Question 1

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

Accepted Answer

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

Question 2

Determine where f is discontinuous. $f(x)=\left\{\begin{array}{ccc}
\sqrt{-x} & \text { if } & x5
\end{array}\right.$

Accepted Answer

The answer of Determine where f is discontinuous. \(f(x)=\left\{\begin{array}{ccc}
\sqrt{-x} &...

Question 3

Use the given graph of $f(x)=x^{2}$ to find a number $\delta$ such that $\left|x^{2}-4\right|<\frac{1}{2}$ whenever $|x-1|<\delta$ .
Round your answer to two decimal places.

Accepted Answer

The answer of Use the given graph of $f(x)=x^{2}$ to...

Question 4

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

Accepted Answer

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 
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 5

Evaluate the function $f(x)=7\left(\frac{\sqrt{x}-\sqrt{2}}{x-2}\right)$ at the given numbers (correct to six decimal places). Use the results to guess the value of the limit $\lim _{x \rightarrow 2} f(x)$ $\begin{array}{|c|c|}
\hline x & f(x) \
\hline 1.6 & \
\hline 1.8 & \
\hline 1.9 & \
\hline 1.99 & \
\hline 1.999 & \
\hline 2.4 & \
\hline 2.2 & \
\hline 2.1 & \
\hline 2.01 & \
\hline 2.001 & \
\hline \text { Limit } & \
\hline
\end{array}$

Accepted Answer

2.612794,2.540047,2.506608,2.

Question 6

Use a graph to find a number $\delta$ such that $|\sqrt{4 x+1}-3|<0.1$ whenever $|x-2|<\delta$ .

Accepted Answer

The answer of Use a graph to find a number...

Question 7

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

Accepted Answer

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

Question 8

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

Accepted Answer

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

Question 9

Use the graph of the function to find each limit.   $\lim _{x \rightarrow 3^{-}} f(x)$ , $\lim _{x \rightarrow 3^{+}} f(x)$ , $\lim _{x \rightarrow 3} f(x)$

Accepted Answer

The answer of Use the graph of the function to...

Question 10

The cost (in dollars) of producing x units of a certain commodity is $C(x)=4,336+12 x+0.07 x^{2}$ Find the average rate of change with respect to x when the production level is changed from $x=105 \text { to } x=107 \text {. }$

Accepted Answer

The answer of The cost (in dollars) of producing x...

Question 11

Find the limit $\lim _{x \rightarrow 3} \frac{x^{2}+x-12}{x^{2}-9}$ , if it exists.

Accepted Answer

The answer of Find the limit \(\lim _{x \rightarrow 3}...

Question 12

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 $t=9 .$

Accepted Answer

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

Question 13

How would you define $f(5)$ in order to make f continuous at 5 ? $f(x)=\frac{x^{3}-125}{x^{2}-25}$

Accepted Answer

The answer of How would you define $f(5)$ in order...

Question 14

How close to 2 do we have to take x so that $5 x+3$ is within a distance of 0.01 from 13?

Accepted Answer

The answer of How close to 2 do we have...

Question 15

A tank holds $1000$ gallons of water, which drains from the bottom of the tank in half an hour. The values in the table show the volume V of water remaining in the tank (in gallons) after $t$ minutes. If $P$ is the point $(10,275)$ on the graph of V, fill the table with the slopes of the secant lines $\mathrm{PQ}$ where Q is the point on the graph with the corresponding $t$ . $\begin{array}{|c|c|c|c|c|c|c|}
\hline t \text { (min) } & 5 & 10 & 15 & 20 & 25 & 30 \
\hline V \text { (gal) } & 686 & 481 & 263 & 171 & 22 & 0 \
\hline
\end{array}$ Enter your answer to two decimal places.
$\begin{array}{|c|c|}
\hline t & \text { Slope } \
\hline 5 & \
\hline 10 & \
\hline 20 & \
\hline 25 & \
\hline 30 & \
\hline
\end{array}$

Accepted Answer

The answer of A tank holds $1000$ gallons of water,...

