Question 1

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

Accepted Answer

The answer of \[\text { If } 6 x -...

Question 2

The graph shows the percentage of households in a certain city watching television during a $24 - h r$ period on a weekday ( $t = 0$ corresponds to 6 a.m.). By computing the slope of the respective tangent line, estimate the rate of change of the percentage of households watching television at a-12 p.m.
Note that $d y = 0.03$

Accepted Answer

The answer of The graph shows the percentage of households...

Question 3

Use the table to evaluate the expression $( f \circ g ) ( 6 )$.
$\begin{array}{|c|c|c|c|c|c|c|}
\hline x & 1 & 2 & 3 & 4 & 5 & 6 \
\hline f(x) & 3 & 2 & 1 & 0 & 1 & 2 \
\hline g(x) & 6 & 5 & 2 & 3 & 4 & 6 \
\hline
\end{array}$

Accepted Answer

A)  5 
B)  2 
C)  3 
D)  4 
E)  6

Question 4

Evaluate the limit.
$$\lim _ { x \rightarrow 1 } ( x + 5 ) ^ { 3 } \left( x ^ { 2 } - 6 \right)$$

Accepted Answer

The answer of Evaluate the limit.
\[\lim _ { x \rightarrow...

Question 5

If a rock is thrown upward on the planet Mars with a velocity of $12 \mathrm {~m} / \mathrm { s }$, its height in meters $t$ seconds later is given by
$$y = 12 t - 1.92 t ^ { 2 }$$
Find the average velocity over the time interval $[ 2,3 ]$.

Accepted Answer

A)  $- 0.6 \mathrm {~m} / \mathrm { s }$ 
B)  $4.4 \mathrm {~m} / \mathrm { s }$ 
C)  $2.4 \mathrm {~m} / \mathrm { s }$ 
D)  $3.4 \mathrm {~m} / \mathrm { s }$ 
E)  $1.4 \mathrm {~m} / \mathrm { s }$

Question 6

It makes sense that the larger the area of a region, the larger the number of species that inhabit the region. Many ecologists have modeled the species-area relation with a power function and, in particular, the number of species $S$ of bats living in caves in central Mexico has been related to the surface area $A$ measured in $m ^ { 2 }$ of the caves by the equation
$$S = 0.7 A ^ { 03 }$$
(a) The cave called mission impossible near puebla, mexico, has suface area of $A = 90 \mathrm {~m} ^ { 2 }$. How many species of bats would expect to find in that cave?
(b) If you discover that 5 species of bats live in cave estimate the area of the cave.

Accepted Answer

The answer of It makes sense that the larger the...

Question 7

The displacement (in feet) of a certain particle moving in a straight line is given by $s = \frac { t ^ { 3 } } { 8 }$
where $t$ is measured in seconds. Find the average velocity over the interval $[ 1,1.19 ]$.
Round your answer to three decimal places.

Accepted Answer

A)  $0.251$ 
B)  $ 0.390$ 
C)  $0.351$ 
D)  $0.241$ 
E)  $ 0.551$
F) $0.451$

Question 8

If the point $( 7,3 )$ is on the graph of an even function, what other point must also be on the graph?
Select the correct answer.

Accepted Answer

A)  $( - 7 , - 3 )$ 
B)  $( 7 , - 3 )$ 
C)  $( 0,0 )$ 
D)  $( - 7,3 )$ 
E)  None of these

Question 9

Find the limit.
$\lim _ { x \rightarrow 2 } \sqrt { \frac { 4 x ^ { 2 } + 1 } { 3 x - 2 } }$

Accepted Answer

A)  0 
B)  $- \frac { 4 } { 3 }$ 
C)  $\frac { \sqrt { 17 } } { 2 }$ 
D)  $\frac { 4 } { 3 }$ 
E)  does not exist

Question 10

The relationship between the Fahrenheit and Celsius temperature scales is given by the linear function.
$$F = \frac { 9 } { 5 } C + 32$$
What is the $F$-intercept and what does it represent?

Accepted Answer

The answer of The relationship between the Fahrenheit and Celsius...

Question 11

Find the numbers, if any, where the function $f ( x ) = \left\{ \begin{array} { c c } 3 x - 2 & \text { if } x \leq 1 \ 0 & \text { if } x > 1 \end{array} \right.$ is discontinuous. Select the correct answer.

Accepted Answer

A)  $- 2$ 
B)  2 
C)  1 
D)  0

Question 12

Let $F ( x ) = \frac { x - 5 } { | x - 5 | }$. Find the following limits.
$$\lim _ { x \rightarrow 5 ^ { + } } F ( x ) , \lim _ { x \rightarrow 5 ^ { - } } F ( x )$$

Accepted Answer

The answer of Let \(F ( x ) = \frac...

