Question 1

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

Accepted Answer

A)  1 
B)  35 
C)  0 
D)  $- 6$ 
E)  $- 35$ 
A)  1 
B)  35 
C)  0 
D)  $- 6$ 
E)  $- 35$ B

Question 2

Find the domain of the function.
$$f ( x ) = \frac { 7 x + 1 } { x ^ { 2 } }$$

Accepted Answer

A)  $( - \infty , 0 )$ 
B)  $\left( - \infty , - \frac { 1 } { 7 } \right) \cup \left( - \frac { 1 } { 7 } , \infty \right)$ 
C)  $\left( - \infty , \frac { 1 } { 7 } \right) \cup \left( \frac { 1 } { 7 } , \infty \right)$ 
D)  $( - \infty , 0 ) \cup ( 0 , \infty )$ 
A)  $( - \infty , 0 )$ 
B)  $\left( - \infty , - \frac { 1 } { 7 } \right) \cup \left( - \frac { 1 } { 7 } , \infty \right)$ 
C)  $\left( - \infty , \frac { 1 } { 7 } \right) \cup \left( \frac { 1 } { 7 } , \infty \right)$ 
D)  $( - \infty , 0 ) \cup ( 0 , \infty )$ D

Question 3

Express the function in the form of $f \circ g$.
$$v ( t ) = \sec \left( t ^ { 4 } \right) \tan \left( t ^ { 4 } \right)$$

Accepted Answer

$$\begin{array} { l } 
f ( t ) = \sec ( t ) \tan ( t ) \
g ( t ) = t ^ { 4 }
\end{array}$$

Question 4

$$\text { If } f ( x ) = 4 x ^ { 2 } + 2 \text {, find and simplify } \frac { f ( 1 + h ) - f ( 1 ) } { h } \text {, where } h \neq 0$$

Accepted Answer

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

Question 5

A cardiac monitor is used to measure the heart rate of a patient after surgery. It compiles the number of heartbeats after $t$ minutes. When the data in the table are graphed, the slope of the tangent line represents the heart rate in beats per minute. The monitor estimates this value by calculating the slope of a secant line. Use the data to estimate the patient's heart rate after 42 minutes using the secant line between the points with $t = 38$ and $t = 42$.
$\begin{array}{|c|c|c|c|c|c|}
\hline t \text { (mins) } & 36 & 38 & 40 & 42 & 44 \
\hline \text { Heartbeats } & 2570 & 2720 & 2840 & 3020 & 3070 \
\hline
\end{array}$

Select the correct answer.

Accepted Answer

A)  74 
B)  80 
C)  85 
D)  70 
E)  76
F) 75 
A)  74 
B)  80 
C)  85 
D)  70 
E)  76
F) 75

Question 6

The monthly cost of driving a car depends on the number of miles driven. Julia found that in October it cost her $\$ 200$ to drive $300 \mathrm { mi }$ and in July it cost her $\$ 350$ to drive $600 \mathrm { mi }$. Express the monthly $\operatorname { cost } C$ as a function of the distance driven $d$ assuming that a linear relationship gives a suitable model.
Select the correct answer.

Accepted Answer

A)  $C = - 50 d + 0.5$ 
B)  $C = 50 d - 0.5$ 
C)  $C = 0.5 d + 50$ 
D)  $C = 2 d + 50$ 
E)  $C = 0.5 d - 50$ 
A)  $C = - 50 d + 0.5$ 
B)  $C = 50 d - 0.5$ 
C)  $C = 0.5 d + 50$ 
D)  $C = 2 d + 50$ 
E)  $C = 0.5 d - 50$

Question 7

$$\text { Use the graph to determine where the function is discontinuous. }$$

Accepted Answer

The answer of \[\text { Use the graph to determine...

Question 8

Find $a$, such that the function $f ( x ) = 4 x + \sqrt { a - x ^ { 2 } }$ has the domain $( - 4,4 )$.

Accepted Answer

A)  $a = - 16$ 
B)  $a = \sqrt { 4 }$ 
C)  $a = - \sqrt { 4 }$ 
D)  $a = 16$ 
E)  $a = 4$ 
A)  $a = - 16$ 
B)  $a = \sqrt { 4 }$ 
C)  $a = - \sqrt { 4 }$ 
D)  $a = 16$ 
E)  $a = 4$

Question 9

Evaluate the limit. $$\lim _ { x \rightarrow 9 } \frac { 3 - \sqrt { x } } { x - 9 }$$

Accepted Answer

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

Question 10

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 11

Find the domain of the function. $$f ( x ) = \sqrt { 49 - x ^ { 2 } }$$

Accepted Answer

The answer of Find the domain of the function. \[f...

