Deck 11: Differential Calculus

Question 1

Decide whether the limit exists. If it exists, find its value.

- $\lim f(x)$
$x \rightarrow 1$

A) Does not exist
B) 2
C) 1
D) 0

Answer 1

Does not exist

Question 2

Decide whether the limit exists. If it exists, find its value.

- $\lim f(x)$
$x \rightarrow 0$

A) -1
B) Does not exist
C) 0
D) 1

Answer 2

0

Question 3

Decide whether the limit exists. If it exists, find its value.

- $\lim f(x)$
$x \rightarrow 0$

A) -1
B) -2
C) 0
D) Does not exist

Answer 3

-2

Question 4

Decide whether the limit exists. If it exists, find its value.

- $\lim f(x)$
$x \rightarrow 0$

A) 0
B) 2
C) -2
D) Does not exist

Question 5

Decide whether the limit exists. If it exists, find its value.

- $\lim f(x)$
$x \rightarrow 0$

A) Does not exist
B) 1
C) 0
D) -1

Question 6

Decide whether the limit exists. If it exists, find its value.

- $\lim f(x)$
$x \rightarrow 0$

A) -1
B) Does not exist
C) 0
D) 1

Question 7

Decide whether the limit exists. If it exists, find its value.

- $\lim f(x)$
$x \rightarrow 0$

A) 0
B) -2
C) Does not exist
D) -1

Question 8

Decide whether the limit exists. If it exists, find its value.

- $\lim f(x)$
$x \rightarrow-\frac{1}{2}$

A) Does not exist
B) -2
C) 0
D) -1

Question 9

Use a calculator to estimate the limit.

- $\lim _{x \rightarrow 5} \frac{x^{2}-11 x+30}{x^{2}-7 x+10}$

A) $\frac{11}{3}$
B) $\frac{-1}{3}$
C) $\frac{1}{3}$
D) Does not exist

Question 10

Use a calculator to estimate the limit.

- $\lim _{x \rightarrow 4} \frac{x^{2}+16}{x+4}$

A) Does not exist
B) 4
C) 0
D) 8

Question 11

Use a calculator to estimate the limit.

- $\lim _{x \rightarrow 3} \sqrt{x^{2}+6 x+9}$

A) $\pm 6$
B) Does not exist
C) 6
D) 36

Question 12

Use a calculator to estimate the limit.

- $\lim _{x \rightarrow 7} \frac{x^{4}-1}{x-1}$

A) 4
B) 0
C) Does not exist
D) 2

Question 13

Use a calculator to estimate the limit.

- $\lim _{x \rightarrow 7} \frac{\sqrt{x-1}}{x-1}$

A) 0
B) -1
C) Does not exist
D) 1

Question 14

Use a calculator to estimate the limit.

- $\lim _{x \rightarrow 0} \frac{x^{3}+12 x^{2}-5 x}{5 x}$

A) Does not exist
B) 5
C) 0
D) -1

Question 15

Use a calculator to estimate the limit.

- $\lim _{x \rightarrow 49} \frac{\sqrt{x}-\sqrt{7}}{x-49}$

A) 0
B) $\frac{\sqrt{7}}{14}$
C) $2 \sqrt{7}$
D) Does not exist

Question 16

Use a calculator to estimate the limit.

- $\lim _{x \rightarrow 1} \frac{\sqrt{x}-4}{x-16}$

A) $\frac{1}{\sqrt{x}+4}$
B) $-\frac{1}{8}$
C) Does not exist
D) $\frac{1}{5}$

Question 17

Use a calculator to estimate the limit.

- $\lim _{x \rightarrow 0} \frac{e^{6 x}-1}{x}$

A) 0
B) Does not exist
C) 6
D) $6 e^{6 x}$

Question 18

Use a calculator to estimate the limit.

- $\lim _{\mathrm{x} \rightarrow 4} \frac{\ln \mathrm{x}-\ln 4}{\mathrm{x}-4}$

A) 4
B) Does not exist
C) 0
D) $\frac{1}{4}$

Question 19

Use the limit properties to find the following limit.

-If $\lim _{x \rightarrow 4} f(x)=19$ and $\lim _{x \rightarrow 4} g(x)=13$ , find $\lim _{x \rightarrow 4}[f(x)-g(x)]$ .

