Question 1

Give an appropriate answer.
-$$f ( x ) = \sqrt { x } \text { for } x _ { 0 } = 13$$

Accepted Answer

A)  $\frac { \sqrt { 13 } } { 26 }$ 
B)  $\frac { 13 } { 2 }$ 
C)  $\frac { \sqrt { 13 } } { 13 }$ 
D)  Does not exist 
A)  $\frac { \sqrt { 13 } } { 26 }$ 
B)  $\frac { 13 } { 2 }$ 
C)  $\frac { \sqrt { 13 } } { 13 }$ 
D)  Does not exist

Question 2

Find the limit $L$ for the given function $f$, the point $x _ { 0 }$, and the positive number $\varepsilon$. Then find a number $\delta > 0$ such that, for all $x _ { t } 0 < \left| x - x _ { 0 } \right| < \delta \Rightarrow | f ( x ) - L | < \varepsilon$.
-

Accepted Answer

A)  0.05 
B)  0.1 
C)  0.5 
D)  4 
A)  0.05 
B)  0.1 
C)  0.5 
D)  4

Question 3

Find all points where the function is discontinuous.
-

Accepted Answer

A)  x = 1, x = 4, x = 5 
B)  None 
C)  x = 1, x = 5 
D)  x = 4 
A)  x = 1, x = 4, x = 5 
B)  None 
C)  x = 1, x = 5 
D)  x = 4

Question 4

Find the limit.
-$\lim _ { x \rightarrow 0 } \left( x ^ { 2 } - 5 \right)$

Accepted Answer

A)  does not exist 
B)  0 
C)  -5 
D)  5 
A)  does not exist 
B)  0 
C)  -5 
D)  5

Question 5

Use a CAS to plot the function near the point x0 being approached. From your plot guess the value of the limit.
-$$\lim _ { x \rightarrow 3 } \frac { x ^ { 2 } - 9 } { \sqrt { x ^ { 2 } + 7 } - 4 }$$

Accepted Answer

A)  3 
B)  8 
C)  $\frac { 1 } { 4 }$ 
D)  4 
A)  3 
B)  8 
C)  $\frac { 1 } { 4 }$ 
D)  4

Question 6

Use the table to estimate the rate of change of y at the specified value of x.
-x = 1. 
$\begin{array} { c | c } 
\mathrm { x } & \mathrm { y } \
\hline 0.900 & - 0.05263 \
0.990 & - 0.00503 \
0.999 & - 0.0005 \
1.000 & 0.0000 \
1.001 & 0.0005 \
1.010 & 0.00498 \
1.100 & 0.04762
\end{array}$

Accepted Answer

A)  0 
B)  1 
C)  0.5 
D)  -0.5 
A)  0 
B)  1 
C)  0.5 
D)  -0.5

Question 7

Find the limit.
-$\lim _ { x \rightarrow 0 } ( \sqrt { x } - 2 )$

Accepted Answer

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

Question 8

Find the limit, if it exists.
-$$\lim _ { x \rightarrow 7 } \frac { x ^ { 2 } + 2 x - 63 } { x - 7 }$$

Accepted Answer

A)  0 
B)  Does not exist 
C)  16 
D)  2 
A)  0 
B)  Does not exist 
C)  16 
D)  2

Question 9

Find the limit, if it exists.
-$\lim _ { x \rightarrow 0 } \frac { x ^ { 3 } + 12 x ^ { 2 } - 5 x } { 5 x }$

Accepted Answer

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

Question 10

Use the graph to evaluate the limit.
-$\lim _ { x \rightarrow 0 } f ( x )$

Accepted Answer

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

Question 11

$\text { Find the limit using } \lim _ { x = 0 } \frac { \sin x } { x } = 1 \text {. }$
-$$\lim _ { x \rightarrow 5 ^ { - } } \frac { \sqrt { 3 x } ( x - 5 ) } { | x - 5 | }$$

Accepted Answer

A)  Does not exist 
B)  0 
C)  $- \sqrt { 15 }$ 
D)  $\sqrt { 15 }$ 
A)  Does not exist 
B)  0 
C)  $- \sqrt { 15 }$ 
D)  $\sqrt { 15 }$

Question 12

Give an appropriate answer.
-Suppose $\lim _ { x \rightarrow \theta } f ( x ) = 1$ and $\lim _ { x \rightarrow \theta } g ( x ) = - 3$. Name the limit rules that are used to accomplish steps (a), (b), and (c) of the following calculation.

