Question 1

Find the limit. $$\lim _ { x \rightarrow 2 } \frac { x ^ { 2 } + 2 x - 12 } { x - 2 }$$

Accepted Answer

6

Question 2

Determine whether $f$ is even, odd, or neither.
$$f ( x ) = \frac { 8 x ^ { 2 } } { x ^ { 4 } + 1 }$$

Accepted Answer

even

Question 3

Graph the function by hand, not by plotting points, but by starting with the graph of one of the
standard functions and then applying the appropriate transformations. $$y = 4 + 2 x - x ^ { 2 }$$

Accepted Answer

The answer of Graph the function by hand, not by...

Question 4

Find a function $g$ that agrees with $f$ for $x \neq 25$ and is continuous on $\Re$.
$$f ( x ) = \frac { 5 - \sqrt { x } } { 25 - x }$$

Accepted Answer

The answer of Find a function $g$ that agrees with...

Question 5

Consider the following function. $$f ( x ) = \left\{ \begin{array} { c c } 
3 - x & x < - 1 \
x & - 1 \leq x < 3 \
( x - 3 ) ^ { 2 } & x \geq 3
\end{array} \right.$$
Determine the values of $a$ for which $\lim _ { x \rightarrow a } f ( x )$ exists.

Accepted Answer

The answer of Consider the following function. \[f ( x...

Question 6

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)  $( - \infty , 0 ) \cup ( 0 , \infty )$

Question 7

Find the value of $\lim _ { x \rightarrow 0 ^ { + } } f ( x )$
$f ( x ) = \frac { 1 } { 1 + 6 ^ { 1 / x } }$

Accepted Answer

A)  0 
B)  $- 0.7$ 
C)  $- 0.7$. 
D)  $- 0.6$. 
E)  $0.16$. 
A)  0 
B)  $- 0.7$ 
C)  $- 0.7$. 
D)  $- 0.6$. 
E)  $0.16$.

Question 8

How would you define $f ( 7 )$ in order to make $f$ continuous at 7 ?
$f ( x ) = \frac { x ^ { 2 } - 2 x - 3 } { x - 7 }$

Accepted Answer

A)  $f ( 3 ) = - 8$ 
B)  $f ( 7 ) = 12$ 
C)  $f ( 7 ) = 0$ 
D)  $f ( 7 ) = - 12$ 
E)  None of these 
A)  $f ( 3 ) = - 8$ 
B)  $f ( 7 ) = 12$ 
C)  $f ( 7 ) = 0$ 
D)  $f ( 7 ) = - 12$ 
E)  None of these

Question 9

Determine whether $f$ is even, odd, or neither.
$f ( x ) = \frac { 4 x ^ { 2 } } { x ^ { 4 } + 5 }$

Accepted Answer

A)  neither 
B)  odd 
C)  even 
A)  neither 
B)  odd 
C)  even

Question 10

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 11

Which of the given functions is discontinuous?

Accepted Answer

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

Question 12

Find the range of the function. $y = 2 + \cos x$

Accepted Answer

The answer of Find the range of the function. \(y...

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

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 15

Find the range of the function. $$y = 2 + \cos x$$
Select the correct answer.

Accepted Answer

A)  $( - \infty , \infty )$ 
B)  $( 2 , \infty )$ 
C)  $[ - 1,1 ]$ 
D)  $( - 1,3 )$ 
E)  $[ 1,3 ]$ 
A)  $( - \infty , \infty )$ 
B)  $( 2 , \infty )$ 
C)  $[ - 1,1 ]$ 
D)  $( - 1,3 )$ 
E)  $[ 1,3 ]$

Question 16

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

A)  both 1 
B)  2 and 1 
C)  2 and $- 1$ 
D)  1 and $- 1$ 
E)  both-1 
A)  both 1 
B)  2 and 1 
C)  2 and $- 1$ 
D)  1 and $- 1$ 
E)  both-1

Question 17

Let
$f ( x ) = \left\{ \begin{array} { c c } 
x - 4 & \text { if } x \leq 5 \
k x ^ { 2 } - 24 x + 46 & \text { if } x > 5
\end{array} \right.$
Find the value of $k$ that will make $f$ continuous on $( - \infty , \infty )$.

