Question 1

If $f$ and $g$ are continuous functions with $f ( 9 ) = 6$ and $\lim _ { x \rightarrow 9 } [ 2 f ( x ) - g ( x ) ] = 9$, find $g ( 9 )$.

Accepted Answer

A)  $g ( 9 ) = 21$ 
B)  $g ( 9 ) = 15$ 
C)  $g ( 9 ) = 12$ 
D)  $g ( 9 ) = 24$ 
E)  $g ( 9 ) = 3$ 
A)  $g ( 9 ) = 21$ 
B)  $g ( 9 ) = 15$ 
C)  $g ( 9 ) = 12$ 
D)  $g ( 9 ) = 24$ 
E)  $g ( 9 ) = 3$

Question 2

Suppose the distance $s$ (in feet) covered by a car moving along a straight road after $t$ sec is given by the function $s = f ( t ) = 3 t ^ { 2 } + 13 t$. Calculate the (instantaneous) velocity of the car when $t = 35$.

Accepted Answer

A)  223 ft/sec 
B)  16 ft/sec 
C)  560 ft/sec 
D)  4130 ft/sec 
A)  223 ft/sec 
B)  16 ft/sec 
C)  560 ft/sec 
D)  4130 ft/sec

Question 3

$$\text { If } f \text { and } g \text { are continuous functions with } f ( 7 ) = 10 \text { and } \lim _ { x \rightarrow 7 } [ 2 f ( x ) - g ( x ) ] = 7 , \text { find } g ( 7 ) \text {. }$$

Accepted Answer

The answer of \[\text { If } f \text {...

Question 4

If $f ( x ) = x ^ { 2 } - x + 6$, evaluate the difference quotient $\frac { f ( a + h ) - f ( a ) } { h }$

Accepted Answer

A)  $2 a + h - 6$ 
B)  $2 a - 6$ 
C)  $2 a - h - 6$ 
D)  $h$ 
E)  none of these 
A)  $2 a + h - 6$ 
B)  $2 a - 6$ 
C)  $2 a - h - 6$ 
D)  $h$ 
E)  none of these

Question 5

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

Accepted Answer

A)  $- \frac { 1 } { 8 }$ 
B)  1 
C)  $- \frac { 1 } { 16 }$ 
D)  8 
E)  does not exist 
A)  $- \frac { 1 } { 8 }$ 
B)  1 
C)  $- \frac { 1 } { 16 }$ 
D)  8 
E)  does not exist

Question 6

Sketch the graph of the function $f$ and evaluate $\lim _ { x \rightarrow 3 } f ( x )$.
$$f ( x ) = \left\{ \begin{array} { l l } 
x - 4 , & \text { if } x \leq 3 \
- 2 x + 5 , & \text { if } x > 3
\end{array} \right.$$

Accepted Answer

A) 
  
B) 
 
-1 
C) 
 
3 
D) 
 
-1 
A) 
  
B) 
 
-1 
C) 
 
3 
D) 
 
-1

Question 7

The following figure shows a portion of the graph of a function $f$ defined on the interval $[ - 1,1 ]$. Sketch the complete graph of $f$ if it is known $f$ is odd.

Accepted Answer

The answer of The following figure shows a portion of...

Question 8

Find the domain. $$g ( u ) = \sqrt { u } - \sqrt { 3 - u }$$

Accepted Answer

The answer of Find the domain. \[g ( u )...

Question 9

Use the graph of $f ( x ) = \frac { x ^ { 2 } + x - 2 } { x + 2 }$ to guess at the limit $\lim _ { x \rightarrow - 2 } \frac { x ^ { 2 } + x - 2 } { x + 2 }$, if it exists.

Accepted Answer

A)  - 2 
B)  - 1 
C)  Does not exist 
D)  - 3 
A)  - 2 
B)  - 1 
C)  Does not exist 
D)  - 3

Question 10

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 11

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 12

Sketch the graph of $y = - 1 - \cos x$ over one period.

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 13

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 14

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...

Question 15

The graphs of $f ( x )$ and $g ( x )$ are given.
a) For what values of $x$ is $f ( x ) = g ( x )$ ?
b) Find the values of $f ( - 2 )$ and $g ( 4 )$.

Accepted Answer

The answer of The graphs of $f ( x )$...

Question 16

For what value of the constant $c$ is the function $f$ continuous on $( - \infty , \infty ) ?$
$f ( x ) = \left\{ \begin{array} { l l l } c x + 5 & \text { for } & x \leq 2 \ c x ^ { 2 } - 5 & \text { for } & x > 2 \end{array} \right.$
Select the correct answer.