Accepted Answer

A)  
(a) Difference Rule
(b) Power Rule
(c) Sum Rule 
B)  
(a) Quotient Rule
(b) Difference Rule, Sum Rule
(c) Constant Multiple Rule and Power Rule 
C)  
(a) Quotient Rule
(b) Difference Rule
(c) Constant Multiple Rule 
D)  
(a) Quotient Rule
(b) Difference Rule, Power Rule
(c) Constant Multiple Rule and Sum Rule 
A)  
(a) Difference Rule
(b) Power Rule
(c) Sum Rule 
B)  
(a) Quotient Rule
(b) Difference Rule, Sum Rule
(c) Constant Multiple Rule and Power Rule 
C)  
(a) Quotient Rule
(b) Difference Rule
(c) Constant Multiple Rule 
D)  
(a) Quotient Rule
(b) Difference Rule, Power Rule
(c) Constant Multiple Rule and Sum Rule

Question 13

Give an appropriate answer.
-$$f ( x ) = \frac { 5 } { x } \text { for } x _ { 0 } = 6$$

Accepted Answer

A)  $- \frac { 5 } { 36 }$ 
B)  Does not exist 
C)  $\frac { 5 } { 6 }$ 
D)  $- 30$ 
A)  $- \frac { 5 } { 36 }$ 
B)  Does not exist 
C)  $\frac { 5 } { 6 }$ 
D)  $- 30$

Question 14

$\text { Find the limit using } \lim _ { x = 0 } \frac { \sin x } { x } = 1 \text {. }$
-$$\lim _ { x \rightarrow 2 ^ { + } } \sqrt { \frac { 7 x ^ { 2 } } { 5 + x } }$$

Accepted Answer

A)  $\sqrt { \frac { 28 } { 3 } }$ 
B)  $\frac { 28 } { 3 }$ 
C)  Does not exist 
D)  $\sqrt { - \frac { 14 } { 3 } }$ 
A)  $\sqrt { \frac { 28 } { 3 } }$ 
B)  $\frac { 28 } { 3 }$ 
C)  Does not exist 
D)  $\sqrt { - \frac { 14 } { 3 } }$

Question 15

Prove the limit statement
-$$\lim _ { x \rightarrow 6 } \frac { 3 x ^ { 2 } - 14 x - 24 } { x - 6 } = 22$$

Accepted Answer

Let @#LAT-DLM& be given. Choose @#LAT-DLM&. Then @#LAT-DLM& implies th

Question 16

Give an appropriate answer.
-Let $\lim _ { x \rightarrow 4 } f ( x ) = 7$ and $\lim _ { x \rightarrow 4 } g ( x ) = - 7$. Find $\lim _ { x \rightarrow 4 } \left[ \frac { - 4 f ( x ) - 7 g ( x ) } { 9 + g ( x ) } \right]$.

Accepted Answer

A)  $- \frac { 77 } { 2 }$ 
B)  $- \frac { 91 } { 9 }$ 
C)  $- 4$ 
D)  $\frac { 21 } { 2 }$ 
A)  $- \frac { 77 } { 2 }$ 
B)  $- \frac { 91 } { 9 }$ 
C)  $- 4$ 
D)  $\frac { 21 } { 2 }$

Question 17

$\text { Find the limit using } \lim _ { x = 0 } \frac { \sin x } { x } = 1 \text {. }$
-$$\lim _ { x \rightarrow 0 } \frac { \sin 5 x } { x }$$

Accepted Answer

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

Question 18

Find all points where the function is discontinuous.
-

Accepted Answer

A)  x = 0 
B)  x = 3 
C)  x = 0, x = 3 
D)  None 
A)  x = 0 
B)  x = 3 
C)  x = 0, x = 3 
D)  None

Question 19

Prove the limit statement
-Ohm's Law for electrical circuits is stated $\mathrm { V } = \mathrm { RI }$, where $\mathrm { V }$ is a constant voltage, $\mathrm { R }$ is the resistance in ohms and $\mathrm { I }$ is the current in amperes. Your firm has been asked to supply the resistors for a circuit in which $\mathrm { V }$ will be 12 volts and $I$ is to be $3 \pm 0.1$ amperes. In what interval does $R$ have to lie for I to be within $0.1$ amps of the target value $I _ { 0 } = 3 ?$

Accepted Answer

A)  $\left( \frac { 31 } { 120 } , \frac { 29 } { 120 } \right)$ 
B)  $\left( \frac { 120 } { 29 } , \frac { 120 } { 31 } \right)$ 
C)  $\left( \frac { 10 } { 29 } , \frac { 10 } { 31 } \right)$ 
D)  $\left( \frac { 120 } { 31 } , \frac { 120 } { 29 } \right)$ 
A)  $\left( \frac { 31 } { 120 } , \frac { 29 } { 120 } \right)$ 
B)  $\left( \frac { 120 } { 29 } , \frac { 120 } { 31 } \right)$ 
C)  $\left( \frac { 10 } { 29 } , \frac { 10 } { 31 } \right)$ 
D)  $\left( \frac { 120 } { 31 } , \frac { 120 } { 29 } \right)$

