Question 1

Find the limit. $\lim _ { x \rightarrow \sin x } \sin$ $x \rightarrow \frac { 3 \pi } { 4 }$

Accepted Answer

A)  $\frac { \sqrt { 3 } } { 2 }$ 
B)  $- \frac { \sqrt { 2 } } { 2 }$ 
C)  $- \frac { 1 } { 2 }$ 
D)  $\frac { \sqrt { 2 } } { 2 }$ 
E)  does not exist 
A)  $\frac { \sqrt { 3 } } { 2 }$ 
B)  $- \frac { \sqrt { 2 } } { 2 }$ 
C)  $- \frac { 1 } { 2 }$ 
D)  $\frac { \sqrt { 2 } } { 2 }$ 
E)  does not exist

Question 2

Find the limit. $\lim _ { x \rightarrow 6 } \frac { x } { x ^ { 2 } + 8 }$

Accepted Answer

A)  $\frac { 1 } { 14 }$ 
B)  $\frac { 1 } { 10 }$ 
C)  $\frac { 3 } { 22 }$ 
D)  $\frac { 3 } { 7 }$ 
E)  $\frac { 3 } { 10 }$ 
A)  $\frac { 1 } { 14 }$ 
B)  $\frac { 1 } { 10 }$ 
C)  $\frac { 3 } { 22 }$ 
D)  $\frac { 3 } { 7 }$ 
E)  $\frac { 3 } { 10 }$

Question 3

Find the limit (if it exists). $\lim x \tan \pi x$
$x \rightarrow \frac { 1 } { 2 }$

Accepted Answer

A)  $- \infty$ 
B)  $\frac { 1 } { 2 }$ 
C)  0 
D)  $\infty$ 
E)  Limit does not exist 
A)  $- \infty$ 
B)  $\frac { 1 } { 2 }$ 
C)  0 
D)  $\infty$ 
E)  Limit does not exist

Question 4

Use the graph to determine the following limits, and discuss the continuity of the function at $x = - 3$.
(i) $\lim _ { x \rightarrow - 3 ^ { + } } f ( x )$
(ii) $\lim _ { x \rightarrow - 3 ^ { - } } f ( x )$
(iii) $\lim _ { x \rightarrow - 3 } f ( x )$

Accepted Answer

A)  $1 , - 1$, does not exist, not continuous 
B)  1,0 , does not exist, not continuous 
C)  0,1 , does not exist, not continuous 
D)  $- 3,0$, does not exist, not continuous 
E)  $0,1,0$, continuous 
A)  $1 , - 1$, does not exist, not continuous 
B)  1,0 , does not exist, not continuous 
C)  0,1 , does not exist, not continuous 
D)  $- 3,0$, does not exist, not continuous 
E)  $0,1,0$, continuous

Question 5

Find the limit (if it exists). $$\lim _ { x \rightarrow 36 ^ { - } } \frac { \sqrt { x } - 6 } { x - 36 }$$

Accepted Answer

A)  0 
B)  $- \frac { 1 } { 12 }$ 
C)  $\frac { 1 } { 72 }$ 
D)  $\frac { 1 } { 12 }$ 
E)  Limit does not exist. 
A)  0 
B)  $- \frac { 1 } { 12 }$ 
C)  $\frac { 1 } { 72 }$ 
D)  $\frac { 1 } { 12 }$ 
E)  Limit does not exist.

Question 6

Use the graph as shown to determine the following limits, and discuss the continuity of the function at $x = 3$ . 
(i) $\lim _ { x \rightarrow 3 ^ { + } } f ( x )$
(ii) $\lim _ { x \rightarrow 3 ^ { - } } f ( x )$
(iii) $\lim _ { x \rightarrow 3 } f ( x )$

Accepted Answer

A)  $1,1,1$, not continuous 
B)  $2,2,2$, continuous 
C)  $4,4,4$, not continuous 
D)  $2,2,2$, not continuous 
E)  $1,1,1$, continuous 
A)  $1,1,1$, not continuous 
B)  $2,2,2$, continuous 
C)  $4,4,4$, not continuous 
D)  $2,2,2$, not continuous 
E)  $1,1,1$, continuous

Question 7

Decide whether the following problem can be solved using precalculus, or whether calculus is required. If the problem can be solved using precalculus, solve it. If the Problem seems to require calculus, use a graphical or numerical approach to estimate the Solution. A cyclist is riding on a path whose elevation is modeled by the function $f ( x ) = 0.2 x$ where $x$ and $f ( x )$ are measured in miles. Find the rate of change of elevation when $x = 5$.
$y = f ( x )$

