Question 1

Use the figure below to approximate the slope of the curve at the point $( x , y )$.

Accepted Answer

A)  $- \frac { 1 } { 2 }$ 
B)  The slope is undefined. 
C)  1 
D)  $\frac { 1 } { 2 }$ 
E)  0 
A)  $- \frac { 1 } { 2 }$ 
B)  The slope is undefined. 
C)  1 
D)  $\frac { 1 } { 2 }$ 
E)  0

Question 2

Find
$\lim _ { x \rightarrow 6 } \frac { \sqrt { x + 7 } } { x - 2 }$
by direct substitution.

Accepted Answer

A)  $\frac { \sqrt { 6 } } { 4 }$ 
B)  $\frac { \sqrt { 13 } } { 2 }$ 
C)  $\frac { \sqrt { 13 } } { 4 }$ 
D)  $\frac { 1 } { 4 }$ 
E)  $\sqrt { 13 }$ 
A)  $\frac { \sqrt { 6 } } { 4 }$ 
B)  $\frac { \sqrt { 13 } } { 2 }$ 
C)  $\frac { \sqrt { 13 } } { 4 }$ 
D)  $\frac { 1 } { 4 }$ 
E)  $\sqrt { 13 }$

Question 3

Find $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ for $f ( x ) = 8 - x$.

Accepted Answer

A)  1 
B)  9 
C)  $- 1$ 
D)  8 
E)  limit does not exist 
A)  1 
B)  9 
C)  $- 1$ 
D)  8 
E)  limit does not exist

Question 4

Find $\lim _ { x \rightarrow - \infty } \left[ \frac { x } { ( 5 + x ) ^ { 2 } } + 7 \right]$ (if it exists).

Accepted Answer

A)  $\frac { 8 } { 25 }$ 
B)  $- 7$ 
C)  7 
D)  $\infty$ 
E)  limit does not exist 
A)  $\frac { 8 } { 25 }$ 
B)  $- 7$ 
C)  7 
D)  $\infty$ 
E)  limit does not exist

Question 5

Find the following limit, if it exists.
$\lim _{x \rightarrow-2} x^{8}$

Accepted Answer

A)  $ -2 $ 
B)  $ -16 $ 
C)  2 
D)  256 
E)  The limit does not exist. 
A)  $ -2 $ 
B)  $ -16 $ 
C)  2 
D)  256 
E)  The limit does not exist.

Question 6

Find the following limit, if it exists.
$$\lim _ { x \rightarrow \infty } \frac { 3 } { x ^ { - 6 } }$$

Accepted Answer

A)  $\frac { 1 } { 2 }$ 
B)  $- 1$ 
C)  3 
D)  The limit does not exist. 
E)  0 
A)  $\frac { 1 } { 2 }$ 
B)  $- 1$ 
C)  3 
D)  The limit does not exist. 
E)  0

Question 7

$\text { Use the graph to find } \lim _{x \rightarrow 2} \frac{3 x^{2}-12}{x-2}$

Accepted Answer

A)  12 
B)  $ \infty $ 
C)  6 
D)  0 
E)  limit does not exist 
A)  12 
B)  $ \infty $ 
C)  6 
D)  0 
E)  limit does not exist

Question 8

Find the following limit, if it exists.
$$\lim _ { x \rightarrow - 3 } x ^ { 5 }$$

Accepted Answer

A)  $- 3$ 
B)  $- 15$ 
C)  3 
D)  $- 243$ 
E)  The limit does not exist. 
A)  $- 3$ 
B)  $- 15$ 
C)  3 
D)  $- 243$ 
E)  The limit does not exist.

Question 9

Find the following limit of the sequence as $n$ approaches infinity, if it exists.
$$a _ { n } = \frac { 1 } { n } \left( n + \frac { 4 } { n } \left[ \frac { n ( n + 1 ) } { 6 } \right] \right)$$

Accepted Answer

A)  $\frac { 2 } { 3 }$ 
B)  $\frac { 8 } { 3 }$ 
C)  $\frac { 5 } { 3 }$ 
D)  $\frac { 11 } { 6 }$ 
E)  The limit does not exist. 
A)  $\frac { 2 } { 3 }$ 
B)  $\frac { 8 } { 3 }$ 
C)  $\frac { 5 } { 3 }$ 
D)  $\frac { 11 } { 6 }$ 
E)  The limit does not exist.