Question 13

Find the limit.
$$\lim _ { x \rightarrow - \infty } \frac { \sqrt { x ^ { 2 } - 25 } } { 2 x - 6 }$$

Accepted Answer

The answer of Find the limit.
\[\lim _ { x \rightarrow...

Question 14

Which of the following graphs is neither even nor odd?

Accepted Answer

A) 
$f ( x ) = \frac { 4 x ^ { 2 } } { x ^ { 4 } + 1 }$ 
B)  $f ( x ) = 8 x ^ { 3 } + 10 x ^ { 2 } + 1$ 
C)  $f ( x ) = x ^ { 3 } - 9 x$

Question 15

Determine whether $f$ is even, odd, or neither.
$$f ( x ) = \frac { 4 x ^ { 2 } } { x ^ { 4 } + 5 }$$

Accepted Answer

The answer of Determine whether $f$ is even, odd, or...

Question 16

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

Accepted Answer

The answer of Evaluate
\[\frac { \cot \left( \frac { \pi...

Question 17

$$\text { Find a number } \delta \text { such that if } | x - 2 | < \delta \text {, then } | 4 x - 8 | < \varepsilon \text {, where } \varepsilon = 0.1 \text {. }$$

Accepted Answer

The answer of \[\text { Find a number } \delta...

Question 18

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

Accepted Answer

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

Question 19

$$\text { What is the equation of this graph? }$$

Accepted Answer

The answer of \[\text { What is the equation of...

Exam 1: Functions and Models

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

Use the table to evaluate the expression $( f \circ g ) ( 6 )$ . $\begin{array}{|c|c|c|c|c|c|c|}\hline x & 1 & 2 & 3 & 4 & 5 & 6 \\\hline f(x) & 3 & 2 & 1 & 0 & 1 & 2 \\\hline g(x) & 6 & 5 & 2 & 3 & 4 & 6 \\\hline\end{array}$

Evaluate the limit. $\lim _ { x \rightarrow 1 } ( x + 5 ) ^ { 3 } \left( x ^ { 2 } - 6 \right)$

If a rock is thrown upward on the planet Mars with a velocity of $12 \mathrm {~m} / \mathrm { s }$ , its height in meters $t$ seconds later is given by $y = 12 t - 1.92 t ^ { 2 }$ Find the average velocity over the time interval $[ 2,3 ]$ .

The displacement (in feet) of a certain particle moving in a straight line is given by $s = \frac { t ^ { 3 } } { 8 }$ where $t$ is measured in seconds. Find the average velocity over the interval $[ 1,1.19 ]$ . Round your answer to three decimal places.

If the point $( 7,3 )$ is on the graph of an even function, what other point must also be on the graph? Select the correct answer.

Find the limit. $\lim _ { x \rightarrow 2 } \sqrt { \frac { 4 x ^ { 2 } + 1 } { 3 x - 2 } }$

The relationship between the Fahrenheit and Celsius temperature scales is given by the linear function. $F = \frac { 9 } { 5 } C + 32$ What is the $F$ -intercept and what does it represent?

Find the numbers, if any, where the function $f ( x ) = \left\{ \begin{array} { c c } 3 x - 2 & \text { if } x \leq 1 \\ 0 & \text { if } x > 1 \end{array} \right.$ is discontinuous. Select the correct answer.

Let $F ( x ) = \frac { x - 5 } { | x - 5 | }$ . Find the following limits. $\lim _ { x \rightarrow 5 ^ { + } } F ( x ) , \lim _ { x \rightarrow 5 ^ { - } } F ( x )$

Find the limit. $\lim _ { x \rightarrow - \infty } \frac { \sqrt { x ^ { 2 } - 25 } } { 2 x - 6 }$

Which of the following graphs is neither even nor odd?

Determine whether $f$ is even, odd, or neither. $f ( x ) = \frac { 4 x ^ { 2 } } { x ^ { 4 } + 5 }$

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

$\text { Find a number } \delta \text { such that if } | x - 2 | < \delta \text {, then } | 4 x - 8 | < \varepsilon \text {, where } \varepsilon = 0.1 \text {. }$

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

$\text { What is the equation of this graph? }$ $\text { What is the equation of this graph? }$

Exam 1: Functions and Models

If 6x−1≤f(x)≤x2−1,find lim⁡x→6f(x)\text { If } 6 x - 1 \leq f ( x ) \leq x ^ { 2 } - 1 \text {,find } \lim _ { x \rightarrow 6 } f ( x ) If 6x−1≤f(x)≤x2−1,find limx→6​f(x)

Evaluate the limit. lim⁡x→1(x+5)3(x2−6)\lim _ { x \rightarrow 1 } ( x + 5 ) ^ { 3 } \left( x ^ { 2 } - 6 \right)limx→1​(x+5)3(x2−6)

If a rock is thrown upward on the planet Mars with a velocity of 12 m/s12 \mathrm {~m} / \mathrm { s }12 m/s , its height in meters ttt seconds later is given by y=12t−1.92t2y = 12 t - 1.92 t ^ { 2 }y=12t−1.92t2 Find the average velocity over the time interval [2,3][ 2,3 ][2,3] .