A) -4
B) 6
C) 32
D) 0

Question 20

Use the limit properties to find the following limit.

-If $\lim _{x \rightarrow 3} f(x)=3$ and $\lim _{x \rightarrow 3} g(x)=2$ , find $\lim _{x \rightarrow 3}[f(x) \cdot g(x)]$ .

A) 6
B) Does not exist
C) 18
D) 5

Question 21

Use the limit properties to find the following limit.

-If $\lim _{x \rightarrow 1} f(x)=10$ and $\lim _{x \rightarrow 1} g(x)=3$ , find $\lim _{x \rightarrow 1} \frac{f(x)}{g(x)}$ .

A) -1
B) $\frac{10}{3}$
C) $-\frac{10}{3}$
D) Does not exist

Question 22

Use the limit properties to find the following limit.

-If $\lim _{x \rightarrow 7} f(x)=16$ , find $\lim _{x \rightarrow 7} \sqrt{f(x)}$ .

A) Does not exist
B) 4
C) 7
D) 16

Question 23

Use the limit properties to find the following limit.

-If $\lim _{x \rightarrow 4} f(x)=5$ and $\lim _{x \rightarrow 4} g(x)=10$ , find $\lim _{x \rightarrow 4} \frac{f(x)+g(x)}{5 g(x)}$ .

A) 1
B) $\begin{gathered}1 \\ -10\end{gathered}$
C) $\frac{3}{10}$
D) -4

Question 24

Use the properties of limits to evaluate the limit if it exists.

- $\lim _{x \rightarrow 1} \frac{x^{4}-1}{x-1}$

A) 4
B) 0
C) 2
D) Does not exist

Question 25

Use the properties of limits to evaluate the limit if it exists.

- $\lim _{x \rightarrow 1} \frac{2 x-7}{4 x+5}$

A) $-\frac{1}{2}$
B) $-\frac{7}{5}$
C) $-\frac{5}{9}$
D) Does not exist

Question 26

Use the properties of limits to evaluate the limit if it exists.

- $\lim _{x \rightarrow 1} \frac{3 x^{2}+7 x-2}{3 x^{2}-4 x-2}$

A) 0
B) Does not exist
C) $-\frac{7}{4}$
D) $-\frac{8}{3}$

Question 27

Use the properties of limits to evaluate the limit if it exists.

- $\lim _{x \rightarrow 6} \frac{x+6}{(x-6)^{2}}$

A) Does not exist
B) 6
C) 0
D) -6

Question 28

Use the properties of limits to evaluate the limit if it exists.

- $\lim _{x \rightarrow 49} \frac{\sqrt{x}-\sqrt{7}}{x-49}$

A) Does not exist
B) $2 \sqrt{7}$
C) 0
D) $\frac{\sqrt{7}}{14}$

Question 29

Use the properties of limits to evaluate the limit if it exists.

- $\lim _{\mathrm{x} \rightarrow 7 } \frac{\sqrt{\mathrm{x}}-7}{\mathrm{x}-49}$

A) 0
B) Does not exist
C) $\frac{1}{6}-\frac{\sqrt{7}}{42}$
D) $\frac{\sqrt{7}}{42}-\frac{1}{6}$

Question 30

Use the properties of limits to evaluate the limit if it exists.

- $\lim _{x \rightarrow 5} \frac{x^{2}-2 x-15}{x+3}$

A) 5
B) -8
C) Does not exist
D) 0

Question 31

Solve the problem.

-A company training program determines that, on average, a new employee can do $P(s)$ pieces of work per day after $s$ days of on-the-job training, where $P(s)=\frac{96+51 s}{s+7}$ . Find $\lim _{s \rightarrow 2} P(s)$ .

A) 28
B) Does not exist
C) 22
D) 99

Question 32

Solve the problem.

-The cost of manufacturing a particular videotape is $c(x)=12,000+12 x$ , where $x$ is the number of tapes produced. The average cost per tape, denoted by $\bar{c}(x)$ , is found by dividing $c(x)$ by $x$ . Find $\lim _{x \rightarrow 1000} \bar{c}(x)$ .

A) 34
B) 24
C) 14
D) Does not exist

Question 33

Solve the problem.