Question 10

Use the graph to find $\lim _ { x \rightarrow 3 } \frac { 4 x ^ { 2 } - 36 } { x - 3 }$.

Accepted Answer

A)  24 
B)  $\infty$ 
C)  12 
D)  0 
E)  limit does not exist 
A)  24 
B)  $\infty$ 
C)  12 
D)  0 
E)  limit does not exist

Question 11

Find the derivative of $ f(x)=7 x^{2}-9 x+4 $

Accepted Answer

A)  $ 14 x-9 $ 
B)  $ 7 x-9 $ 
C)  $ 14 x-5 $ 
D)  $ 7 x-5 $ 
E)  $ 14 x $ 
A)  $ 14 x-9 $ 
B)  $ 7 x-9 $ 
C)  $ 14 x-5 $ 
D)  $ 7 x-5 $ 
E)  $ 14 x $

Question 12

Using the summation formulas and properties, evaluate the following expression.
$$\sum _ { i = 1 } ^ { 30 } 8$$

Accepted Answer

A)  3720 
B)  465 
C)  232 
D)  240 
E)  38 
A)  3720 
B)  465 
C)  232 
D)  240 
E)  38

Question 13

Use the derivative of $f ( x ) = 5 x ^ { 3 } + 15 x$ to determine any points on the graph of $f ( x )$ at which the tangent line is horizontal.

Accepted Answer

A)  $( 1,20 )$ 
B)  $( 1,20 )$ and $( - 1 , - 20 )$ 
C)  $( 5,700 )$ and $( - 5,700 )$ 
D)  $( 0,0 )$ 
E)  $f ( x )$ has no points with a horizontal tangent line. 
A)  $( 1,20 )$ 
B)  $( 1,20 )$ and $( - 1 , - 20 )$ 
C)  $( 5,700 )$ and $( - 5,700 )$ 
D)  $( 0,0 )$ 
E)  $f ( x )$ has no points with a horizontal tangent line.

Question 14

Find $\lim _{x \rightarrow 3}[g(x)-f(x)]$ for $ f(x)=3 x^{3} $ and $ g(x)=\frac{\sqrt{x^{2}+2}}{5 x^{2}} $.

Accepted Answer

A)  $ \frac{\sqrt{11}}{45}-81 $ 
B)  $ \frac{\sqrt{3}}{5}-81 $ 
C)  $ \frac{9 \sqrt{11}}{5} $ 
D)  $ \frac{\sqrt{3}}{5}+81 $ 
E)  limit does not exist 
A)  $ \frac{\sqrt{11}}{45}-81 $ 
B)  $ \frac{\sqrt{3}}{5}-81 $ 
C)  $ \frac{9 \sqrt{11}}{5} $ 
D)  $ \frac{\sqrt{3}}{5}+81 $ 
E)  limit does not exist

Question 15

Find an equation of the tangent line to the graph of the following function at the point $( 3 , - 30 )$.
$$- 3 x ^ { 2 } - 3$$

Accepted Answer

A)  $y = - 18 x - 84$ 
B)  $y = - 18 x + 84$ 
C)  $y = - 18 x + 24$ 
D)  $y = 180 x + 570$ 
E)  $y = 180 x - 570$ 
A)  $y = - 18 x - 84$ 
B)  $y = - 18 x + 84$ 
C)  $y = - 18 x + 24$ 
D)  $y = 180 x + 570$ 
E)  $y = 180 x - 570$

Question 16

Find the following limit, if it exists.
$$\lim _ { x \rightarrow - 3 } \frac { 9 - 6 x - 3 x ^ { 2 } } { 3 + x }$$

Accepted Answer

A)  $- 6$ 
B)  12 
C)  6 
D)  $- 12$ 
E)  The limit does not exist. 
A)  $- 6$ 
B)  12 
C)  6 
D)  $- 12$ 
E)  The limit does not exist.

Question 17

Use the first six terms to predict the limit of the sequence $ a_{n}=\frac{8 n^{3}+6}{n+3} $ (assume $ n $ begins with 1).