The displacement (in feet) of a certain particle moving in a straight line is given by s=t38s = \frac { t ^ { 3 } } { 8 }s=8t3​ where ttt is measured in seconds. Find the average velocity over the interval [1,1.19][ 1,1.19 ][1,1.19] . Round your answer to three decimal places.

If the point (7,3) ( 7,3 ) (7,3) is on the graph of an even function, what other point must also be on the graph? Select the correct answer.

Find the limit. lim⁡x→24x2+13x−2\lim _ { x \rightarrow 2 } \sqrt { \frac { 4 x ^ { 2 } + 1 } { 3 x - 2 } }limx→2​3x−24x2+1​​

The relationship between the Fahrenheit and Celsius temperature scales is given by the linear function. F=95C+32F = \frac { 9 } { 5 } C + 32F=59​C+32 What is the FFF -intercept and what does it represent?

Find the numbers, if any, where the function f(x) ={3x−2 if x≤10 if x>1f ( x ) = \left\{ \begin{array} { c c } 3 x - 2 & \text { if } x \leq 1 \\ 0 & \text { if } x > 1 \end{array} \right.f(x) ={3x−20​ if x≤1 if x>1​ is discontinuous. Select the correct answer.

Let F(x)=x−5∣x−5∣F ( x ) = \frac { x - 5 } { | x - 5 | }F(x)=∣x−5∣x−5​ . Find the following limits. lim⁡x→5+F(x),lim⁡x→5−F(x)\lim _ { x \rightarrow 5 ^ { + } } F ( x ) , \lim _ { x \rightarrow 5 ^ { - } } F ( x )limx→5+​F(x),limx→5−​F(x)

Find the limit. lim⁡x→−∞x2−252x−6\lim _ { x \rightarrow - \infty } \frac { \sqrt { x ^ { 2 } - 25 } } { 2 x - 6 }limx→−∞​2x−6x2−25​​

Which of the following graphs is neither even nor odd?

Determine whether fff is even, odd, or neither. f(x)=4x2x4+5f ( x ) = \frac { 4 x ^ { 2 } } { x ^ { 4 } + 5 }f(x)=x4+54x2​

Evaluate cot⁡(π4+h)−1h.\frac { \cot \left( \frac { \pi } { 4 } + h \right) - 1 } { h } .hcot(4π​+h)−1​.

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

What is the equation of this graph? \text { What is the equation of this graph? } What is the equation of this graph?

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

Evaluate the limit. $\lim _ { x \rightarrow 1 } ( x + 5 ) ^ { 3 } \left( x ^ { 2 } - 6 \right)$

If a rock is thrown upward on the planet Mars with a velocity of $12 \mathrm {~m} / \mathrm { s }$ , its height in meters $t$ seconds later is given by $y = 12 t - 1.92 t ^ { 2 }$ Find the average velocity over the time interval $[ 2,3 ]$ .

The displacement (in feet) of a certain particle moving in a straight line is given by $s = \frac { t ^ { 3 } } { 8 }$ where $t$ is measured in seconds. Find the average velocity over the interval $[ 1,1.19 ]$ . Round your answer to three decimal places.

If the point $( 7,3 )$ is on the graph of an even function, what other point must also be on the graph? Select the correct answer.

Find the limit. $\lim _ { x \rightarrow 2 } \sqrt { \frac { 4 x ^ { 2 } + 1 } { 3 x - 2 } }$

The relationship between the Fahrenheit and Celsius temperature scales is given by the linear function. $F = \frac { 9 } { 5 } C + 32$ What is the $F$ -intercept and what does it represent?

Find the numbers, if any, where the function $f ( x ) = \left\{ \begin{array} { c c } 3 x - 2 & \text { if } x \leq 1 \\ 0 & \text { if } x > 1 \end{array} \right.$ is discontinuous. Select the correct answer.

Let $F ( x ) = \frac { x - 5 } { | x - 5 | }$ . Find the following limits. $\lim _ { x \rightarrow 5 ^ { + } } F ( x ) , \lim _ { x \rightarrow 5 ^ { - } } F ( x )$

Find the limit. $\lim _ { x \rightarrow - \infty } \frac { \sqrt { x ^ { 2 } - 25 } } { 2 x - 6 }$

Determine whether $f$ is even, odd, or neither. $f ( x ) = \frac { 4 x ^ { 2 } } { x ^ { 4 } + 5 }$

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

$\text { What is the equation of this graph? }$ $\text { What is the equation of this graph? }$