-A certain object, after being heated, cools at such a rate that its temperature $T$ (in ${ }^{\circ} \mathrm{C}$ ) decreases $6 \%$ each minute. If the object is originally heated to $100^{\circ} \mathrm{C}$ , find $\lim _{t \rightarrow 9}$ where $t$ is the time in minutes.

A) $100.0^{\circ} \mathrm{C}$
B) $0.0^{\circ} \mathrm{C}$
C) $57.3^{\circ} \mathrm{C}$
D) $0{ }^{\circ} \mathrm{C}$

Question 34

Solve the problem.

- The graph shows the consumption of an essential commodity, $\mathrm{F}(\mathrm{t})$ , in millions of units. We assume that the price is rising rapidly. Here $t$ is time in months after the price began rising. Use the price to find $\lim _{t \rightarrow 11} F(t)$ .

A) 9
B) 8
C) Does not exist
D) 7

Question 35

Use the given graph to determine the limit, if it exists.

- $\lim f(x)$ and $\lim f(x)$ .
$x-0^{-} \quad x-0^{+}$

A) $1 ;-1$
B) $-1 ; 1$
C) $1 ; 1$
D) $-1 ;-1$

Question 36

Use the given graph to determine the limit, if it exists.

- $\lim f(x)$ and $\lim f(x)$ .
$x-2^{-} \quad x-2^{+}$

A) $5 ;-2$
B) $1 ; 1$
C) $-2 ; 5$
D) does not exist

Question 37

Use the given graph to determine the limit, if it exists.

- $\lim f(x)$
$x \rightarrow 2^{-}$

A) -1
B) 4
C) 1.3
D) 2.3

Question 38

Use the given graph to determine the limit, if it exists.

- $\lim f(x)$
$x \rightarrow 1^{-}$

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

Question 39

Use the given graph to determine the limit, if it exists.

- $\lim f(x)$ $x \rightarrow 1^{+}$

A) 3
B) $3 \frac{1}{2}$
C) 4
D) does not exist

Question 40

Use the properties of limits to find the indicated limit.

- $\lim (4-5 x)$
$x \rightarrow \theta^{-}$

A) 0
B) 4
C) -1
D) does not exist

Question 41

Use the properties of limits to find the indicated limit.

- $\lim \left(x^{2}+5 x-5\right)$
$x \rightarrow 2^{+}$

A) 19
B) -1
C) 9
D) -11

Question 42

Use the properties of limits to find the indicated limit.

- $\lim _{x \rightarrow 0.5^{-}} \sqrt{\frac{x+7}{x+5}}$

A) $\sqrt{\frac{15}{11}}$
B) $\sqrt{\frac{13}{9}}$
C) $\frac{13}{9}$
D) does not exist

Question 43

Use the properties of limits to find the indicated limit.

- $\lim _{x \rightarrow 4^{+}} \sqrt{\frac{4 x^{2}}{8+x}}$

A) $\sqrt{\frac{4}{3}}$
B) $\sqrt{\frac{16}{3}}$
C) $\frac{16}{3}$
D) does not exist

Question 44

Use the properties of limits to find the indicated limit.

- $\lim _{x \rightarrow 7^{+}} \sqrt{x^{2}+12 x+36}$

A) $\pm 13$
B) 13
C) 169
D) does not exist

Question 45

Use the properties of limits to find the indicated limit.

- $\lim f(x)$ ,
$x \rightarrow 9^{-}$
Where $f(x)= \begin{cases}9 & (x<-3) \\ x^{2} & (-3<x<3) \\ 9 & (x>3)\end{cases}$

A) 9
B) -9
C) 3
D) does not exist

Question 46

Use a calculator to estimate the limit.

- $\lim _{x \rightarrow 3^{+}} \frac{5}{x-3}$

A) 0
B) $\infty$
C) $-\infty$
D) does not exist

Question 47

Use a calculator to estimate the limit.

- $\lim _{x \rightarrow 0^{-}} \frac{1}{x^{2}+2 x}$

A) $\infty$
B) $-\infty$
C) 0
D) does not exist

Question 48

Use a calculator to estimate the limit.

- $\lim _{x \rightarrow 9^{+}} \frac{x-6}{x+9}$

A) 0
B) $-\infty$
C) $\infty$
D) does not exist

Question 49

Solve the problem.