Accepted Answer

A)  calculus, 2 
B)  precalculus, $0.2$ 
C)  calculus, $0.2$ 
D)  precalculus, 2 
E)  precalculus, $0.45$ 
A)  calculus, 2 
B)  precalculus, $0.2$ 
C)  calculus, $0.2$ 
D)  precalculus, 2 
E)  precalculus, $0.45$

Question 8

Find  the limit. $\lim _ { x \rightarrow 14 ^ { + } } \frac { x - 3 } { x - 14 }$

Accepted Answer

A)  1 
B)  $- \infty$ 
C)  0 
D)  $\infty$ 
E)  $- 1$ 
A)  1 
B)  $- \infty$ 
C)  0 
D)  $\infty$ 
E)  $- 1$

Question 9

$\text { Consider the length of the graph of } f ( x ) = 5 / x \text { from } ( 1,5 ) \text { to } ( 5,1 )$ Approximate the length of the curve by finding the sum of the lengths of four line segments, as shown in following figure. Round your answer to two decimal places.

Accepted Answer

A)  $6.11$ 
B)  $8.12$ 
C)  $5.66$ 
D)  $8.49$ 
E)  $7.11$ 
A)  $6.11$ 
B)  $8.12$ 
C)  $5.66$ 
D)  $8.49$ 
E)  $7.11$

Question 10

Complete the table and use the result to estimate the limit. $\lim _ { x \rightarrow 0 } \frac { \cos ( 3 x ) - 1 } { 3 x }$
$\begin{array}{|c|c|c|c|c|c|c|}
\hline x & -0.1 & -0.01 & -0.001 & 0.001 & 0.01 & 0.1 \
\hline f(x) & & & & & & \
\hline
\end{array}$

Accepted Answer

A)  $- 1$ 
B)  $- 0.5$ 
C)  0 
D)  $0.5$ 
E)  1 
A)  $- 1$ 
B)  $- 0.5$ 
C)  0 
D)  $0.5$ 
E)  1

Question 11

Find the value of c guaranteed by the Intermediate Value Theorem. $$f ( x ) = x ^ { 2 } - 2 x + 8 , [ 2,6 ] , f ( c ) = 11$$

Accepted Answer

A)  0 
B)  3 
C)  5 
D)  1 
E)  4 
A)  0 
B)  3 
C)  5 
D)  1 
E)  4

Question 12

$\text { Find the } x \text {-values (if any) at which } f ( x ) = \frac { x } { x ^ { 2 } - 2 x } \text { is not continuous. }$

Accepted Answer

A)  $f ( x )$ is not continuous at $x = 0$ and $f ( x )$ has a removable discontinuity at $x = 0$. 
B)  $f ( x )$ is not continuous at $x = 0,2$ and both the discontinuities are nonremovable. 
C)  $f ( x )$ is not continuous at $x = 2$ and $f ( x )$ has a removable discontinuity at $x = 2$. 
D)  $f ( x )$ is not continuous at $x = 0,2$ and $f ( x )$ has a removable discontinuity at $x = 0$. 
E)  $f ( x )$ is continuous for all real $x$. 
A)  $f ( x )$ is not continuous at $x = 0$ and $f ( x )$ has a removable discontinuity at $x = 0$. 
B)  $f ( x )$ is not continuous at $x = 0,2$ and both the discontinuities are nonremovable. 
C)  $f ( x )$ is not continuous at $x = 2$ and $f ( x )$ has a removable discontinuity at $x = 2$. 
D)  $f ( x )$ is not continuous at $x = 0,2$ and $f ( x )$ has a removable discontinuity at $x = 0$. 
E)  $f ( x )$ is continuous for all real $x$.

Question 13

Find the $x$-values (if any) at which the function
$f ( x ) = 13 x ^ { 2 } - 15 x - 15$ is not continuous. Which of the discontinuities are removable?

Accepted Answer

A)  $x = 4$, removable 
B)  $x = 0$, removable 
C)  $x = \frac { 15 } { 26 }$, not removable. 
D)  continuous everywhere 
E)  $x = \frac { 15 } { 26 }$, removable. 
A)  $x = 4$, removable 
B)  $x = 0$, removable 
C)  $x = \frac { 15 } { 26 }$, not removable. 
D)  continuous everywhere 
E)  $x = \frac { 15 } { 26 }$, removable.

Question 14

A ring has a inner circumference of 9 centimeters. If the ring's inner circumference can vary between 8 centimeters and 10 centimeters how can the radius vary? Round your answer to Five decimal places.