Question 20

Use the graph to evaluate the limit.
-$\lim _ { x \rightarrow 0 } f ( x )$

Accepted Answer

A)  -3 
B)  0 
C)  does not exist 
D)  3 
A)  -3 
B)  0 
C)  does not exist 
D)  3

Give an appropriate answer. - $f ( x ) = \sqrt { x } \text { for } x _ { 0 } = 13$

Find all points where the function is discontinuous. -

Find the limit. - $\lim _ { x \rightarrow 0 } \left( x ^ { 2 } - 5 \right)$

Use a CAS to plot the function near the point x0 being approached. From your plot guess the value of the limit. - $\lim _ { x \rightarrow 3 } \frac { x ^ { 2 } - 9 } { \sqrt { x ^ { 2 } + 7 } - 4 }$

Use the table to estimate the rate of change of y at the specified value of x. -x = 1. 0.900 -0.05263 0.990 -0.00503 0.999 -0.0005 1.000 0.0000 1.001 0.0005 1.010 0.00498 1.100 0.04762

Find the limit. - $\lim _ { x \rightarrow 0 } ( \sqrt { x } - 2 )$

Find the limit, if it exists. - $\lim _ { x \rightarrow 7 } \frac { x ^ { 2 } + 2 x - 63 } { x - 7 }$

Find the limit, if it exists. - $\lim _ { x \rightarrow 0 } \frac { x ^ { 3 } + 12 x ^ { 2 } - 5 x } { 5 x }$

Use the graph to evaluate the limit. - $\lim _ { x \rightarrow 0 } f ( x )$ $Use the graph to evaluate the limit. - \lim _ { x \rightarrow 0 } f ( x )$

$\text { Find the limit using } \lim _ { x = 0 } \frac { \sin x } { x } = 1 \text {. }$ - $\lim _ { x \rightarrow 5 ^ { - } } \frac { \sqrt { 3 x } ( x - 5 ) } { | x - 5 | }$

Give an appropriate answer. - $f ( x ) = \frac { 5 } { x } \text { for } x _ { 0 } = 6$

$\text { Find the limit using } \lim _ { x = 0 } \frac { \sin x } { x } = 1 \text {. }$ - $\lim _ { x \rightarrow 2 ^ { + } } \sqrt { \frac { 7 x ^ { 2 } } { 5 + x } }$

Prove the limit statement - $\lim _ { x \rightarrow 6 } \frac { 3 x ^ { 2 } - 14 x - 24 } { x - 6 } = 22$

Give an appropriate answer. -Let $\lim _ { x \rightarrow 4 } f ( x ) = 7$ and $\lim _ { x \rightarrow 4 } g ( x ) = - 7$ . Find $\lim _ { x \rightarrow 4 } \left[ \frac { - 4 f ( x ) - 7 g ( x ) } { 9 + g ( x ) } \right]$ .

$\text { Find the limit using } \lim _ { x = 0 } \frac { \sin x } { x } = 1 \text {. }$ - $\lim _ { x \rightarrow 0 } \frac { \sin 5 x } { x }$

Find all points where the function is discontinuous. -

Use the graph to evaluate the limit. - $\lim _ { x \rightarrow 0 } f ( x )$ $Use the graph to evaluate the limit. - \lim _ { x \rightarrow 0 } f ( x )$

Functions

Differentiation

Applications of Derivatives

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Exam 2: Limits and Derivatives

Give an appropriate answer. - f(x)=x for x0=13f ( x ) = \sqrt { x } \text { for } x _ { 0 } = 13f(x)=x​ for x0​=13

Find all points where the function is discontinuous. -

Find the limit. - lim⁡x→0(x2−5)\lim _ { x \rightarrow 0 } \left( x ^ { 2 } - 5 \right)limx→0​(x2−5)

Use a CAS to plot the function near the point x0 being approached. From your plot guess the value of the limit. - lim⁡x→3x2−9x2+7−4\lim _ { x \rightarrow 3 } \frac { x ^ { 2 } - 9 } { \sqrt { x ^ { 2 } + 7 } - 4 }limx→3​x2+7​−4x2−9​

Use the table to estimate the rate of change of y at the specified value of x. -x = 1. 0.900 -0.05263 0.990 -0.00503 0.999 -0.0005 1.000 0.0000 1.001 0.0005 1.010 0.00498 1.100 0.04762