Question 16

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

Accepted Answer

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

Question 17

Complete the table by computing $f(x)$ 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. $\lim _{x \rightarrow-1} \frac{x+1}{x^{2}-x-2}$ 
$\begin{array}{|l|l|l|l|l|l|l|}
 \hline  \text {x} &-0.9 & -0.99 & -0.999 & -1.001 & -1.01 & -1.1  \
\hline  \text {f(x)} &&\
\hline 
\end{array}$

Accepted Answer

None

Question 18

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

Accepted Answer

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

Question 19

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.

Accepted Answer

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 $16$ 
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 20

Consider the following function. $f(x)=\left\{\begin{array}{cc}
3-x & x<-1 \
x & -1 \leq x<3 \
(x-3)^{2} & x \geq 3
\end{array}\right.$ Determine the values of a for which $\lim _{x \rightarrow a} f(x)$ exists.

Accepted Answer

The answer of Consider the following function. \(f(x)=\left\{\begin{array}{cc}
3-x & x<-1...

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

Determine where f is discontinuous. $f(x)=\left\{\begin{array}{ccc}\sqrt{-x} & \text { if } & x<0 \\5-x & \text { if } & 0 \leq x<5 \\(5-x)^{2} & \text { if } & x>5\end{array}\right.$

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

Evaluate the function $f(x)=7\left(\frac{\sqrt{x}-\sqrt{2}}{x-2}\right)$ at the given numbers (correct to six decimal places). Use the results to guess the value of the limit $\lim _{x \rightarrow 2} f(x)$ x f(x) 1.6 1.8 1.9 1.99 1.999 2.4 2.2 2.1 2.01 2.001 Limit

Use a graph to find a number $\delta$ such that $|\sqrt{4 x+1}-3|<0.1$ whenever $|x-2|<\delta$ .

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

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

The cost (in dollars) of producing x units of a certain commodity is $C(x)=4,336+12 x+0.07 x^{2}$ Find the average rate of change with respect to x when the production level is changed from $x=105 \text { to } x=107 \text {. }$

Find the limit $\lim _{x \rightarrow 3} \frac{x^{2}+x-12}{x^{2}-9}$ , if it exists.

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 $t=9 .$

How would you define $f(5)$ in order to make f continuous at 5 ? $f(x)=\frac{x^{3}-125}{x^{2}-25}$

How close to 2 do we have to take x so that $5 x+3$ is within a distance of 0.01 from 13?

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

Complete the table by computing $f(x)$ 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. $\lim _{x \rightarrow-1} \frac{x+1}{x^{2}-x-2}$ x -0.9 -0.99 -0.999 -1.001 -1.01 -1.1 f(x)

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

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.

Consider the following function. $f(x)=\left\{\begin{array}{cc}3-x & x<-1 \\x & -1 \leq x<3 \\(x-3)^{2} & x \geq 3\end{array}\right.$ Determine the values of a for which $\lim _{x \rightarrow a} f(x)$ exists.

Functions and Limits

Applications of Differentiation

Integrals

Applications of Integration

Inverse Functions

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Final Exam

Filters

Exam 2: Derivatives

Find the limit lim⁡h→−1(h4−h3−4h+6)\lim _{h \rightarrow-1}\left(h^{4}-h^{3}-4 h+6\right)limh→−1​(h4−h3−4h+6) .

Determine where f is discontinuous. f(x)={−x if x<05−x if 0≤x<5(5−x)2 if x>5f(x)=\left\{\begin{array}{ccc}\sqrt{-x} & \text { if } & x<0 \\5-x & \text { if } & 0 \leq x<5 \\(5-x)^{2} & \text { if } & x>5\end{array}\right.f(x)=⎩⎨⎧​−x​5−x(5−x)2​ if if if ​x<00≤x<5x>5​

Use the given graph of f(x)=x2f(x)=x^{2}f(x)=x2 to find a number δ\deltaδ such that ∣x2−4∣<12\left|x^{2}-4\right|<\frac{1}{2}​x2−4​<21​ whenever ∣x−1∣<δ|x-1|<\delta∣x−1∣<δ . Round your answer to two decimal places.