Accepted Answer

A) Radius can vary between 6.48456 centimeters and 10.13212 centimeters. 
B) Radius can vary between 1.59577 centimeters and 1.78412 centimeters. 
C) Radius can vary between 1.27324 centimeters and 1.59155 centimeters. 
D) Radius can vary between 2.54648 centimeters and 3.18310 centimeters. 
E) Radius can vary between 0.43239 centimeter and 2.43239 centimeters. 
A) Radius can vary between 6.48456 centimeters and 10.13212 centimeters. 
B) Radius can vary between 1.59577 centimeters and 1.78412 centimeters. 
C) Radius can vary between 1.27324 centimeters and 1.59155 centimeters. 
D) Radius can vary between 2.54648 centimeters and 3.18310 centimeters. 
E) Radius can vary between 0.43239 centimeter and 2.43239 centimeters.

Question 15

$\text { Let } f ( x ) = \left\{ \begin{array} { l l } 
x ^ { 2 } + 5 , & x \neq 1 \
1 , & x = 1
\end{array} \right.$ Determine the following limit.m(Hint: Use the graph to calculate the limit.)
$\lim _ { x \rightarrow 1 } f ( x )$

Accepted Answer

A)  6 
B)  25 
C)  1 
D)  5 
E)  does not exist. 
A)  6 
B)  25 
C)  1 
D)  5 
E)  does not exist.

Question 16

$\text { A petrol car is parked } 35 \text { feet from a long warehouse (see figure). The revolving }$ light on top of the car turns at a rate of $\frac { 1 } { 2 }$ revolution per second. The rate at which the light beam moves along the wall is $r = 35 \pi \mathrm { sec } ^ { 2 } \theta \mathrm { ft } / \mathrm { sec }$. Find the rate $r$ when $\theta$ is $\frac { \pi } { 6 }$.

Accepted Answer

A)  $r = \frac { 140 } { 3 } \mathrm { tt } / \mathrm { sec }$ 
B)  $r = \frac { 70 \sqrt { 3 } \pi } { 3 } \mathrm { ft } / \mathrm { sec }$ 
C)  $r = \frac { 70 \sqrt { 3 } } { 3 } \mathrm { ft } / \mathrm { sec }$ 
D) $r = \frac { 140 \pi } { 3 } \mathrm { ft } / \mathrm { sec }$ 
E) $r = \frac { 70 \pi } { 3 } \mathrm { ft } / \mathrm { sec }$ 
A)  $r = \frac { 140 } { 3 } \mathrm { tt } / \mathrm { sec }$ 
B)  $r = \frac { 70 \sqrt { 3 } \pi } { 3 } \mathrm { ft } / \mathrm { sec }$ 
C)  $r = \frac { 70 \sqrt { 3 } } { 3 } \mathrm { ft } / \mathrm { sec }$ 
D) $r = \frac { 140 \pi } { 3 } \mathrm { ft } / \mathrm { sec }$ 
E) $r = \frac { 70 \pi } { 3 } \mathrm { ft } / \mathrm { sec }$

Question 17

Determine the limit (if it exists). $$\lim _ { x \rightarrow 0 } \frac { \sin x ( 1 - \cos x ) } { 2 x ^ { 8 } }$$

Accepted Answer

A)  8 
B)  1 
C)  0 
D)  2 
E)  does not exist 
A)  8 
B)  1 
C)  0 
D)  2 
E)  does not exist

Question 18

Use the rectangles in the graph given below to approximate the area of the region bounded by $y = 4 / x , y = 0 , x = 1$, and $x = 4$ Round your answer to three decimal places.

Accepted Answer

A)  $2.481$ units $^ { 2 }$ 
B)  $6.371$ units $^ { 2 }$ 
C)  $3.585$ units $^ { 2 }$ 
D)  $6.872$ units $^ { 2 }$ 
E)  $6.903$ units $^ { 2 }$ 
A)  $2.481$ units $^ { 2 }$ 
B)  $6.371$ units $^ { 2 }$ 
C)  $3.585$ units $^ { 2 }$ 
D)  $6.872$ units $^ { 2 }$ 
E)  $6.903$ units $^ { 2 }$

Question 19

$\text { Find all vertical asymptotes (if any) of the function } f ( x ) = \frac { x ^ { 2 } + 4 x + 3 } { x ^ { 3 } - 4 x ^ { 2 } - x + 4 } \text {. }$