Accepted Answer

A)  $ \frac{8}{3} $ 
B)  183 
C)  126 
D)  0 
E)  the sequence diverges 
A)  $ \frac{8}{3} $ 
B)  183 
C)  126 
D)  0 
E)  the sequence diverges

Question 18

Evaluate $\sum _ { j = 1 } ^ { 10 } \left( j ^ { 3 } - 4 j ^ { 2 } \right)$ using the summation formulas and properties.

Accepted Answer

A)  3,025 
B)  2,625 
C)  3,021 
D)  2,985 
E)  1,485 
A)  3,025 
B)  2,625 
C)  3,021 
D)  2,985 
E)  1,485

Question 19

Estimate the following limit numerically, if it exists.
$$\lim _ { x \rightarrow 1 } \frac { x - 1 } { x ^ { 2 } + 4 x - 5 }$$

Accepted Answer

A)  $\frac { 1 } { 6 }$ 
B)  0 
C)  $- \frac { 1 } { 4 }$ 
D)  $\frac { 1 } { 4 }$ 
E)  The limit does not exist. 
A)  $\frac { 1 } { 6 }$ 
B)  0 
C)  $- \frac { 1 } { 4 }$ 
D)  $\frac { 1 } { 4 }$ 
E)  The limit does not exist.

Question 20

Use the graph below to find $\lim _ { x \rightarrow - 2 } \frac { | x + 2 | } { x + 2 }$, if it exists.

Accepted Answer

A)  $- 1$ 
B)  $- 2$ 
C)  2 
D)  1 
E)  The limit does not exist. 
A)  $- 1$ 
B)  $- 2$ 
C)  2 
D)  1 
E)  The limit does not exist.

Use the figure below to approximate the slope of the curve at the point $( x , y )$ .

Find $\lim _ { x \rightarrow 6 } \frac { \sqrt { x + 7 } } { x - 2 }$ by direct substitution.

Find $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ for $f ( x ) = 8 - x$ .

Find $\lim _ { x \rightarrow - \infty } \left[ \frac { x } { ( 5 + x ) ^ { 2 } } + 7 \right]$ (if it exists).

Find the following limit, if it exists. $\lim _{x \rightarrow-2} x^{8}$

Find the following limit, if it exists. $\lim _ { x \rightarrow \infty } \frac { 3 } { x ^ { - 6 } }$

$\text { Use the graph to find } \lim _{x \rightarrow 2} \frac{3 x^{2}-12}{x-2}$ $\text { Use the graph to find } \lim _{x \rightarrow 2} \frac{3 x^{2}-12}{x-2}$

Find the following limit, if it exists. $\lim _ { x \rightarrow - 3 } x ^ { 5 }$

Find the following limit of the sequence as $n$ approaches infinity, if it exists. $a _ { n } = \frac { 1 } { n } \left( n + \frac { 4 } { n } \left[ \frac { n ( n + 1 ) } { 6 } \right] \right)$

Use the graph to find $\lim _ { x \rightarrow 3 } \frac { 4 x ^ { 2 } - 36 } { x - 3 }$ . $Use the graph to find \lim _ { x \rightarrow 3 } \frac { 4 x ^ { 2 } - 36 } { x - 3 } .$

Find the derivative of $f(x)=7 x^{2}-9 x+4$

Using the summation formulas and properties, evaluate the following expression. $\sum _ { i = 1 } ^ { 30 } 8$

Use the derivative of $f ( x ) = 5 x ^ { 3 } + 15 x$ to determine any points on the graph of $f ( x )$ at which the tangent line is horizontal.

Find $\lim _{x \rightarrow 3}[g(x)-f(x)]$ for $f(x)=3 x^{3}$ and $g(x)=\frac{\sqrt{x^{2}+2}}{5 x^{2}}$ .

Find an equation of the tangent line to the graph of the following function at the point $( 3 , - 30 )$ . $- 3 x ^ { 2 } - 3$

Find the following limit, if it exists. $\lim _ { x \rightarrow - 3 } \frac { 9 - 6 x - 3 x ^ { 2 } } { 3 + x }$

Use the first six terms to predict the limit of the sequence $a_{n}=\frac{8 n^{3}+6}{n+3}$ (assume $n$ begins with 1).

