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 
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)  $1.4 \mathrm {~m} / \mathrm { s }$ 
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) $0.451$ 
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 
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 
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 
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)  $f ( x ) = x ^ { 3 } - 9 x$ 
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...

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

Use the table to evaluate the expression $( f \circ g ) ( 6 )$ . x 1 2 3 4 5 6 f(x) 3 2 1 0 1 2 g(x) 6 5 2 3 4 6

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?

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

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

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

Filters

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)

Use the table to evaluate the expression (f∘g)(6)( f \circ g ) ( 6 )(f∘g)(6) . x 1 2 3 4 5 6 f(x) 3 2 1 0 1 2 g(x) 6 5 2 3 4 6

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?

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

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

Filters

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

Use the table to evaluate the expression $( f \circ g ) ( 6 )$ . x 1 2 3 4 5 6 f(x) 3 2 1 0 1 2 g(x) 6 5 2 3 4 6

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

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