-Use the graph of the function $f$ to find $\lim _{x \rightarrow ?} f(x)$ .

A) $\infty$
B) $-\infty$
C) 1
D) does not exist

Question 50

Solve the problem.

-Use the graph of the function $f$ to find $\lim _{x \rightarrow ?} f(x)$ .

A) $\infty$
B) $-\infty$
C) 0
D) does not exist

Question 51

Solve the problem.

-Use the graph of the function $f$ to find $\lim _{x \rightarrow \infty} f(x)$ .

A) $\infty$
B) 0
C) $-\infty$
D) does not exist

Question 52

Solve the problem.

-Use the graph of the function $f$ to find $\lim _{x \rightarrow ?} f(x)$ .

A) $\infty$
B) 0
C) -2
D) does not exist

Question 53

Solve the problem.

-Use the graph of the function $f$ to find $\lim _{x \rightarrow ?} f(x)$ .

A) $\infty$
B) $-\infty$
C) does not exist
D) -2

Question 54

Solve the problem.

-Use the graph of the function $f$ to find $\lim _{x \rightarrow ?} f(x)$ .

A) 2
B) $-\infty$
C) $\infty$
D) does not exist

Question 55

Use a calculator to estimate the limit.

- $\lim _{x \rightarrow \infty} \frac{6+x}{x^{4}}$

A) $-\infty$
B) $\infty$
C) 0
D) does not exist

Question 56

Use a calculator to estimate the limit.

- $\lim _{x \rightarrow \infty} x^{5} e^{x}$

A) 5
B) 1
C) $\infty$
D) 0

Question 57

Use a calculator to estimate the limit.

- $\lim _{x \rightarrow} x^{5} e^{-x}$

A) 1
B) 0
C) e
D) $\infty$

Question 58

Use a calculator to estimate the limit.

- $\lim _{x \rightarrow \infty} \frac{\ln |x|}{x^{4}}$

A) $-\infty$
B) 0
C) 1
D) $\infty$

Question 59

Use the properties of limits to find the indicated limit.

- $\lim _{x \rightarrow \infty} \frac{5 x^{2}+6}{x^{2}-5 x}$

A) $-\frac{1}{5}$
B) 5
C) 0
D) does not exist

Question 60

Use the properties of limits to find the indicated limit.

- $\lim _{x \rightarrow \infty} \frac{x^{2}+5 x+17}{x^{3}+2 x^{2}+12}$

A) 1
B) $\infty$
C) 0
D) $\frac{17}{12}$

Question 61

Use the properties of limits to find the indicated limit.

- $\lim _{x \rightarrow \infty} \frac{-10 x^{2}-7 x+15}{-4 x^{2}+5 x+7}$

A) $\infty$
B) 1
C) $\frac{15}{7}$
D) $\frac{5}{2}$

Question 62

Use the properties of limits to find the indicated limit.

- $\lim _{x \rightarrow \infty} \frac{2 x^{3}-5 x^{2}+3 x}{-x^{3}-2 x+6}$

A) 2
B) $\frac{3}{2}$
C) -2
D) $\infty$

Question 63

Use the properties of absolute value to find the indicated limit.

- $\lim _{x \rightarrow} \frac{x-7}{|x-7|}$

A) 0
B) 1
C) -1
D) does not exist

Question 64

Use the properties of absolute value to find the indicated limit.

- $\lim _{x \rightarrow \infty} \frac{x+4}{|x+4|}$

A) 0
B) -1
C) 1
D) does not exist

Question 65

Use the properties of absolute value to find the indicated limit.

- $\lim _{x \rightarrow \infty} \frac{x}{|x|+4}$

A) 0
B) -1
C) 1
D) does not exist

Question 66

Solve the problem.

-A company training program determines that, on average, a new employee can do $\mathrm{P}(\mathrm{s})$ pieces of work per day after $s$ days of on-the-job training, where $P(s)=\frac{85+49 s}{s+5}$ . Find $\lim _{s \rightarrow ?} P(s)$ .

A) 22 pieces of work per day
B) 49 pieces of work per day
C) 85 pieces of work per day
D) does not exist

Question 67

Solve the problem.

