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Applied Calculus For Business, Economics, and Finance

Warren B. Gordon, Walter O. Wang, April Allen Materowski

Chapter 2

An Introduction to Calculus

Educators


Problem 1

Refer to Figures $8,9,$ and $10 .$ In each case, choose another point on the tangent line to determine the slope of the curve at $P$.

Kaylee M.
Numerade Educator

Problem 2

Refer to Figures $8,9,$ and $10 .$ In each case, choose another point on the tangent line to determine the slope of the curve at $P$.

Kaylee M.
Numerade Educator

Problem 3

Refer to Figures $8,9,$ and $10 .$ In each case, choose another point on the tangent line to determine the slope of the curve at $P$.

Kaylee M.
Numerade Educator

Problem 4

a) sketch the graph of the given function, and then draw the tangent line at the point $P$. (b) Using your sketch, approximate the slope of the curve at $P,$ (c) Use (1) to determine the exact value of the slope at $P$.
$$f(x)=x^{2}+3 \quad P(1,4)$$

Kaylee M.
Numerade Educator

Problem 5

a) sketch the graph of the given function, and then draw the tangent line at the point $P$. (b) Using your sketch, approximate the slope of the curve at $P,$ (c) Use (1) to determine the exact value of the slope at $P$.
$$f(x)=-2 x^{2}+3 x+3$$$\quad$$$P(2,1)$$

Kaylee M.
Numerade Educator

Problem 6

a) sketch the graph of the given function, and then draw the tangent line at the point $P$. (b) Using your sketch, approximate the slope of the curve at $P,$ (c) Use (1) to determine the exact value of the slope at $P$.
$$f(x)=\sqrt{x+1} \quad P(3,2)$$

Kaylee M.
Numerade Educator

Problem 7

a) sketch the graph of the given function, and then draw the tangent line at the point $P$. (b) Using your sketch, approximate the slope of the curve at $P,$ (c) Use (1) to determine the exact value of the slope at $P$.
$$f(x)=x^{3} \quad P(2,8)$$

Kaylee M.
Numerade Educator

Problem 8

a) sketch the graph of the given function, and then draw the tangent line at the point $P$. (b) Using your sketch, approximate the slope of the curve at $P,$ (c) Use (1) to determine the exact value of the slope at $P$.
$$f(x)=-x^{2}+2 x-1$$$\quad$$$P(1,0)$$

Kaylee M.
Numerade Educator

Problem 9

Given the curve whose equation is $f(x)=x^{2}+3 .$ Let $P$ be the point (1,4)
(a) Determine the slope of the secant line joining $P$ to $Q,$ if $Q$ has as its $x$ -coordinate: (i) 1.01 (ii) 1.001 (iii) 1.0001 (iv) 0.99 (v) 0.999 (vi) 0.9999.
(b) What limiting value does the slope of the secant line appear to be approaching as $Q$ approaches $P ?$

Kaylee M.
Numerade Educator

Problem 10

Given the curve whose equation is $f(x)=\sqrt{x+4} .$ Let $P$ be the point(5,3). (a) Determine the slope of the secant line joining $P$ to $Q,$ if $Q$ has as its $x$ -coordinate: (i) 5.01 (ii) 5.001 (iii) 5.0001 (iv) 4.99 (v) 4.999 (vi) 4.9999. (b) What limiting value does the slope of the secant line appear to be approaching as $Q$ approaches $P ?$

Kaylee M.
Numerade Educator

Problem 11

Given the curve whose equation is $f(x)=x^{0.3} .$ Let $P$ be the point (1,1). (a) Determine the slope of the secant line joining $P$ to $Q,$ if $Q$ has as its $x$ -coordinate: (i) 1.001 (ii) 1.00001 (iii) 0.999 (iv) 0.9999. (b) What limiting value does the slope of the secant line appear to be approaching as $Q$ approaches $P ?$

Kaylee M.
Numerade Educator

Problem 12

Determine the derivative at the given point on the curve using equation (2).
$y=x^{2}$ at the point (3,9).

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Problem 13

Determine the derivative at the given point on the curve using equation (2).
$f(x)=3-2 x-x^{2}$ at the point (-1,4).

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Problem 14

Determine the derivative at the given point on the curve using equation (2).
$f(x)$ as defined in Exercise 4.

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Problem 15

Determine the derivative at the given point on the curve using equation (2).
$f(x)$ as defined in Exercise 5.

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Numerade Educator

Problem 16

Determine the derivative at the given point on the curve using equation (2).
$f(x)$ as defined in Exercise 6.

Wendi Z.
Numerade Educator

Problem 17

Determine the derivative at the given point on the curve using equation (2).
$f(x)$ as defined in Exercise 7.

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Numerade Educator

Problem 18

Find $f^{\prime}(x)$.
$$f(x)=2 x^{2}-7 x+9$$.

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Numerade Educator

Problem 19

Find $f^{\prime}(x)$.
$$f(x)=\sqrt{x}$$.

Check back soon!

Problem 20

Find $f^{\prime}(x)$.
$$f(x)=-3 x^{2}+7 x-11$$.

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Problem 21

Find $f^{\prime}(x)$.
(a) $f(x)=53$ (b) Give a geometric explanation for your result.

