Deck 2: Derivatives

Question 1

If $g(x)=\sqrt{4-3 x}$ find the domain of $g^{\prime}(x)$

A) $(0, \infty)$
B) $\left(-\infty, \frac{4}{3}\right)$
C) $(-\infty, 0) \cup(0, \infty)$
D) $\left[-\frac{4}{3}, \frac{4}{3}\right]$
E) $\left[-\infty, \frac{4}{3}\right)$

Answer 1

$\left(-\infty, \frac{4}{3}\right)$

Question 2

Choose an equation from the following that expresses the fact that a function f is continuous at the number 6 .

A) $\lim _{x \rightarrow 0} f(x)=f(6)$
B) $\lim _{x \rightarrow 0} f(x)=6$
C) $\lim _{x \rightarrow 6} f(x)=f(6)$
D) $\lim _{x \rightarrow 6} f(x)=\infty$
E) $\lim _{x \rightarrow 6} f(x)=-\infty$

Answer 2

$\lim _{x \rightarrow 6} f(x)=f(6)$

Question 3

The cost (in dollars) of producing x units of a certain commodity is Find the average rate of change with respect to x when the production level is changed from

Answer 3

Question 4

At what point is the function $f(x)=|4-x|$ not differentiable.

A) 4
B) 0
C) 1
D) -1
E) -4

Question 5

Determine where f is discontinuous. $f(x)=\left\{\begin{array}{ccc}\sqrt{-x} & \text { if } & x<0 \\11-x & \text { if } & 0 \leq x<11 \\(11-x)^{2} & \text { if } & x>11\end{array}\right.$

A) $0 \text { and } 11$
B) 0 only
C) $-11$ only
D) 11 only
E) $0 \text { and }-11$

Question 6

Find an equation of the tangent line to the curve $y=x^{3}-2 x$ at the point $(2,6)$ .

A) $y=2+2 x$
B) $y=x-6$
C) $y=6+2 x$
D) $y=2-2 x$
E) None of these

Question 7

Find the slope of the tangent line to the curve $y=5 x^{2}$ at the point $(-4,22)$ .

A) $-40$
B) 40
C) $-4$
D) 4
E) 25

Question 8

Find an equation of the tangent line to the parabola $y=5 x^{3}$ at the point $(-5,-145)$ .

A) $y=375 x-1730$
B) $y=395 x-1730$ .
C) $y=375 x+1730$
D) $y=395 x+1730$ .
E) $y=395 x-1730$

Question 9

Use continuity to evaluate the limit. $\lim _{x \rightarrow 9_{x}} \sin (x+10 \sin x)$

A) $\infty$
B) $9 \pi$
C) $-1$
D) 0
E) 1

Question 10

Let $F(x)=\frac{x^{3}-1}{|x-1|}$ Find the following limits. $\lim _{x \rightarrow 1^{+}} F(x), \lim _{x \rightarrow 1^{-}} F(x)$

A) 3 and 1
B) 3 and $-3$
C) both 1
D) both 3
E) 3 and $-1$

Question 11

For the function f whose graph is shown, state the following.

Question 12

The symbol $\lfloor\rfloor$ can be used to denote the greatest integer function, which is defined by $\lfloor x\rfloor=$ the greatest integer n such that $n \leq x \text {. }$ Use the graph of the function to find the indicated limit, if it exists. $\lim _{x \rightarrow 0.7}\lfloor x\rfloor$

A) 1
B) 0.7
C) 0
D) -0.7

Question 13

Use the definition of the derivative to find $f^{\prime}(3) \text {, where } f(x)=x^{3}-2 x \text {. }$

A) $-25$
B) $25$
C) 27
D) 18
E) does not exist

Question 14

If $g(x)=\sqrt{3-5 x}$ find the defination of derivative to find $g^{\prime}(x)$

A) $g^{\prime}(x)=-\frac{5}{2}(3-5 x)^{-1 / 2}$
B) $g^{\prime}(x)=-\frac{5}{2}(3-5 x)^{1 / 2}$
C) $g^{\prime}(x)=-(3-5 x)^{-1 / 2}$
D) $g^{\prime}(x)=-\frac{1}{2}(3-5 x)^{-1 / 2}$
E) None of these

Question 15

The cost (in dollars) of producing x units of a certain commodity is $C(x)=4,571+17 x+0.02 x^{2}$ Find the instantaneous rate of change with respect to x when $x=103 \text {. }$ (This is called the marginal cost.)