Accepted Answer

A)  $x = 4,1$ 
B)  $x = 4,1 , - 1$ 
C)  $x = - 4 , - 1$ 
D)  $x = 1$ 
E)  $x = - 1$ 
A)  $x = 4,1$ 
B)  $x = 4,1 , - 1$ 
C)  $x = - 4 , - 1$ 
D)  $x = 1$ 
E)  $x = - 1$

Question 20

Find all the vertical asymptotes (if any) of the graph of the function $$f ( x ) = \frac { 1 + x } { x ^ { 2 } ( 1 - x ) }$$

Accepted Answer

A)  $x = - 1$ 
B)  $x = 1$ 
C)  $x = 0$ 
D)  $x = 1 , x = 0$ 
E)  no vertical asymptotes 
A)  $x = - 1$ 
B)  $x = 1$ 
C)  $x = 0$ 
D)  $x = 1 , x = 0$ 
E)  no vertical asymptotes

Find the limit. $\lim _ { x \rightarrow \sin x } \sin$ $x \rightarrow \frac { 3 \pi } { 4 }$

Find the limit. $\lim _ { x \rightarrow 6 } \frac { x } { x ^ { 2 } + 8 }$

Find the limit (if it exists). $\lim x \tan \pi x$ $x \rightarrow \frac { 1 } { 2 }$

Find the limit (if it exists). $\lim _ { x \rightarrow 36 ^ { - } } \frac { \sqrt { x } - 6 } { x - 36 }$

Find the limit. $\lim _ { x \rightarrow 14 ^ { + } } \frac { x - 3 } { x - 14 }$

Complete the table and use the result to estimate the limit. $\lim _ { x \rightarrow 0 } \frac { \cos ( 3 x ) - 1 } { 3 x }$ x -0.1 -0.01 -0.001 0.001 0.01 0.1 f(x)

Find the value of c guaranteed by the Intermediate Value Theorem. $f ( x ) = x ^ { 2 } - 2 x + 8 , [ 2,6 ] , f ( c ) = 11$

$\text { Find the } x \text {-values (if any) at which } f ( x ) = \frac { x } { x ^ { 2 } - 2 x } \text { is not continuous. }$

Find the $x$ -values (if any) at which the function $f ( x ) = 13 x ^ { 2 } - 15 x - 15$ is not continuous. Which of the discontinuities are removable?

A ring has a inner circumference of 9 centimeters. If the ring's inner circumference can vary between 8 centimeters and 10 centimeters how can the radius vary? Round your answer to Five decimal places.

Determine the limit (if it exists). $\lim _ { x \rightarrow 0 } \frac { \sin x ( 1 - \cos x ) } { 2 x ^ { 8 } }$

Use the rectangles in the graph given below to approximate the area of the region bounded by $y = 4 / x , y = 0 , x = 1$ , and $x = 4$ Round your answer to three decimal places.

$\text { Find all vertical asymptotes (if any) of the function } f ( x ) = \frac { x ^ { 2 } + 4 x + 3 } { x ^ { 3 } - 4 x ^ { 2 } - x + 4 } \text {. }$

Find all the vertical asymptotes (if any) of the graph of the function $f ( x ) = \frac { 1 + x } { x ^ { 2 } ( 1 - x ) }$

Graphs and Models

The Derivative and the Tangent Line Problem

Extrema on an Interval

Antiderivatives and Indefinite Integration

Slope Fields and Eulers Method

Area of a Region Between Two Curves

Basic Integration Rules

Sequences

Conics and Calculus

Vectors in the Plane

Vector-Valued Functions

Introduction to Functions of Several Variables

Iterated Integrals and Area in the Plane

Vector Fields

Exact First-Order Equations

Filters

Exam 2: A Preview of Calculus

Find the limit. lim⁡x→sin⁡xsin⁡\lim _ { x \rightarrow \sin x } \sinlimx→sinx​sin x→3π4x \rightarrow \frac { 3 \pi } { 4 }x→43π​

Find the limit. lim⁡x→6xx2+8\lim _ { x \rightarrow 6 } \frac { x } { x ^ { 2 } + 8 }limx→6​x2+8x​

Find the limit (if it exists). lim⁡xtan⁡πx\lim x \tan \pi xlimxtanπx x→12x \rightarrow \frac { 1 } { 2 }x→21​

Find the limit (if it exists). lim⁡x→36−x−6x−36\lim _ { x \rightarrow 36 ^ { - } } \frac { \sqrt { x } - 6 } { x - 36 }limx→36−​x−36x​−6​

Find the limit. lim⁡x→14+x−3x−14\lim _ { x \rightarrow 14 ^ { + } } \frac { x - 3 } { x - 14 }limx→14+​x−14x−3​