Question 12

$$\text { Find the limit } \lim _ { x \rightarrow 0 } \frac { \sqrt { x + 6 } - \sqrt { 6 } } { x } \text {, if it exists. }$$

Accepted Answer

The answer of \[\text { Find the limit } \lim...

Question 13

Evaluate the limit. $$\lim _ { x \rightarrow 9 } \frac { 3 - \sqrt { x } } { x - 9 }$$

Accepted Answer

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

Question 14

Suppose that the graph of is given $f$ is given. Describe how the graph of the function $y = f ( x - 5 ) - 5$ can be obtained from the graph of $f$.

Accepted Answer

Shift the graph 5 un

Question 15

Evaluate the limit. $$\lim _ { x \rightarrow 0 } \frac { 3 - \sqrt { 3 - x ^ { 2 } } } { x }$$

Accepted Answer

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

Question 16

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

Accepted Answer

The answer of \[\text { Find the limit } \lim...

Question 17

$$\text { Find the interval(s) where } f ( x ) = \sqrt { x ^ { 2 } - 2 x + 3 } \text { is continuous. }$$

Accepted Answer

The answer of \[\text { Find the interval(s) where }...

Question 18

Find an equation of the tangent line to the given curve at the indicated point.
$$\frac { 1 } { 7 } y ^ { 2 } - x ^ { 3 } - x ^ { 2 } = 0 ; \quad ( 1 , \sqrt { 14 } )$$

Accepted Answer

The answer of Find an equation of the tangent line...

Question 19

Use continuity to evaluate the limit.
$$\lim _ { x \rightarrow 3 x } \sin ( x + 4 \sin x )$$

Accepted Answer

The answer of Use continuity to evaluate the limit.
\[\lim _...

Question 20

$$\text { Which of the given functions is discontinuous? }$$

Accepted Answer

\[f ( x ) = \left\{ \begin{arr

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

Find the domain of the function. $f ( x ) = \frac { 7 x + 1 } { x ^ { 2 } }$

Express the function in the form of $f \circ g$ . $v ( t ) = \sec \left( t ^ { 4 } \right) \tan \left( t ^ { 4 } \right)$

$\text { If } f ( x ) = 4 x ^ { 2 } + 2 \text {, find and simplify } \frac { f ( 1 + h ) - f ( 1 ) } { h } \text {, where } h \neq 0$

$\text { Use the graph to determine where the function is discontinuous. }$ $\text { Use the graph to determine where the function is discontinuous. }$

Find $a$ , such that the function $f ( x ) = 4 x + \sqrt { a - x ^ { 2 } }$ has the domain $( - 4,4 )$ .

Evaluate the limit. $\lim _ { x \rightarrow 9 } \frac { 3 - \sqrt { x } } { x - 9 }$

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 $[ 2,3 ]$ .

Find the domain of the function. $f ( x ) = \sqrt { 49 - x ^ { 2 } }$

$\text { Find the limit } \lim _ { x \rightarrow 0 } \frac { \sqrt { x + 6 } - \sqrt { 6 } } { x } \text {, if it exists. }$

Evaluate the limit. $\lim _ { x \rightarrow 9 } \frac { 3 - \sqrt { x } } { x - 9 }$

Suppose that the graph of is given $f$ is given. Describe how the graph of the function $y = f ( x - 5 ) - 5$ can be obtained from the graph of $f$ .

Evaluate the limit. $\lim _ { x \rightarrow 0 } \frac { 3 - \sqrt { 3 - x ^ { 2 } } } { x }$

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

$\text { Find the interval(s) where } f ( x ) = \sqrt { x ^ { 2 } - 2 x + 3 } \text { is continuous. }$

Use continuity to evaluate the limit. $\lim _ { x \rightarrow 3 x } \sin ( x + 4 \sin x )$

$\text { Which of the given functions is discontinuous? }$

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)≤x2−16 x - 1 \leq f ( x ) \leq x ^ { 2 } - 16x−1≤f(x)≤x2−1 , find lim⁡x→6f(x)\lim _ { x \rightarrow 6 } f ( x )limx→6​f(x)

Find the domain of the function. f(x)=7x+1x2f ( x ) = \frac { 7 x + 1 } { x ^ { 2 } }f(x)=x27x+1​

Express the function in the form of f∘gf \circ gf∘g . v(t)=sec⁡(t4)tan⁡(t4)v ( t ) = \sec \left( t ^ { 4 } \right) \tan \left( t ^ { 4 } \right)v(t)=sec(t4)tan(t4)

If f(x)=4x2+2, find and simplify f(1+h)−f(1)h, where h≠0\text { If } f ( x ) = 4 x ^ { 2 } + 2 \text {, find and simplify } \frac { f ( 1 + h ) - f ( 1 ) } { h } \text {, where } h \neq 0 If f(x)=4x2+2, find and simplify hf(1+h)−f(1)​, where h=0

Use the graph to determine where the function is discontinuous. \text { Use the graph to determine where the function is discontinuous. } Use the graph to determine where the function is discontinuous.