Evaluate $\sum _ { j = 1 } ^ { 10 } \left( j ^ { 3 } - 4 j ^ { 2 } \right)$ using the summation formulas and properties.

Estimate the following limit numerically, if it exists. $\lim _ { x \rightarrow 1 } \frac { x - 1 } { x ^ { 2 } + 4 x - 5 }$

Use the graph below to find $\lim _ { x \rightarrow - 2 } \frac { | x + 2 | } { x + 2 }$ , if it exists. $Use the graph below to find \lim _ { x \rightarrow - 2 } \frac { | x + 2 | } { x + 2 } , if it exists.$

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Additional Topics in Trigonometry

Linear Systems and Matrices

Sequences, Series, and Probability

Topics in Analytic Geometry

Analytic Geometry in Three Dimensions

Filters

Exam 11: Calculus Practice Problems

Use the figure below to approximate the slope of the curve at the point (x,y)( x , y )(x,y) .

Find lim⁡x→6x+7x−2\lim _ { x \rightarrow 6 } \frac { \sqrt { x + 7 } } { x - 2 }limx→6​x−2x+7​​ by direct substitution.

Find lim⁡h→0f(x+h)−f(x)h\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }limh→0​hf(x+h)−f(x)​ for f(x)=8−xf ( x ) = 8 - xf(x)=8−x .

Find lim⁡x→−∞[x(5+x)2+7]\lim _ { x \rightarrow - \infty } \left[ \frac { x } { ( 5 + x ) ^ { 2 } } + 7 \right]limx→−∞​[(5+x)2x​+7] (if it exists).

Find the following limit, if it exists. lim⁡x→−2x8\lim _{x \rightarrow-2} x^{8}limx→−2​x8

Find the following limit, if it exists. lim⁡x→∞3x−6\lim _ { x \rightarrow \infty } \frac { 3 } { x ^ { - 6 } }limx→∞​x−63​

Use the graph to find lim⁡x→23x2−12x−2\text { Use the graph to find } \lim _{x \rightarrow 2} \frac{3 x^{2}-12}{x-2} Use the graph to find limx→2​x−23x2−12​

Find the following limit, if it exists. lim⁡x→−3x5\lim _ { x \rightarrow - 3 } x ^ { 5 }limx→−3​x5

Find the following limit of the sequence as nnn approaches infinity, if it exists. an=1n(n+4n[n(n+1)6])a _ { n } = \frac { 1 } { n } \left( n + \frac { 4 } { n } \left[ \frac { n ( n + 1 ) } { 6 } \right] \right)an​=n1​(n+n4​[6n(n+1)​])

Use the graph to find lim⁡x→34x2−36x−3\lim _ { x \rightarrow 3 } \frac { 4 x ^ { 2 } - 36 } { x - 3 }limx→3​x−34x2−36​ .

Find the derivative of f(x)=7x2−9x+4 f(x)=7 x^{2}-9 x+4 f(x)=7x2−9x+4

Using the summation formulas and properties, evaluate the following expression. ∑i=1308\sum _ { i = 1 } ^ { 30 } 8∑i=130​8

Use the derivative of f(x)=5x3+15xf ( x ) = 5 x ^ { 3 } + 15 xf(x)=5x3+15x to determine any points on the graph of f(x)f ( x )f(x) at which the tangent line is horizontal.

Find lim⁡x→3[g(x)−f(x)]\lim _{x \rightarrow 3}[g(x)-f(x)]limx→3​[g(x)−f(x)] for f(x)=3x3 f(x)=3 x^{3} f(x)=3x3 and g(x)=x2+25x2 g(x)=\frac{\sqrt{x^{2}+2}}{5 x^{2}} g(x)=5x2x2+2​​ .

Find an equation of the tangent line to the graph of the following function at the point (3,−30)( 3 , - 30 )(3,−30) . −3x2−3- 3 x ^ { 2 } - 3−3x2−3

Find the following limit, if it exists. lim⁡x→−39−6x−3x23+x\lim _ { x \rightarrow - 3 } \frac { 9 - 6 x - 3 x ^ { 2 } } { 3 + x }limx→−3​3+x9−6x−3x2​

Use the first six terms to predict the limit of the sequence an=8n3+6n+3 a_{n}=\frac{8 n^{3}+6}{n+3} an​=n+38n3+6​ (assume n n n begins with 1).

