educator

Problem 13

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

Clarissa N.
Numerade Educator

Problem 14

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

Clarissa N.
Numerade Educator

Problem 15

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

Clarissa N.
Numerade Educator

Problem 16

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

Clarissa N.
Numerade Educator

Problem 17

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

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Numerade Educator

Problem 18

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

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Numerade Educator

Problem 19

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

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Numerade Educator

Problem 20

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

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Problem 21

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

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Numerade Educator

Problem 22

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

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Problem 23

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

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Numerade Educator

Problem 24

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

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Problem 25

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

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Problem 26

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

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Problem 27

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

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Numerade Educator

Problem 28

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

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Numerade Educator

Problem 29

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

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Numerade Educator

Problem 30

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

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Numerade Educator

Problem 31

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

Carson M.
Numerade Educator

Problem 32

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

Clarissa N.
Numerade Educator

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 N.
Numerade Educator

Problem 34

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

Clarissa N.
Numerade Educator

Problem 35

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

Clarissa N.
Numerade Educator

Problem 36

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

Clarissa N.
Numerade Educator

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 N.
Numerade Educator

Problem 38

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

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Numerade Educator

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 N.
Numerade Educator

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 N.
Numerade Educator

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 N.
Numerade Educator

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 N.
Numerade Educator

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 N.
Numerade Educator

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 N.
Numerade Educator

Problem 45

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

Clarissa N.
Numerade Educator

Problem 46

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

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Numerade Educator

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 N.
Numerade Educator

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 N.
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 N.
Numerade Educator

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 N.
Numerade Educator

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 N.
Numerade Educator

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 N.
Numerade Educator

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 N.
Numerade Educator

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 N.
Numerade Educator

Problem 55

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

Clarissa N.
Numerade Educator

Problem 56

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

Clarissa N.
Numerade Educator

Problem 57

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

Clarissa N.
Numerade Educator

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 N.
Numerade Educator

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 N.
Numerade Educator

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 N.
Numerade Educator

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 N.
Numerade Educator

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 N.
Numerade Educator

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 N.
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.

Clarissa N.
Numerade Educator

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 N.
Numerade Educator

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 $

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Numerade Educator

Problem 67

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

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Numerade Educator

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 N.
Numerade Educator

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 N.
Numerade Educator

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 N.
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 '. $

FL
Frank L.
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'. $

FL
Frank L.
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 N.
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 N.
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 N.
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 N.
Numerade Educator

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 N.
Numerade Educator

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 N.
Numerade Educator

Problem 79

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

Clarissa N.
Numerade Educator

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 N.
Numerade Educator

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 N.
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 N.
Numerade Educator

Problem 83

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

Clarissa N.
Numerade Educator

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 N.
Numerade Educator

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 N.
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 N.
Numerade Educator