Accepted Answer

A)  $c = 1$ 
B)  $c = 5$ 
C)  $c = - 2$ 
D)  $c = - 5$ 
E)  $c = 2$ 
A)  $c = 1$ 
B)  $c = 5$ 
C)  $c = - 2$ 
D)  $c = - 5$ 
E)  $c = 2$

Question 17

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 18

The position of a car is given by the values in the following table. $\begin{array}{|c|c|c|c|c|c|}
\hline t \text { (seconds) } & 0 & 1 & 2 & 3 & 4 \
\hline s \text { (meters) } & 0 & 21.9 & 25.8 & 69.2 & 92.2 \
\hline
\end{array}$

Find the average velocity for the time period beginning when $t = 2$ and lasting 2 seconds.

Accepted Answer

A)  $46.1 \mathrm { ft } / \mathrm { s }$ 
B)  $31.9 \mathrm { ft } / \mathrm { s }$ 
C)  $33.5 \mathrm { ft } / \mathrm { s }$ 
D)  $33.8 \mathrm { ft } / \mathrm { s }$ 
E)  $33.2 \mathrm { ft } / \mathrm { s }$ 
A)  $46.1 \mathrm { ft } / \mathrm { s }$ 
B)  $31.9 \mathrm { ft } / \mathrm { s }$ 
C)  $33.5 \mathrm { ft } / \mathrm { s }$ 
D)  $33.8 \mathrm { ft } / \mathrm { s }$ 
E)  $33.2 \mathrm { ft } / \mathrm { s }$

Question 19

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.

Accepted Answer

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

Question 20

Find the limit. $$\lim _ { x \rightarrow \frac { 10 } { x } } \tan ^ { - 1 } \left( \frac { 5 } { x } \right)$$

Accepted Answer

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

If $f$ and $g$ are continuous functions with $f ( 9 ) = 6$ and $\lim _ { x \rightarrow 9 } [ 2 f ( x ) - g ( x ) ] = 9$ , find $g ( 9 )$ .

Suppose the distance $s$ (in feet) covered by a car moving along a straight road after $t$ sec is given by the function $s = f ( t ) = 3 t ^ { 2 } + 13 t$ . Calculate the (instantaneous) velocity of the car when $t = 35$ .

$\text { If } f \text { and } g \text { are continuous functions with } f ( 7 ) = 10 \text { and } \lim _ { x \rightarrow 7 } [ 2 f ( x ) - g ( x ) ] = 7 , \text { find } g ( 7 ) \text {. }$

If $f ( x ) = x ^ { 2 } - x + 6$ , evaluate the difference quotient $\frac { f ( a + h ) - f ( a ) } { h }$

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

Sketch the graph of the function $f$ and evaluate $\lim _ { x \rightarrow 3 } f ( x )$ . $f ( x ) = \left\{ \begin{array} { l l } x - 4 , & \text { if } x \leq 3 \\- 2 x + 5 , & \text { if } x > 3\end{array} \right.$

The following figure shows a portion of the graph of a function $f$ defined on the interval $[ - 1,1 ]$ . Sketch the complete graph of $f$ if it is known $f$ is odd.

Find the domain. $g ( u ) = \sqrt { u } - \sqrt { 3 - u }$

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

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.

Sketch the graph of $y = - 1 - \cos x$ over one period.

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

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

The graphs of $f ( x )$ and $g ( x )$ are given. a) For what values of $x$ is $f ( x ) = g ( x )$ ? b) Find the values of $f ( - 2 )$ and $g ( 4 )$ .

For what value of the constant $c$ is the function $f$ continuous on $( - \infty , \infty ) ?$ $f ( x ) = \left\{ \begin{array} { l l l } c x + 5 & \text { for } & x \leq 2 \\ c x ^ { 2 } - 5 & \text { for } & x > 2 \end{array} \right.$ Select the correct answer.

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 position of a car is given by the values in the following table. t (seconds) 0 1 2 3 4 s (meters) 0 21.9 25.8 69.2 92.2 Find the average velocity for the time period beginning when $t = 2$ and lasting 2 seconds.

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.