-A company training program determines that, on average, a new employee can do $\mathrm{P}(\mathrm{s})$ pieces of work per day after $s$ days of on-the-job training, where $P(s)=\frac{80+48 s}{s+5}$ . Find $\lim _{s \rightarrow 3} P(s)$ .

A) 75 pieces of work per day
B) 28 pieces of work per day
C) 45 pieces of work per day
D) does not exist

Question 68

Solve the problem.

-The cost of manufacturing a particular videotape (in dollars) is $c(x)=9000+8 x$ , where $x$ is the number of tapes produced. The average cost per tape (in dollars), denoted by $\bar{c}(x)$ , is found by dividing $\mathrm{c}(\mathrm{x})$ by $\mathrm{x}$ . Find $\lim _{\mathrm{x} \rightarrow 1000} \overline{\mathrm{c}}(\mathrm{x})$ .

A) $\$ 24$
B) $\$ 17$
C) $\$ 10$
D) does not exist

Question 69

Solve the problem.

-The cost of manufacturing a particular videotape (in dollars) is $c(x)=10,000+10 x$ , where $x$ is the number of tapes produced. The average cost per tape (in dollars), denoted by $\bar{c}(x)$ , is found by dividing $\mathrm{c}(\mathrm{x})$ by $\mathrm{x}$ . Find $\lim _{\mathrm{x} \rightarrow \mathrm{?}} \overline{\mathrm{c}}(\mathrm{x})$ .

A) $\$ 10$
B) $\$ 5$
C) $\$ 8$
D) does not exist

Question 70

Find the average rate of change for the function over the given interval.

- $y=x^{2}+6 x$ between $x=7$ and $x=9$

A) $\frac{135}{2}$
B) 15
C) $\frac{44}{9}$
D) 22

Question 71

Find the average rate of change for the function over the given interval.

- $y=9 x^{3}+6 x^{2}-1$ between $x=-6$ and $x=5$

A) 273
B) $\frac{3003}{5}$
C) $\frac{1274}{11}$
D) $\frac{1274}{5}$

Question 72

Find the average rate of change for the function over the given interval.

- $y=\sqrt{2 x}$ between $x=2$ and $x=8$

A) $-\frac{3}{10}$
B) 7
C) 2
D) $\frac{1}{3}$

Question 73

Find the average rate of change for the function over the given interval.

- $y=\frac{3}{x-2}$ between $x=4$ and $x=7$

A) 7
B) $\frac{1}{3}$
C) 2
D) $-\frac{3}{10}$

Question 74

Find the average rate of change for the function over the given interval.

- $y=4 x^{2}$ between $x=0$ to $x=\frac{7}{4}$

A) 7
B) $-\frac{3}{10}$
C) 2
D) $\frac{1}{3}$

Question 75

Find the average rate of change for the function over the given interval.

- $y=-3 x^{2}-x$ between $x=5$ and $x=6$

A) $\frac{1}{2}$
B) -2
C) -34
D) $-\frac{1}{6}$

Question 76

Find the average rate of change for the function over the given interval.

- $y=x^{3}+x^{2}-8 x-7$ between $x=0$ and $x=2$

A) -2
B) $\frac{1}{2}$
C) -28
D) $-\frac{1}{6}$

Question 77

Find the average rate of change for the function over the given interval.

- $y=\sqrt{2 x-1}$ between $x=1$ and $x=5$

A) -2
B) $\frac{1}{2}$
C) -28
D) $-\frac{1}{6}$

Question 78

Find the average rate of change for the function over the given interval.

- $y=\frac{3}{x+2}$ between $x=1$ and $x=4$

A) -2
B) $\frac{1}{2}$
C) -28
D) $-\frac{1}{6}$

Question 79

Find the average rate of change for the function over the given interval.

- $y=5 x+7$ between $x=-1$ and $x=0$

A) $-\frac{1}{6}$
B) -28
C) $\frac{1}{2}$
D) 5

Question 80

The graph shows the total sales in thousand of dollars from the distribution of x thousand catalogs. Find the average rate of change of sales with respect to the number of catalogs distributed for the change in x.


-10 to 20

A) $\frac{1}{2}$
B) 1
C) $\frac{3}{2}$
D) 2