A) $4.12$
B) $21.12$
C) $23.18$
D) $19.06$ .
E) $19.06$

Question 16

If a cylindrical tank holds gallons of water, which can be drained from the bottom of the tank in an hour, then Torricelli's Law gives the volume of water remaining in the tank after t minutes as Find the rate at which the water is flowing out of the tank (the instantaneous rate of change of V with respect to t) as a function of t.

Question 17

Find the vertical asymptotes of the function. $y=\frac{4 x^{2}+1}{3 x-4 x^{2}}$

A) $x=-\frac{4}{3}$
B) $x=\frac{4}{3}$
C) $x=3$
D) $x=0, \frac{4}{3}$
E) None of these

Question 18

If the tangent line to $y=f(x) \text { at }(8,4)$ passes through the point $(4,-32) \text {, find } f^{\prime}(8)$

A) $f^{\prime}(8)=29$
B) $f^{\prime}(8)=19$
C) $f^{\prime}(8)=9$
D) $f^{\prime}(8)=34$
E) $f^{\prime}(8)=-9$

Question 19

If an equation of the tangent line to the curve $y=f(x)$ at the point where $a=3 \text { is } y=4 x-9 \text {, }$ $\text { find } f(3) \text { and } f^{\prime}(3) \text {. }$

A) $\begin{array}{l}f(2)=3 \\f^{\prime}(2)=3\end{array}$
B) $\begin{array}{l}f(2)=-3 \\f^{\prime}(2)=3\end{array}$
C) $\begin{array}{l}f(2)=3 \\f^{\prime}(2)=4\end{array}$
D) $\begin{array}{l}f(2)=9 \\f^{\prime}(2)=4\end{array}$
E) None of these

Question 20

Find the derivative of the function.

Question 21

A machinist is required to manufacture a circular metal disk with area 1000 $\mathrm{cm}^{2}$ . If the machinist is allowed an error tolerance of $\pm 15 \mathrm{~cm}^{2}$ in the area of the disk, how close to the ideal radius must the machinist control the radius? Round your answer to the nearest hundred thousandth.

A) $\delta \leq 0.13131 \mathrm{~cm}$
B) $\delta \leq 0.13281 \mathrm{~cm}$
C) $\delta \leq 0.13231 \mathrm{~cm}$
D) $\delta \leq 0.13331 \mathrm{~cm}$
E) $\delta \leq 0.13431 \mathrm{~cm}$

Question 22

Write an equation that expresses the fact that a function f is continuous at the number .

Question 23

For what value of the constant c is the function f continuous on $(-\infty, \infty) ?$ $f(x)=\left\{\begin{array}{lll}c x+7 & \text { for } & x \leq 2 \\c x^{2}-5 & \text { for } & x>2\end{array}\right.$

A) $c=-6$
B) $c=1$
C) $c=2$
D) $c=6$
E) $c=-2$

Question 24

How would you define $f(3)$ in order to make f continuous at 3 ? $f(x)=\frac{x^{2}-3 x-7}{x-3}$

A) $f(3)=1$
B) $f(3)=3$
C) $f(3)=-3$
D) $f(3)=0$
E) None of these

Question 25

Find a function g that agrees with f for and is continuous on .

Question 26

Use the given graph of to find a number such that whenever .
Round your answer to two decimal places.

Question 27

If f and g are continuous functions with $f(9)=2 \text { and } \lim _{x \rightarrow 9}[2 f(x)-g(x)]=9, \text { find } g(9) \text {. }$

A) $g(9)=8$
B) $g(9)=4$
C) $g(9)=13$
D) $g(9)=-5$
E) $g(9)=11$

Question 28

How close to do we have to take x so that

Question 29

Use a graph to find a number such that whenever .
Round your answer to the nearest thousandth.

Question 30

How would you define in order to make f continuous at ?

Question 31

If f and g are continuous functions with

Question 32

Use a graph to find a number such that whenever .

Question 33

Find the point at which the given function is discontinuous.