Find the limit. $\lim _ { x \rightarrow \frac { 10 } { x } } \tan ^ { - 1 } \left( \frac { 5 } { x } \right)$

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

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Exam 1: Functions and Models

If fff and ggg are continuous functions with f(9)=6f ( 9 ) = 6f(9)=6 and lim⁡x→9[2f(x)−g(x)]=9\lim _ { x \rightarrow 9 } [ 2 f ( x ) - g ( x ) ] = 9limx→9​[2f(x)−g(x)]=9 , find g(9)g ( 9 )g(9) .

Suppose the distance sss (in feet) covered by a car moving along a straight road after ttt sec is given by the function s=f(t)=3t2+13ts = f ( t ) = 3 t ^ { 2 } + 13 ts=f(t)=3t2+13t . Calculate the (instantaneous) velocity of the car when t=35t = 35t=35 .

If f(x)=x2−x+6f ( x ) = x ^ { 2 } - x + 6f(x)=x2−x+6 , evaluate the difference quotient f(a+h)−f(a)h\frac { f ( a + h ) - f ( a ) } { h }hf(a+h)−f(a)​

If 1≤f(x)≤x2+6x+61 \leq f ( x ) \leq x ^ { 2 } + 6 x + 61≤f(x)≤x2+6x+6 , for all xxx , find lim⁡x→−1f(x)\lim _ { x \rightarrow - 1 } f ( x )limx→−1​f(x)

The following figure shows a portion of the graph of a function fff defined on the interval [−1,1][ - 1,1 ][−1,1] . Sketch the complete graph of fff if it is known fff is odd.

Find the domain. g(u)=u−3−ug ( u ) = \sqrt { u } - \sqrt { 3 - u }g(u)=u​−3−u​

Use the graph of f(x)=x2+x−2x+2f ( x ) = \frac { x ^ { 2 } + x - 2 } { x + 2 }f(x)=x+2x2+x−2​ to guess at the limit lim⁡x→−2x2+x−2x+2\lim _ { x \rightarrow - 2 } \frac { x ^ { 2 } + x - 2 } { x + 2 }limx→−2​x+2x2+x−2​ , if it exists.

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​

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.

Sketch the graph of y=−1−cos⁡xy = - 1 - \cos xy=−1−cosx over one period.

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​​

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)

The graphs of f(x)f ( x )f(x) and g(x)g ( x )g(x) are given. a) For what values of xxx is f(x)=g(x)f ( x ) = g ( x )f(x)=g(x) ? b) Find the values of f(−2)f ( - 2 )f(−2) and g(4)g ( 4 )g(4) .

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 position of a car is given by the values in the following table. t (seconds) 0 1 2 3 4 s (meters) 0 21.9 25.8 69.2 92.2 Find the average velocity for the time period beginning when t=2t = 2t=2 and lasting 2 seconds.

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.

Find the limit. lim⁡x→10xtan⁡−1(5x)\lim _ { x \rightarrow \frac { 10 } { x } } \tan ^ { - 1 } \left( \frac { 5 } { x } \right)limx→x10​​tan−1(x5​)

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

If $f$ and $g$ are continuous functions with $f ( 9 ) = 6$ and $\lim _ { x \rightarrow 9 } [ 2 f ( x ) - g ( x ) ] = 9$ , find $g ( 9 )$ .

Suppose the distance $s$ (in feet) covered by a car moving along a straight road after $t$ sec is given by the function $s = f ( t ) = 3 t ^ { 2 } + 13 t$ . Calculate the (instantaneous) velocity of the car when $t = 35$ .

If $f ( x ) = x ^ { 2 } - x + 6$ , evaluate the difference quotient $\frac { f ( a + h ) - f ( a ) } { h }$

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

The following figure shows a portion of the graph of a function $f$ defined on the interval $[ - 1,1 ]$ . Sketch the complete graph of $f$ if it is known $f$ is odd.

Find the domain. $g ( u ) = \sqrt { u } - \sqrt { 3 - u }$

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

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.

Sketch the graph of $y = - 1 - \cos x$ over one period.

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

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

The graphs of $f ( x )$ and $g ( x )$ are given. a) For what values of $x$ is $f ( x ) = g ( x )$ ? b) Find the values of $f ( - 2 )$ and $g ( 4 )$ .

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 position of a car is given by the values in the following table. t (seconds) 0 1 2 3 4 s (meters) 0 21.9 25.8 69.2 92.2 Find the average velocity for the time period beginning when $t = 2$ and lasting 2 seconds.

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.

Find the limit. $\lim _ { x \rightarrow \frac { 10 } { x } } \tan ^ { - 1 } \left( \frac { 5 } { x } \right)$