Accepted Answer

A)  5 
B)  46 
C)  $- 4$ 
D)  3 
A)  5 
B)  46 
C)  $- 4$ 
D)  3

Question 18

The graph of the function $f ( x ) = x ^ { 2 } - 11 x + 7$ has been stretched horizontally by a factor of 2 . Find the function for the transformed graph.

Accepted Answer

A)  $g ( x ) = \frac { x ^ { 2 } - 11 x + 7 } { 2 }$ 
B)  $g ( x ) = 2 x ^ { 2 } - 22 x + 14$ 
C)  $g ( x ) = \frac { x ^ { 2 } - 22 x + 28 } { 4 }$ 
D)  $g ( x ) = 4 x ^ { 2 } - 22 x + 7$ 
A)  $g ( x ) = \frac { x ^ { 2 } - 11 x + 7 } { 2 }$ 
B)  $g ( x ) = 2 x ^ { 2 } - 22 x + 14$ 
C)  $g ( x ) = \frac { x ^ { 2 } - 22 x + 28 } { 4 }$ 
D)  $g ( x ) = 4 x ^ { 2 } - 22 x + 7$

Question 19

You are given $\lim _ { x \rightarrow a } f ( x ) = L$ and a tolerance $\varepsilon$. Find a number $\delta$ such that $| f ( x ) - L | < \varepsilon$ whenever $0 < | x - a | < \delta$.
$\lim _ { x \rightarrow 3 } 4 x = 12 ; \quad \varepsilon = 0.01$

Accepted Answer

The answer of You are given \(\lim _ { x...

Question 20

Estimate the value of the following limit by graphing the function $f ( x ) = \frac { ( 5 \sin x ) } { ( \sin \pi x ) }$. $\lim _ { x \rightarrow 0 } \frac { 5 \sin x } { \sin \pi x }$ Round your answer correct to two decimal places.

Accepted Answer

The answer of Estimate the value of the following limit...

Find the limit. $\lim _ { x \rightarrow 2 } \frac { x ^ { 2 } + 2 x - 12 } { x - 2 }$

Determine whether $f$ is even, odd, or neither. $f ( x ) = \frac { 8 x ^ { 2 } } { x ^ { 4 } + 1 }$

Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions and then applying the appropriate transformations. $y = 4 + 2 x - x ^ { 2 }$

Find a function $g$ that agrees with $f$ for $x \neq 25$ and is continuous on $\Re$ . $f ( x ) = \frac { 5 - \sqrt { x } } { 25 - x }$

Consider the following function. $f ( x ) = \left\{ \begin{array} { c c } 3 - x & x < - 1 \\x & - 1 \leq x < 3 \\( x - 3 ) ^ { 2 } & x \geq 3\end{array} \right.$ Determine the values of $a$ for which $\lim _ { x \rightarrow a } f ( x )$ exists.

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

Find the value of $\lim _ { x \rightarrow 0 ^ { + } } f ( x )$ $f ( x ) = \frac { 1 } { 1 + 6 ^ { 1 / x } }$

How would you define $f ( 7 )$ in order to make $f$ continuous at 7 ? $f ( x ) = \frac { x ^ { 2 } - 2 x - 3 } { x - 7 }$

Determine whether $f$ is even, odd, or neither. $f ( x ) = \frac { 4 x ^ { 2 } } { x ^ { 4 } + 5 }$

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

Which of the given functions is discontinuous?

Find the range of the function. $y = 2 + \cos x$

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

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

Find the range of the function. $y = 2 + \cos x$ Select the correct answer.

Let $F ( x ) = \frac { x - 5 } { | x - 5 | }$ . Find the following limits. $\lim _ { x \rightarrow 5 ^ { + } } F ( x ) , \lim _ { x \rightarrow 5 ^ { - } } F ( x )$

Let $f ( x ) = \left\{ \begin{array} { c c } x - 4 & \text { if } x \leq 5 \\k x ^ { 2 } - 24 x + 46 & \text { if } x > 5\end{array} \right.$ Find the value of $k$ that will make $f$ continuous on $( - \infty , \infty )$ .

The graph of the function $f ( x ) = x ^ { 2 } - 11 x + 7$ has been stretched horizontally by a factor of 2 . Find the function for the transformed graph.