Complete the table and use the result to estimate the limit. lim⁡x→0cos⁡(3x)−13x\lim _ { x \rightarrow 0 } \frac { \cos ( 3 x ) - 1 } { 3 x }limx→0​3xcos(3x)−1​ x -0.1 -0.01 -0.001 0.001 0.01 0.1 f(x)

Find the value of c guaranteed by the Intermediate Value Theorem. f(x)=x2−2x+8,[2,6],f(c)=11f ( x ) = x ^ { 2 } - 2 x + 8 , [ 2,6 ] , f ( c ) = 11f(x)=x2−2x+8,[2,6],f(c)=11

Find the x-values (if any) at which f(x)=xx2−2x is not continuous. \text { Find the } x \text {-values (if any) at which } f ( x ) = \frac { x } { x ^ { 2 } - 2 x } \text { is not continuous. } Find the x-values (if any) at which f(x)=x2−2xx​ is not continuous.

Find the xxx -values (if any) at which the function f(x)=13x2−15x−15f ( x ) = 13 x ^ { 2 } - 15 x - 15f(x)=13x2−15x−15 is not continuous. Which of the discontinuities are removable?

A ring has a inner circumference of 9 centimeters. If the ring's inner circumference can vary between 8 centimeters and 10 centimeters how can the radius vary? Round your answer to Five decimal places.

Determine the limit (if it exists). lim⁡x→0sin⁡x(1−cos⁡x)2x8\lim _ { x \rightarrow 0 } \frac { \sin x ( 1 - \cos x ) } { 2 x ^ { 8 } }limx→0​2x8sinx(1−cosx)​

Use the rectangles in the graph given below to approximate the area of the region bounded by y=4/x,y=0,x=1y = 4 / x , y = 0 , x = 1y=4/x,y=0,x=1 , and x=4x = 4x=4 Round your answer to three decimal places.

Find all the vertical asymptotes (if any) of the graph of the function f(x)=1+xx2(1−x)f ( x ) = \frac { 1 + x } { x ^ { 2 } ( 1 - x ) }f(x)=x2(1−x)1+x​

Graphs and Models

The Derivative and the Tangent Line Problem

Extrema on an Interval

Antiderivatives and Indefinite Integration

Slope Fields and Eulers Method

Area of a Region Between Two Curves

Basic Integration Rules

Sequences

Conics and Calculus

Vectors in the Plane

Vector-Valued Functions

Introduction to Functions of Several Variables

Iterated Integrals and Area in the Plane

Vector Fields

Exact First-Order Equations

Filters

Find the limit. $\lim _ { x \rightarrow \sin x } \sin$ $x \rightarrow \frac { 3 \pi } { 4 }$

Find the limit. $\lim _ { x \rightarrow 6 } \frac { x } { x ^ { 2 } + 8 }$

Find the limit (if it exists). $\lim x \tan \pi x$ $x \rightarrow \frac { 1 } { 2 }$

Find the limit (if it exists). $\lim _ { x \rightarrow 36 ^ { - } } \frac { \sqrt { x } - 6 } { x - 36 }$

Find the limit. $\lim _ { x \rightarrow 14 ^ { + } } \frac { x - 3 } { x - 14 }$

Complete the table and use the result to estimate the limit. $\lim _ { x \rightarrow 0 } \frac { \cos ( 3 x ) - 1 } { 3 x }$ x -0.1 -0.01 -0.001 0.001 0.01 0.1 f(x)

Find the value of c guaranteed by the Intermediate Value Theorem. $f ( x ) = x ^ { 2 } - 2 x + 8 , [ 2,6 ] , f ( c ) = 11$

$\text { Find the } x \text {-values (if any) at which } f ( x ) = \frac { x } { x ^ { 2 } - 2 x } \text { is not continuous. }$

Find the $x$ -values (if any) at which the function $f ( x ) = 13 x ^ { 2 } - 15 x - 15$ is not continuous. Which of the discontinuities are removable?

Determine the limit (if it exists). $\lim _ { x \rightarrow 0 } \frac { \sin x ( 1 - \cos x ) } { 2 x ^ { 8 } }$

Use the rectangles in the graph given below to approximate the area of the region bounded by $y = 4 / x , y = 0 , x = 1$ , and $x = 4$ Round your answer to three decimal places.

Find all the vertical asymptotes (if any) of the graph of the function $f ( x ) = \frac { 1 + x } { x ^ { 2 } ( 1 - x ) }$