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Numerade Educator

Problem 22

Find $f^{\prime}(x)$.
$$f(x)=m x+b$$

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Problem 23

Find $f^{\prime}(x)$.
$$f(x)=3 / x$$

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Problem 24

Find $f^{\prime}(x)$.
$$f(x)=-2 / x$$

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Numerade Educator

Problem 25

Short segments of the tangent lines are given at various points along a curve. Use this information to sketch the curve.
See Figure 11.

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Numerade Educator

Problem 26

Short segments of the tangent lines are given at various points along a curve. Use this information to sketch the curve.
See Figure 12.

Kaylee M.
Numerade Educator

Problem 27

Given $f(x)=3 x^{2}-12 x+5 .$ At which point will the curve have slope
(a) $0 ;$ (b) $6 ;$ (c) $-6 ?$

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Numerade Educator

Problem 28

Given $f(x)=x^{3}-12 x .$ At which points will its tangent line (a) be horizontal; (b) have slope $15 ;$ (c) have slope $36 ?$

Majid B.
Numerade Educator

Problem 29

Given $f(x)=x^{3}-12 x .$ At which points will its tangent line (a) be horizontal; (b) have slope 15; (c) have slope 36?

Majid B.
Numerade Educator

Problem 30

Given $f(x)=\sqrt{x}(\text { a) }$ At which point will the tangent line be vertical? (b) What can you say about the derivative at this point?

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Numerade Educator

Problem 31

$$f(x)=\left\{\begin{aligned}4 x-2 & \text { if } x \leq 1 \\x+1 & \text { if } x>1\end{aligned}\right.$$ (a) Sketch the graph of this function. (b) Determine $f^{\prime}(x)$ if $x<1$(c) Determine $f^{\prime}(x)$ if $x>1$ (d) What can you conclude about $f^{\prime}(1) ?$

Kaylee M.
Numerade Educator

Problem 32

$$f(x)=\left\{\begin{array}{l}x^{2} \text { if } x \geq 0 \\x \text { if } x<0\end{array}\right.$$ (a) Sketch the graph of this function. (b) Determine $f^{\prime}(x)$ if $x<0$ (c) Determine $f^{\prime}(x)$ if $x>0$ (d) What can you say about $f^{\prime}(0) ?$

Kaylee M.
Numerade Educator

Problem 33

Given $f(x)=|2 x-5|$ (a) At what point is the function not differentiable? (b) What is the derivative to the left of this point? (c) What is the derivative to the right of this point?

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Numerade Educator

Problem 34

$f(x)=\{\begin{array}{c}9 x+5 \text { if } x \geq 1 \\ x^{2}+7 x+6 \text { if } x<1\end{array} \text { (a) What is } f^{\prime}(x) \text { if } x>1 ?$ (b) What is \right. $f^{\prime}(x)$ if $x<1 ?(\mathrm{c})$ What is the slope of the curve just to the left of $x=1 ?$ (d) What is $f^{\prime}(1) ?$

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Numerade Educator

Problem 35

Find the point on the curve $y=x^{3}$ at which the tangent line at (2,8) crosses the curve. (You may want to use the result of Exercise $17 .$ )

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Problem 36

Consider the two functions: $f(x)=x^{1 / 3}$ and $g(x)=x^{4 / 3}$ near $x=0 .$ Using $h=-0.1,-.0 .01,-0.001$ and $-0.0001,(\text { as } h \text { approaches } 0$ from the left), and $h=0.1, .0 .01,0.001$ and 0.0001 (as $h$ approaches 0 from the right). Find the slope of the secant lines passing through $P(0,0)$ and $Q(h, f h)$ ). Does $m_{\tan }(x)$ exist at (0,0)$?$ Why not? (b) Now repeat the process for $g(x) .$ What is the difference in the behavior at $P(0,0)$ for the two functions?

Wendi Z.
Numerade Educator

Problem 37

Suppose that, in the development of the definition of the derivative, we wrote $(x_{2}, f(x_{2}) \text { for } Q \text { instead of }(x+h, f(x+h)) .$ Show that the definition of. the derivative will then have the following alternate form: $f^{\prime}(x)=\lim _{x_{2} \rightarrow x} \frac{f\left(x_{2}\right)-f(x)}{x_{2}-x}$

Wendi Z.
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Problem 38

Use the alternate form of the derivative given in Exercise $37,$ to compute $f^{\prime}(x)$ for the function defined in: (a) Exercise 21; (b) Exercise 22; (c) Exercise 23; (d) Exercise 24.

Wendi Z.
Numerade Educator

Problem 39

Let $y 1(x)=x^{2}+1,$ determine the equation of the secant line through each of the following $x$ -values and $x=2$ :
(a) $x=2.1,$ call the equation $y 2(x)$ and enter it the $Y=$ screen
(b) $x=2.05,$ call the equation $y 3(x)$ and enter it the $Y=$ screen
(c) $x=2.025,$ call the equation $y 4(x)$ and enter it the $Y=$ screen
(d) choose an appropriate window so the curve and all these secant lines can be seen.
(e) Have the calculator add the tangent line at $x=2$
What is happening to the secant lines as $x$ approaches $2 ?$

Wendi Z.
Numerade Educator