You are given $\lim _ { x \rightarrow a } f ( x ) = L$ and a tolerance $\varepsilon$ . Find a number $\delta$ such that $| f ( x ) - L | < \varepsilon$ whenever $0 < | x - a | < \delta$ . $\lim _ { x \rightarrow 3 } 4 x = 12 ; \quad \varepsilon = 0.01$

Estimate the value of the following limit by graphing the function $f ( x ) = \frac { ( 5 \sin x ) } { ( \sin \pi x ) }$ . $\lim _ { x \rightarrow 0 } \frac { 5 \sin x } { \sin \pi x }$ Round your answer correct to two decimal places.

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

Find the limit. lim⁡x→2x2+2x−12x−2\lim _ { x \rightarrow 2 } \frac { x ^ { 2 } + 2 x - 12 } { x - 2 }limx→2​x−2x2+2x−12​

Determine whether fff is even, odd, or neither. f(x)=8x2x4+1f ( x ) = \frac { 8 x ^ { 2 } } { x ^ { 4 } + 1 }f(x)=x4+18x2​

Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions and then applying the appropriate transformations. y=4+2x−x2y = 4 + 2 x - x ^ { 2 }y=4+2x−x2

Find a function ggg that agrees with fff for x≠25x \neq 25x=25 and is continuous on ℜ\Reℜ . f(x)=5−x25−xf ( x ) = \frac { 5 - \sqrt { x } } { 25 - x }f(x)=25−x5−x​​

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

Find the value of lim⁡x→0+f(x)\lim _ { x \rightarrow 0 ^ { + } } f ( x )limx→0+​f(x) f(x)=11+61/xf ( x ) = \frac { 1 } { 1 + 6 ^ { 1 / x } }f(x)=1+61/x1​

How would you define f(7)f ( 7 )f(7) in order to make fff continuous at 7 ? f(x)=x2−2x−3x−7f ( x ) = \frac { x ^ { 2 } - 2 x - 3 } { x - 7 }f(x)=x−7x2−2x−3​

Determine whether fff is even, odd, or neither. f(x)=4x2x4+5f ( x ) = \frac { 4 x ^ { 2 } } { x ^ { 4 } + 5 }f(x)=x4+54x2​

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) .

Which of the given functions is discontinuous?

Find the range of the function. y=2+cos⁡xy = 2 + \cos xy=2+cosx

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

Find the range of the function. y=2+cos⁡xy = 2 + \cos xy=2+cosx 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)

The graph of the function f(x)=x2−11x+7f ( x ) = x ^ { 2 } - 11 x + 7f(x)=x2−11x+7 has been stretched horizontally by a factor of 2 . Find the function for the transformed 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

Find the limit. $\lim _ { x \rightarrow 2 } \frac { x ^ { 2 } + 2 x - 12 } { x - 2 }$

Determine whether $f$ is even, odd, or neither. $f ( x ) = \frac { 8 x ^ { 2 } } { x ^ { 4 } + 1 }$

Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions and then applying the appropriate transformations. $y = 4 + 2 x - x ^ { 2 }$

Find a function $g$ that agrees with $f$ for $x \neq 25$ and is continuous on $\Re$ . $f ( x ) = \frac { 5 - \sqrt { x } } { 25 - x }$

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

Find the value of $\lim _ { x \rightarrow 0 ^ { + } } f ( x )$ $f ( x ) = \frac { 1 } { 1 + 6 ^ { 1 / x } }$

How would you define $f ( 7 )$ in order to make $f$ continuous at 7 ? $f ( x ) = \frac { x ^ { 2 } - 2 x - 3 } { x - 7 }$

Determine whether $f$ is even, odd, or neither. $f ( x ) = \frac { 4 x ^ { 2 } } { x ^ { 4 } + 5 }$

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

Find the range of the function. $y = 2 + \cos x$

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

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

Find the range of the function. $y = 2 + \cos x$ Select the correct answer.

Let $F ( x ) = \frac { x - 5 } { | x - 5 | }$ . Find the following limits. $\lim _ { x \rightarrow 5 ^ { + } } F ( x ) , \lim _ { x \rightarrow 5 ^ { - } } F ( x )$

The graph of the function $f ( x ) = x ^ { 2 } - 11 x + 7$ has been stretched horizontally by a factor of 2 . Find the function for the transformed graph.