Evaluate ∑j=110(j3−4j2)\sum _ { j = 1 } ^ { 10 } \left( j ^ { 3 } - 4 j ^ { 2 } \right)∑j=110​(j3−4j2) using the summation formulas and properties.

Estimate the following limit numerically, if it exists. lim⁡x→1x−1x2+4x−5\lim _ { x \rightarrow 1 } \frac { x - 1 } { x ^ { 2 } + 4 x - 5 }limx→1​x2+4x−5x−1​

Use the graph below to find lim⁡x→−2∣x+2∣x+2\lim _ { x \rightarrow - 2 } \frac { | x + 2 | } { x + 2 }limx→−2​x+2∣x+2∣​ , if it exists.

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Additional Topics in Trigonometry

Linear Systems and Matrices

Sequences, Series, and Probability

Topics in Analytic Geometry

Analytic Geometry in Three Dimensions

Filters

Use the figure below to approximate the slope of the curve at the point $( x , y )$ .

Find $\lim _ { x \rightarrow 6 } \frac { \sqrt { x + 7 } } { x - 2 }$ by direct substitution.

Find $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ for $f ( x ) = 8 - x$ .

Find $\lim _ { x \rightarrow - \infty } \left[ \frac { x } { ( 5 + x ) ^ { 2 } } + 7 \right]$ (if it exists).

Find the following limit, if it exists. $\lim _{x \rightarrow-2} x^{8}$

Find the following limit, if it exists. $\lim _ { x \rightarrow \infty } \frac { 3 } { x ^ { - 6 } }$

$\text { Use the graph to find } \lim _{x \rightarrow 2} \frac{3 x^{2}-12}{x-2}$ $\text { Use the graph to find } \lim _{x \rightarrow 2} \frac{3 x^{2}-12}{x-2}$

Find the following limit, if it exists. $\lim _ { x \rightarrow - 3 } x ^ { 5 }$

Find the following limit of the sequence as $n$ approaches infinity, if it exists. $a _ { n } = \frac { 1 } { n } \left( n + \frac { 4 } { n } \left[ \frac { n ( n + 1 ) } { 6 } \right] \right)$

Use the graph to find $\lim _ { x \rightarrow 3 } \frac { 4 x ^ { 2 } - 36 } { x - 3 }$ . $Use the graph to find \lim _ { x \rightarrow 3 } \frac { 4 x ^ { 2 } - 36 } { x - 3 } .$

Find the derivative of $f(x)=7 x^{2}-9 x+4$

Using the summation formulas and properties, evaluate the following expression. $\sum _ { i = 1 } ^ { 30 } 8$

Use the derivative of $f ( x ) = 5 x ^ { 3 } + 15 x$ to determine any points on the graph of $f ( x )$ at which the tangent line is horizontal.

Find $\lim _{x \rightarrow 3}[g(x)-f(x)]$ for $f(x)=3 x^{3}$ and $g(x)=\frac{\sqrt{x^{2}+2}}{5 x^{2}}$ .

Find an equation of the tangent line to the graph of the following function at the point $( 3 , - 30 )$ . $- 3 x ^ { 2 } - 3$

Find the following limit, if it exists. $\lim _ { x \rightarrow - 3 } \frac { 9 - 6 x - 3 x ^ { 2 } } { 3 + x }$

Use the first six terms to predict the limit of the sequence $a_{n}=\frac{8 n^{3}+6}{n+3}$ (assume $n$ begins with 1).

Evaluate $\sum _ { j = 1 } ^ { 10 } \left( j ^ { 3 } - 4 j ^ { 2 } \right)$ using the summation formulas and properties.

Estimate the following limit numerically, if it exists. $\lim _ { x \rightarrow 1 } \frac { x - 1 } { x ^ { 2 } + 4 x - 5 }$

Use the graph below to find $\lim _ { x \rightarrow - 2 } \frac { | x + 2 | } { x + 2 }$ , if it exists. $Use the graph below to find \lim _ { x \rightarrow - 2 } \frac { | x + 2 | } { x + 2 } , if it exists.$