Problem 1

(a) How is the number e defined?

(b) Use a calculator to estimate the values of the limits

$$\lim _{h \rightarrow 0} \frac{2.7^{b}-1}{h} \quad \text { and } \quad \lim _{h \rightarrow 0} \frac{2.8^{b}-1}{h}$$

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 $g(x)=x^{e}$ ? 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 28

$3-32$ Differentiate the function.

$$y=a e^{x}+\frac{b}{v}+\frac{c}{v^{2}}$$

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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 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.

$$y=x^{4}+2 x^{2}-x, \quad(1,2)$$

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Problem 35

$35-36$ Find equations of the tangent line and normal line to the curve at the given point.

$$y=x^{4}+2 e^{x}, \quad(0,2)$$

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Problem 36

$35-36$ Find equations of the tangent line and normal line to the curve at the given point.

$$y=x^{2}-x^{4}, \quad(1,0)$$

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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.

$$y=3 x^{2}-x^{3}, \quad(1,2)$$

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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

$39-40$ Find $f^{\prime \prime}(x) .$ Compare the graphs of $f$ and $f^{\prime}$ and use them to explain why your answer is reasonable.

$$f(x)=x^{4}-2 x^{3}+x^{2}$$

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Problem 40

$39-40$ Find $f^{\prime \prime}(x) .$ Compare the graphs of $f$ and $f^{\prime}$ and use them to explain why your answer is reasonable.

$$f(x)=x^{5}-2 x^{3}+x-1$$

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Problem 41

(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 $f^{\prime}$ . (See Example 1 in Section 2.8.)

(c) Calculate $f^{\prime \prime}(x)$ and use this expression, with a graphing device, to graph $f^{\prime}$ . Compare with your sketch in part (b).

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Problem 42

(a) Use a graphing calculator or computer to graph the function $g(x)=e^{x}-3 x^{2}$ 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 $g^{\prime}$ . (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 43

$43-44$ Find the first and second derivatives of the function.

$$f(x)=10 x^{10}+5 x^{5}-x$$

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Problem 44

$43-44$ Find the first and second derivatives of the function.

$$G(r)=\sqrt{r+\sqrt[3]{r}}$$

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Problem 45

$45-46$ 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 $f^{*}$ .

$$f(x)=2 x-5 x^{3 / 4}$$

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Problem 46

$45-46$ 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 $f^{*}$ .

$$f(x)=e^{x}-x^{3}$$

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Problem 47

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 s, and

(c) the acceleration when the velocity is 0 .

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Problem 48

The equation of motion of a particle is $s=t^{4}-2 t^{3}+t^{2}-t,$ where $s$ is in meters and $t$ is in

seconds.

(a) Find the velocity and acceleration as functions of $t$

(b) Find the acceleration after 1 s.

(c) Graph the position, velocity, and acceleration functions on the same screen.

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Problem 49

Boyle s Law states that when a sample of gas is compressed at a constant pressure, 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 $\mathrm{m}^{3}$ at $25^{\circ} \mathrm{C}$ is 50 $\mathrm{kPa}$ . Write $V$ as a function of $P .$

(b). Calculate $d V / d P$ when $P=50 \mathrm{kP}$ . What is the meaning of the derivative? What are its units?

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Problem 50

Car tires need to be inflated properly because overinflation or underinflation can cause premature treadware. The data in the table show tire life $L($ in thousands of miles) for a certain type of tire at various pressures $P\left($ in 1 $\mathrm{b} / \mathrm{in}^{2}\right)$

$$

\begin{array}{|c|c|c|c|c|c|c|}\hline P & {26} & {28} & {31} & {35} & {38} & {42} & {45} \\ \hline L & {50} & {66} & {78} & {81} & {74} & {70} & {59} \\ \hline\end{array} $$

(a) Use a graphing calculator or computer to model tire life with a quadratic function of the pressure.

(b) Use the model to estimate $d L / d P$ 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?

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

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Problem 53

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

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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 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 $m$ th derivative of each function by calculating the first few derivatives and observing the pattern that occurs.

$$(a)f(x)=x^{\circ} \quad \text { (b) } f(x)=1 / x$$

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Problem 63

Find a second-degree polynomial $P$ such that $P(2)=5$

$P^{\prime}(2)=3,$ and $P^{\prime \prime}(2)=2$

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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 Chanter 9$)$

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Problem 65

Find a cubic function $y=a x^{3}+b x^{2}+c x+d$ whose graph as 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}{x^{2}+1} & {\text { if } x<1} \\ {x+1} & {\text { if } x \geqslant 1}\end{array}\right.

$$

$$

f \text { differentiable at } 1 ? \text { Sketch the graphs of } f \text { and } f^{\prime}

$$

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Problem 68

At what numbers is the following function $g$ differentiable?

$$

g(x)=\left\{\begin{array}{ll}{2 x} & {\text { if } x \leqslant 0} \\ {2 x-x^{2}} & {\text { if } 0<x<2} \\ {2-x} & {\text { if } x \geqslant 2}\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 $h^{\prime}$ 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 $r=0$ with eauation $v=2 x+1$ 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 $c$ such that the line $y=\frac{3}{2} x+6$ is tangent to the curve $y=c \sqrt{x} .$

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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 $m$ and $b$ that make $f$ differentiable everywhere.

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Problem 76

A tangent line is drawn to the hyperbola $x y=c$ 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.

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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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