Find the limit. - lim⁡x→0(x−2)\lim _ { x \rightarrow 0 } ( \sqrt { x } - 2 )limx→0​(x​−2)

Find the limit, if it exists. - lim⁡x→7x2+2x−63x−7\lim _ { x \rightarrow 7 } \frac { x ^ { 2 } + 2 x - 63 } { x - 7 }limx→7​x−7x2+2x−63​

Find the limit, if it exists. - lim⁡x→0x3+12x2−5x5x\lim _ { x \rightarrow 0 } \frac { x ^ { 3 } + 12 x ^ { 2 } - 5 x } { 5 x }limx→0​5xx3+12x2−5x​

Use the graph to evaluate the limit. - lim⁡x→0f(x)\lim _ { x \rightarrow 0 } f ( x )limx→0​f(x)

Give an appropriate answer. - f(x)=5x for x0=6f ( x ) = \frac { 5 } { x } \text { for } x _ { 0 } = 6f(x)=x5​ for x0​=6

Find the limit using lim⁡x=0sin⁡xx=1. \text { Find the limit using } \lim _ { x = 0 } \frac { \sin x } { x } = 1 \text {. } Find the limit using limx=0​xsinx​=1. - lim⁡x→2+7x25+x\lim _ { x \rightarrow 2 ^ { + } } \sqrt { \frac { 7 x ^ { 2 } } { 5 + x } }limx→2+​5+x7x2​​

Prove the limit statement - lim⁡x→63x2−14x−24x−6=22\lim _ { x \rightarrow 6 } \frac { 3 x ^ { 2 } - 14 x - 24 } { x - 6 } = 22limx→6​x−63x2−14x−24​=22

Find the limit using lim⁡x=0sin⁡xx=1. \text { Find the limit using } \lim _ { x = 0 } \frac { \sin x } { x } = 1 \text {. } Find the limit using limx=0​xsinx​=1. - lim⁡x→0sin⁡5xx\lim _ { x \rightarrow 0 } \frac { \sin 5 x } { x }limx→0​xsin5x​

Find all points where the function is discontinuous. -

Use the graph to evaluate the limit. - lim⁡x→0f(x)\lim _ { x \rightarrow 0 } f ( x )limx→0​f(x)

Functions

Differentiation

Applications of Derivatives

Integration

Applications of Definite Integrals

Integrals and Transcendental Functions

Techniques of Integration

First-Order Differential Equations

Infinite Sequences and Series

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Give an appropriate answer. - $f ( x ) = \sqrt { x } \text { for } x _ { 0 } = 13$

Find the limit. - $\lim _ { x \rightarrow 0 } \left( x ^ { 2 } - 5 \right)$

Use a CAS to plot the function near the point x0 being approached. From your plot guess the value of the limit. - $\lim _ { x \rightarrow 3 } \frac { x ^ { 2 } - 9 } { \sqrt { x ^ { 2 } + 7 } - 4 }$

Find the limit. - $\lim _ { x \rightarrow 0 } ( \sqrt { x } - 2 )$

Find the limit, if it exists. - $\lim _ { x \rightarrow 7 } \frac { x ^ { 2 } + 2 x - 63 } { x - 7 }$

Find the limit, if it exists. - $\lim _ { x \rightarrow 0 } \frac { x ^ { 3 } + 12 x ^ { 2 } - 5 x } { 5 x }$

Use the graph to evaluate the limit. - $\lim _ { x \rightarrow 0 } f ( x )$ $Use the graph to evaluate the limit. - \lim _ { x \rightarrow 0 } f ( x )$

Give an appropriate answer. - $f ( x ) = \frac { 5 } { x } \text { for } x _ { 0 } = 6$

$\text { Find the limit using } \lim _ { x = 0 } \frac { \sin x } { x } = 1 \text {. }$ - $\lim _ { x \rightarrow 2 ^ { + } } \sqrt { \frac { 7 x ^ { 2 } } { 5 + x } }$

Prove the limit statement - $\lim _ { x \rightarrow 6 } \frac { 3 x ^ { 2 } - 14 x - 24 } { x - 6 } = 22$

$\text { Find the limit using } \lim _ { x = 0 } \frac { \sin x } { x } = 1 \text {. }$ - $\lim _ { x \rightarrow 0 } \frac { \sin 5 x } { x }$

Use the graph to evaluate the limit. - $\lim _ { x \rightarrow 0 } f ( x )$ $Use the graph to evaluate the limit. - \lim _ { x \rightarrow 0 } f ( x )$