Find aaa , such that the function f(x)=4x+a−x2f ( x ) = 4 x + \sqrt { a - x ^ { 2 } }f(x)=4x+a−x2​ has the domain (−4,4)( - 4,4 )(−4,4) .

Evaluate the limit. lim⁡x→93−xx−9\lim _ { x \rightarrow 9 } \frac { 3 - \sqrt { x } } { x - 9 }limx→9​x−93−x​​

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] .

Find the domain of the function. f(x)=49−x2f ( x ) = \sqrt { 49 - x ^ { 2 } }f(x)=49−x2​

Find the limit lim⁡x→0x+6−6x, if it exists. \text { Find the limit } \lim _ { x \rightarrow 0 } \frac { \sqrt { x + 6 } - \sqrt { 6 } } { x } \text {, if it exists. } Find the limit limx→0​xx+6​−6​​, if it exists.

Evaluate the limit. lim⁡x→93−xx−9\lim _ { x \rightarrow 9 } \frac { 3 - \sqrt { x } } { x - 9 }limx→9​x−93−x​​

Suppose that the graph of is given fff is given. Describe how the graph of the function y=f(x−5)−5y = f ( x - 5 ) - 5y=f(x−5)−5 can be obtained from the graph of fff .

Evaluate the limit. lim⁡x→03−3−x2x\lim _ { x \rightarrow 0 } \frac { 3 - \sqrt { 3 - x ^ { 2 } } } { x }limx→0​x3−3−x2​​

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

Find the interval(s) where f(x)=x2−2x+3 is continuous. \text { Find the interval(s) where } f ( x ) = \sqrt { x ^ { 2 } - 2 x + 3 } \text { is continuous. } Find the interval(s) where f(x)=x2−2x+3​ is continuous.

Find an equation of the tangent line to the given curve at the indicated point. 17y2−x3−x2=0;(1,14)\frac { 1 } { 7 } y ^ { 2 } - x ^ { 3 } - x ^ { 2 } = 0 ; \quad ( 1 , \sqrt { 14 } )71​y2−x3−x2=0;(1,14​)

Use continuity to evaluate the limit. lim⁡x→3xsin⁡(x+4sin⁡x)\lim _ { x \rightarrow 3 x } \sin ( x + 4 \sin x )limx→3x​sin(x+4sinx)

Which of the given functions is discontinuous? \text { Which of the given functions is discontinuous? } Which of the given functions is discontinuous?

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

Find the domain of the function. $f ( x ) = \frac { 7 x + 1 } { x ^ { 2 } }$

Express the function in the form of $f \circ g$ . $v ( t ) = \sec \left( t ^ { 4 } \right) \tan \left( t ^ { 4 } \right)$

$\text { If } f ( x ) = 4 x ^ { 2 } + 2 \text {, find and simplify } \frac { f ( 1 + h ) - f ( 1 ) } { h } \text {, where } h \neq 0$

$\text { Use the graph to determine where the function is discontinuous. }$ $\text { Use the graph to determine where the function is discontinuous. }$

Find $a$ , such that the function $f ( x ) = 4 x + \sqrt { a - x ^ { 2 } }$ has the domain $( - 4,4 )$ .

Evaluate the limit. $\lim _ { x \rightarrow 9 } \frac { 3 - \sqrt { x } } { x - 9 }$

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 $[ 2,3 ]$ .

Find the domain of the function. $f ( x ) = \sqrt { 49 - x ^ { 2 } }$

$\text { Find the limit } \lim _ { x \rightarrow 0 } \frac { \sqrt { x + 6 } - \sqrt { 6 } } { x } \text {, if it exists. }$

Evaluate the limit. $\lim _ { x \rightarrow 9 } \frac { 3 - \sqrt { x } } { x - 9 }$

Suppose that the graph of is given $f$ is given. Describe how the graph of the function $y = f ( x - 5 ) - 5$ can be obtained from the graph of $f$ .

Evaluate the limit. $\lim _ { x \rightarrow 0 } \frac { 3 - \sqrt { 3 - x ^ { 2 } } } { x }$

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

$\text { Find the interval(s) where } f ( x ) = \sqrt { x ^ { 2 } - 2 x + 3 } \text { is continuous. }$

Use continuity to evaluate the limit. $\lim _ { x \rightarrow 3 x } \sin ( x + 4 \sin x )$

$\text { Which of the given functions is discontinuous? }$