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Chapter 3

DIFFERENTIATION RULES

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Problem 1

(a) How is the number e defined?
(b) Use a calculator to estimate the values of the limits
$$\lim _{h \rightarrow 0} \frac{2.7^{n}-1}{h} \quad and \quad \lim _{h \rightarrow 0} \frac{2.8^{n}-1}{h}$$
correct to two decimal places. What can you conclude about the value of e?

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Problem 2

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

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Problem 3

$3-32$ Differentiate the function.
$f(x)=186.5$

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Problem 4

$3-32$ Differentiate the function.
$f(x)=\sqrt{30}$

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Problem 5

$3-32$ Differentiate the function.
$f(t)=2-\frac{2}{3} t$

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Problem 6

$3-32$ Differentiate the function.
$F(x)=\frac{3}{4} x^{8}$

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Problem 7

$3-32$ Differentiate the function.
$f(x)=x^{3}-4 x+6$

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Problem 8

$3-32$ Differentiate the function.
$f(t)=\frac{1}{2} t^{6}-3 t^{4}+1$

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Problem 9

$3-32$ Differentiate the function.
$f(t)=\frac{1}{4}\left(t^{4}+8\right)$

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Problem 10

$3-32$ Differentiate the function.
$h(x)=(x-2)(2 x+3)$

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Problem 11

$3-32$ Differentiate the function.
$y=x^{-2 / 5}$

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Problem 12

$3-32$ Differentiate the function.
$y=5 e^{x}+3$

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Problem 13

$3-32$ Differentiate the function.
$V(r)=\frac{4}{3} \pi r^{3}$

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Problem 14

$3-32$ Differentiate the function.
$R(t)=5 t^{-3 / 5}$

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Problem 15

$3-32$ Differentiate the function.
$A(s)=-\frac{12}{s^{5}}$

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Problem 16

$3-32$ Differentiate the function.
$B(y)=c y^{-6}$

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Problem 17

$3-32$ Differentiate the function.
$G(x)=\sqrt{x}-2 e^{x}$

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Problem 18

$3-32$ Differentiate the function.
$y=\sqrt[3]{x}$

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Problem 19

$3-32$ Differentiate the function.
$F(x)=\left(\frac{1}{2} x\right)^{5}$

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Problem 20

$3-32$ Differentiate the function.
$f(t)=\sqrt{t}-\frac{1}{\sqrt{t}}$

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Problem 21

$3-32$ Differentiate the function.
$y=a x^{2}+b x+c$

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Problem 22

$3-32$ Differentiate the function.
$y=\sqrt{x}(x-1)$

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Problem 23

$3-32$ Differentiate the function.
$y=\frac{x^{2}+4 x+3}{\sqrt{x}}$

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Problem 24

$3-32$ Differentiate the function.
$y=\frac{x^{2}-2 \sqrt{x}}{4}$

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Problem 25

$3-32$ Differentiate the function.
$y=4 \pi^{2}$

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Problem 26

$3-32$ Differentiate the function.
$g(u)=\sqrt{2} u+\sqrt{3 u}$

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Problem 27

$3-32$ Differentiate the function.
$H(x)=\left(x+x^{-1}\right)^{3}$

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Problem 28

$3-32$ Differentiate the function.
$y=a e^{v}+\frac{b}{v}+\frac{c}{v^{2}}$

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Problem 29

$3-32$ Differentiate the function.
$\mathrm{u}=\sqrt[5]{\mathrm{t}}+4 \sqrt{\mathrm{t}^{5}}$

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Problem 30

$3-32$ Differentiate the function.
$v=\left(\sqrt{x}+\frac{1}{\sqrt[3]{x}}\right)^{2}$

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Problem 31

$3-32$ Differentiate the function.
$z=\frac{\mathrm{A}}{\mathrm{y}^{10}}+\mathrm{Be}^{y}$

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Problem 32

$3-32$ Differentiate the function.
$y=e^{x+1}+1$

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Problem 33

$33-34$ Find an equation of the tangent line to the curve at the given point.
$y=\sqrt[4]{x}, \quad(1,1)$

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Problem 34

$33-34$ Find an equation of the tangent line to the curve at the given point.
$y=x^{4}+2 x^{2}-x_{1} \quad(1,2)$

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Problem 35

$35-36$ Find equations of the tangent line and normal line to the curve at the given point.
$y=x^{4}+2 e^{x},(0,2)$

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Problem 36

$35-36$ Find equations of the tangent line and normal line to the curve at the given point.
$y=(1+2 x)^{2}, \quad(1,9)$

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Problem 37

$37-38$ Find 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 screen.
$y=3 x^{2}-x^{3},(1,2)$

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Problem 38

$37-38$ Find 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 screen.
$y=x-\sqrt{x}, \quad(1,0)$

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Problem 39

$39-42$ Find $f^{\prime}(x) .$ Compare the graphs of $f$ and $f^{\prime}$ and use them to explain why your answer is reasonable.
$f(x)=e^{x}-5 x$

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Problem 40

$39-42$ Find $f^{\prime}(x) .$ Compare the graphs of $f$ and $f^{\prime}$ and use them to explain why your answer is reasonable.
$f(x)=3 x^{5}-20 x^{3}+50 x$

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Problem 41

$39-42$ Find $f^{\prime}(x) .$ Compare the graphs of $f$ and $f^{\prime}$ and use them to explain why your answer is reasonable.
$f(x)=3 x^{15}-5 x^{3}+3$