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

Use a graph to find a number δ\deltaδ such that ∣4x+1−3∣<0.1|\sqrt{4 x+1}-3|<0.1∣4x+1​−3∣<0.1 whenever ∣x−2∣<δ|x-2|<\delta∣x−2∣<δ .

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

Let F(x)=x3−1∣x−1∣F(x)=\frac{x^{3}-1}{|x-1|}F(x)=∣x−1∣x3−1​ Find the following limits. lim⁡x→1+F(x),lim⁡x→1−F(x)\lim _{x \rightarrow 1^{+}} F(x), \lim _{x \rightarrow 1^{-}} F(x)limx→1+​F(x),limx→1−​F(x)

Use the graph of the function to find each limit. lim⁡x→3−f(x)\lim _{x \rightarrow 3^{-}} f(x)limx→3−​f(x) , lim⁡x→3+f(x)\lim _{x \rightarrow 3^{+}} f(x)limx→3+​f(x) , lim⁡x→3f(x)\lim _{x \rightarrow 3} f(x)limx→3​f(x)

The cost (in dollars) of producing x units of a certain commodity is C(x)=4,336+12x+0.07x2C(x)=4,336+12 x+0.07 x^{2}C(x)=4,336+12x+0.07x2 Find the average rate of change with respect to x when the production level is changed from x=105 to x=107. x=105 \text { to } x=107 \text {. }x=105 to x=107.

Find the limit lim⁡x→3x2+x−12x2−9\lim _{x \rightarrow 3} \frac{x^{2}+x-12}{x^{2}-9}limx→3​x2−9x2+x−12​ , if it exists.

If a ball is thrown into the air with a velocity of 58 ft/s58 ~~\mathrm{ft} / \mathrm{s}58 ft/s , its height (in feet) after t seconds is given by H=58t−18t2H=58 t-18 t^{2}H=58t−18t2 . Find the velocity when t=9.t=9 .t=9.

How would you define f(5)f(5)f(5) in order to make f continuous at 5 ? f(x)=x3−125x2−25f(x)=\frac{x^{3}-125}{x^{2}-25}f(x)=x2−25x3−125​

How close to 2 do we have to take x so that 5x+35 x+35x+3 is within a distance of 0.01 from 13?

Find the limit. lim⁡t→∞t2+3t3+t2−7\lim _{t \rightarrow \infty} \frac{t^{2}+3}{t^{3}+t^{2}-7}limt→∞​t3+t2−7t2+3​

If g(x)=4−3xg(x)=\sqrt{4-3 x}g(x)=4−3x​ find the domain of g′(x)g^{\prime}(x)g′(x)

Is there a number a such that lim⁡x→−33x2+ax+a+3x2+x−16\lim _{x \rightarrow-3} \frac{3 x^{2}+a x+a+3}{x^{2}+x-16}limx→−3​x2+x−163x2+ax+a+3​ exists? If so, find the value of a and the value of the limit.

Functions and Limits

Applications of Differentiation

Integrals

Applications of Integration

Inverse Functions

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Final Exam

Filters

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

Determine where f is discontinuous. $f(x)=\left\{\begin{array}{ccc}\sqrt{-x} & \text { if } & x<0 \\5-x & \text { if } & 0 \leq x<5 \\(5-x)^{2} & \text { if } & x>5\end{array}\right.$

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

Use a graph to find a number $\delta$ such that $|\sqrt{4 x+1}-3|<0.1$ whenever $|x-2|<\delta$ .

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

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

The cost (in dollars) of producing x units of a certain commodity is $C(x)=4,336+12 x+0.07 x^{2}$ Find the average rate of change with respect to x when the production level is changed from $x=105 \text { to } x=107 \text {. }$

Find the limit $\lim _{x \rightarrow 3} \frac{x^{2}+x-12}{x^{2}-9}$ , if it exists.

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 $t=9 .$

How would you define $f(5)$ in order to make f continuous at 5 ? $f(x)=\frac{x^{3}-125}{x^{2}-25}$

How close to 2 do we have to take x so that $5 x+3$ is within a distance of 0.01 from 13?

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

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

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.