Question 34

For determine whether f is continuous from the right, from the left, or neither.

Question 35

Find the limit. $\lim _{x \rightarrow-\infty} \frac{\sqrt{x^{2}-49}}{2 x-6}$

A) 7
B) $-7$
C) $\frac{1}{2}$
D) 6
E) does not exist

Question 36

Evaluate the limit. $\lim _{x \rightarrow 1}(x+4)^{3}\left(x^{2}-9\right)$

A) $-1010$
B) $135$
C) $-1000$
D) $-990$
E) $-189$

Question 37

Determine where f is discontinuous.

Question 38

Use the definition of the limit to find values of that corresponds to . Round your answer to the nearest thousandth.

Question 39

Which of the given functions is discontinuous?

A) $f(x)=\left\{\begin{array}{cc}\frac{1}{x-2}, & x \geq 5 \\\frac{1}{3}, & x<5\end{array}\right.$
B) $f(x)=\left\{\begin{array}{cc}\frac{1}{x-5}, & x \neq 5 \\3, & x=5\end{array}\right.$

Question 40

If $1 \leq f(x) \leq x^{2}+6 x+6$ for all x, find $\lim _{x \rightarrow-1} f(x)$ .

A) $-\frac{1}{8}$
B) 1
C) $-\frac{1}{16}$
D) 8
E) does not exist

Question 41

If $\lim _{x \rightarrow 3^{-}} f(x)=4.1$ , then if $\lim _{x \rightarrow 3} f(x)$ exists, find the value where it exists.

A) $4.1$
B) 1
C) $4 .4$
D) 2
E) 3

Question 42

If $6 x-1 \leq f(x) \leq x^{2}-1,$ find $\lim _{x \rightarrow 6} f(x)$

A) 1
B) 35
C) 0
D) $-6$
E) $-35$

Question 43

Find the limit $\lim _{h \rightarrow-1}\left(h^{4}-h^{3}-4 h+6\right)$ .

A) -1
B) -12
C) 6
D) 12

Question 44

Find the limit $\lim _{x \rightarrow 2} \frac{x-1}{x^{2}-4 x+7}$ .

A) $\frac{1}{7}$
B) $\frac{1}{3}$
C) $-1$
D) 1

Question 45

Find the limit.

Question 46

Find the limit .

Question 47

Find the limit $\lim _{x \rightarrow 3} \frac{x^{2}-x-6}{x-3}$ , if it exists.

A) 3
B) 5
C) 2
D) Does not exist

Question 48

Evaluate the limit, if it exists. $\lim _{h \rightarrow 0} \frac{(x-h)^{7}-x^{7}}{h}$

A) $7 x^{6}$
B) $-7 x^{6}$
C) $-7$
D) 1
E) does not exist

Question 49

Evaluate the limit.

Question 50

If for all x find the limit.

Question 51

Evaluate the limit.

Question 52

Find the limit by evaluating the derivative of a suitable function at an appropriate value of x. (Hint: Use the definition of the derivative.) $\lim _{h \rightarrow 0} \frac{2(5+h)^{2}+5(5+h)-75}{h}$

A) 25
B) 75
C) 0
D) $\infty$

Question 53

Is there a number a such that $\lim _{x \rightarrow-3} \frac{3 x^{2}+a x+a+3}{x^{2}+x-16}$ exists? If so, find the value of a and the value of the limit.

A) $a=-15$ , limit equals $-1.5$
B) $a=15$ , limit equals $1.5$
C) $a=15$ , limit equals $-1.5$
D) $a=-15$ , limit equals $1.5$
E) $a=15$ , limit equals $16$

Question 54

Find the limit.

Question 55

Find the limit. $\lim _{t \rightarrow \infty} \frac{t^{2}+3}{t^{3}+t^{2}-7}$

A) $\infty$
B) $-3$
C) 0
D) 3
E) 7

Question 56

Find the limit. $\lim _{x \rightarrow 3} \frac{x^{2}+3 x-18}{x-3}$

A) 18
B) 0
C) 9
D) 5
E) 11

Question 57

Evaluate $\lim _{h \rightarrow 0} \frac{\cot \left(\frac{\pi}{4}+h\right)-1}{h}$ .