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Problem 42

$39-42$ Find $f^{\prime}(x) .$ Compare the graphs of $f$ and $f^{\prime}$ and use them to explain why your answer is reasonable.
$f(x)=x+\frac{1}{x}$

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Problem 43

(a) Use a graphing calculator or computer to graph the function $f(x)=x^{4}-3 x^{3}-6 x^{2}+7 x+30$ 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'. (See
Example 1 in Section $2.8 .$ .
(c) Calculate $f^{\prime}(x)$ and use this expression, with a graphing device, to graph $f^{\prime}$ '. Compare with your sketch in part (b)

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Problem 44

(a) Use a graphing calculator or computer to graph the function $g(x)=e^{x}-3 x^{2}$ in the viewing rectangle $[-1,4]$ $\quad$ by $[-8,8]$
(b) Using the graph in part (a) to estimate slopes, make a rough sketch, by hand, of the graph of $g^{\prime}$ . (See Example 1 in Section $2.8 .$ .
(c) Calculate $g^{\prime}(x)$ and use this expression, with a graphing device, to graph $g^{\prime}$ . Compare with your sketch in part (b).

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Problem 45

$45-46$ Find the first and second derivatives of the function.
$f(x)=x^{4}-3 x^{3}+16 x$

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Problem 46

$45-46$ Find the first and second derivatives of the function.
$G(r)=\sqrt{r}+\sqrt[3]{r}$

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Problem 47

$47-48$ Find the first and second derivatives of the function. Check to see that your answers are reasonable by comparing the graphs of $f, f^{\prime},$ and $f^{\prime \prime}$
$f(x)=2 x-5 x^{3 / 4}$

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Problem 48

$47-48$ Find the first and second derivatives of the function. Check to see that your answers are reasonable by comparing the graphs of $f, f^{\prime},$ and $f^{\prime \prime}$
$f(x)=e^{x}-x^{3}$

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Problem 49

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

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Problem 50

The equation of motion of a particle is $\mathrm{s}=2 \mathrm{t}^{3}-7 \mathrm{t}^{2}+4 \mathrm{t}+1,$ where $\mathrm{s}$ is in meters and $\mathrm{t}$ is in seconds.
(a) Find the velocity and acceleration as functions of t.
(b) Find the acceleration after 1 $\mathrm{S}$ .
(c) Graph the position, velocity, and acceleration functions on the same screen.

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Problem 51

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

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Problem 52

For what values of $x$ does the graph of $f(x)=x^{3}+3 x^{2}+x+3$ have a horizontal tangent?

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Problem 53

Show that the curve $y=6 x^{3}+5 x-3$ has no tangent line with slope $4 .$

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Problem 54

Find an equation of the tangent line to the curve $y=x \sqrt{x}$ that is parallel to the line $y=1+3 x$

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Problem 55

Find equations of both lines that are tangent to the curve $y=1+x^{3}$ and are parallel to the line $12 x-y=1$

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Problem 56

At what point on the curve $y=1+2 e^{x}-3 x$ is the tangent line parallel to the line $3 x-y=5 ?$ Illustrate by graphing the curve and both lines.

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Problem 57

Find an equation of the normal line to the parabola $y=x^{2}-5 x+4$ that is parallel to the line $x-3 y=5$

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Problem 58

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

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Problem 59

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

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Problem 60

(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.

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Problem 61

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

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Problem 62

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

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Problem 63

Find a second-degree polynomial $\mathrm{P}$ such that $\mathrm{P}(2)=5$ $\mathrm{P}^{\prime}(2)=3,$ and $\mathrm{P}^{\prime \prime}(2)=2$

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Problem 64

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

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Problem 65

Find a cubic function $y=a x^{3}+b x^{2}+c x+d$ whose graph has horizontal tangents at the points $(-2,6)$ and $(2,0)$ .

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Problem 66

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

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Problem 67

Let
$$f(x)=\left\{\begin{array}{ll}{2-x} & {\text { if } x \leq 1} \\ {x^{2}-2 x+2} & {\text { if } x>1}\end{array}\right.$$
Is f differentiable at 17 Sketch the graphs of $f$ and $f^{\prime}$

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Problem 68

At what numbers is the following function $g$ differentiable?
$g(x)=\left\{\begin{array}{ll}{-1-2 x} & {\text { if } x<-1} \\ {x^{2}} & {\text { if }-1 \leq x \leq 1} \\ {x} & {\text { if } x>1}\end{array}\right.$
Give a formula for $g^{\prime}$ and sketch the graphs of $g$ and $g^{\prime}$

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Problem 69

(a) For what values of $x$ is the function $f(x)=\left|x^{2}-9\right|$ differentiable? Find a formula for $f^{\prime}$ .
(b) Sketch the graphs of $f$ and $f^{\prime}$ .

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Problem 70

Where is the function $h(x)=|x-1|+|x+2|$ differentiable? Give a formula for h' and sketch the graphs of $h$ and $h^{\prime}$

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Problem 71

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

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Problem 72

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

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Problem 73

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

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Problem 74

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

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Problem 75

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

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Problem 76

A tangent line is drawn to the hyperbola xy $=\mathrm{c}$ at a point $\mathrm{P}$ .
(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 $\mathrm{P}$ is located on the hyperbola.

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Problem 77

Evaluate lim $\frac{x^{1000}-1}{x-1}$

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Problem 78

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?

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Problem 79

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

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Problem 80

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

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