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# Applied Calculus For Business, Economics, and Finance

## 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.

### 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.

### 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.

### 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.

### 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 $P$.

Kaylee M.

### 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.

### 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.

### 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.

### Problem 12

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

Majid B.

### 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).

Majid B.

### Problem 14

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

Kaylee M.

### Problem 15

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

Kaylee M.

### Problem 16

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

Wendi Z.

### Problem 17

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

Kaylee M.

### Problem 18

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

Majid B.

### Problem 19

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

Check back soon!

### Problem 20

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

Majid B.

### Problem 21

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

Majid B.

### Problem 22

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

Check back soon!

### Problem 23

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

Majid B.

### Problem 24

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

Majid B.

### 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.

Kaylee M.

### 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.

### Problem 27

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

Majid B.

### 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.

### 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.

### 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?

Majid B.

### 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.

### 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.

### 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?

Kaylee M.

### 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 $f^{\prime}(1) ?$

Kaylee M.

### 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 $17 .$ )

Wendi Z.

### 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.

### 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.

### 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.
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
(e) Have the calculator add the tangent line at $x=2$
What is happening to the secant lines as $x$ approaches $2 ?$