A) 0
B) -2
C) 2
D) Undefined

Question 58

Evaluate the limit.

Question 59

Evaluate the limit.

Question 60

Find the limit $\lim _{x \rightarrow 0} \frac{\sqrt{x+14}-\sqrt{14}}{x}$ , if it exists.

A) Does not exist
B) $\frac{\sqrt{14}}{28}$
C) $\frac{\sqrt{14}}{2}$
D) $\frac{\sqrt{14}}{14}$

Question 61

Evaluate the function at the given numbers (correct to six decimal places). Use the results to guess the value of the limit

Question 62

Complete the table by computing at the given values of x, accurate to five decimal places. Use the results to guess at the indicated limit, if it exists, to three decimal places. x
4.9
4.99
4.999
5.001
5.01
5.1

Question 63

Use the graph of the function to state the value of $\lim _{x \rightarrow 0} f(x)$ if it exists. $f(x)=\frac{1}{1+5^{2 / x}}$

A) $\frac{1}{3}$
B) 1
C) $\frac{1}{2}$
D) ∞
E) does not exist

Question 64

Write the expression as a derivative of a function of x.

Question 65

By graphing the function and zooming in toward the point where the graph crosses the y-axis, estimate the value of

Question 66

Use the graph of $f(x)=\frac{x^{2}-x-2}{x-2}$ to guess at the limit $\lim _{x \rightarrow 2} \frac{x^{2}-x-2}{x-2}$ , if it exists.

A) 3
B) Does not exist
C) 2
D) 1

Question 67

Use the graph of the function to state the value of $\lim _{x \rightarrow 0} f(x)$ , if it exists. $f(x)=\frac{x^{2}+x}{4 \sqrt{x^{3}+x^{2}}}$

A) -∞
B) $\frac{1}{4}$
C) $-\frac{1}{4}$
D) ∞
E) does not exist

Question 68

Use the graph of the function to find each limit. , ,

Question 69

Complete the table by computing at the given values of x, accurate to five decimal places. Use the results to guess at the indicated limit, if it exists, to three decimal places.

Question 70

If a ball is thrown into the air with a velocity of $58 ~~\mathrm{ft} / \mathrm{s}$ , its height (in feet) after t seconds is given by $H=58 t-18 t^{2}$ . Find the velocity when $t=9 .$

A) $-20$
B) $-23$
C) $-18$
D) $-22$
E) $-25$

Question 71

Consider the following function. Determine the values of a for which exists.

Question 72

Find the vertical asymptotes of the function.

Question 73

Estimate the value of the following limit by graphing the function $f(x)=\frac{(5 \sin x)}{(\sin \pi x)}$ . $\lim _{x \rightarrow 0} \frac{5 \sin x}{\sin \pi x}$ Round your answer correct to two decimal places.

A) $1.79$
B) $1.49$
C) $1.69$
D) 3.18
E) $1.59$

Question 74

If a rock is thrown upward on the planet Mars with a velocity of $15 \mathrm{~m} / \mathrm{s}$ , its height in meters t seconds later is given by $y=15 t-1.86 t^{2}$ . Find the average velocity over the time interval $[1,4]$ .

A) $2.7$
B) $6 .7$
C) $7.7$
D) $5.7$
E) $4.7$

Question 75

Sketch the graph of the function f and evaluate $\lim _{x \rightarrow 3} f(x)$ $f(x)=\left\{\begin{array}{ll}x-2, & \text { if } x \leq 3 \\-2 x+7, & \text { if } x>3\end{array}\right.$

A)
1
B)
3
C)
3
D)
1

Question 76

Use the graph of the function to find each limit. , ,

Question 77

Sketch the graph of the function f and evaluate $\lim _{x \rightarrow-3^{+}} f(x)$ $f(x)=\left\{\begin{array}{ll}x+5, & \text { if } x \leq-3 \\-2 x-1, & \text { if } x>-3\end{array}\right.$

A)
Does not exist
B)
2
C)
Does not exist
D)
5

Question 78

How close to 2 do we have to take x so that is within a distance of 0.01 from 13?

Question 79

Complete the table by computing at the given values of x, accurate to five decimal places. Use the results to guess at the indicated limit, if it exists, to three decimal places. x

Question 80

Find the limit , if it exists.