Question 1

Find the derivative of the function. $f(x)=14-4 x+5 x^{2}$

Accepted Answer

$10 x-4$

Question 2

Complete the table by computing $f(x)$ at the given values of x, accurate to five decimal places. Use the results to guess at the indicated limit, if it exists, to three decimal places. $\lim _{x \rightarrow 5} \frac{\sqrt{x+20}-5}{x-5}$ 
$\begin{array}{|l|l|l|l|l|l|l|}
 \hline  \text {x} &4.9 & 4.99 & 4.999 & 5.001 & 5.01 & 5.1  \
\hline  \text {f(x)} &&\
\hline 
\end{array}$

Accepted Answer

$\begin{array}{|l|l|l|l|l|l|l|}
 \hline  \text {x} &4.9 & 4.99 & 4.999 & 5.001 & 5.01 & 5.1  \
\hline  \text {f(x)} &0.1001 & 0.10001 & 0.1 & 0.1 & 0.09999 & 0.0999\
\hline 
\end{array}$

Question 3

Use the definition of the derivative to find $f^{\prime}(3) \text {, where } f(x)=x^{3}-2 x \text {. }$

Accepted Answer

A)  $-25$ 
B)  $25$ 
C)  27 
D)  18 
E)  does not exist 
A)  $-25$ 
B)  $25$ 
C)  27 
D)  18 
E)  does not exist B

Question 4

The position of a car is given by the values in the following table. $t$ $\begin{array}{|c|c|c|c|c|c|}
\hline t & 0 & 1 & 2 & 3 & 4 \
 \text { (seconds) } & & & & & \
\hline s & 0 & 23.7 & 27 & 58.3 & 90.5 \
\text { (meters) } & & & & & \
\hline
\end{array}$Find the average velocity for the time period beginning when $t=2$ and lasting 2 seconds.

Accepted Answer

A)  $45.25$ 
B)  $35$ 
C)  31.75 
D)  $33.7$ 
E)  33.5 
A)  $45.25$ 
B)  $35$ 
C)  31.75 
D)  $33.7$ 
E)  33.5

Question 5

Find the vertical asymptotes of the function. $y=\frac{4 x^{2}+1}{3 x-4 x^{2}}$

Accepted Answer

The answer of Find the vertical asymptotes of the function....

Question 6

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

Accepted Answer

The answer of If $1 \leq f(x) \leq x^{2}+6 x+6$...

Question 7

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

Question 8

Find the vertical asymptotes of the function. $y=\frac{4 x^{2}+1}{3 x-4 x^{2}}$

Accepted Answer

A)  $x=-\frac{4}{3}$ 
B)  $x=\frac{4}{3}$ 
C)  $x=3$ 
D)  $x=0, \frac{4}{3}$ 
E)  None of these 
A)  $x=-\frac{4}{3}$ 
B)  $x=\frac{4}{3}$ 
C)  $x=3$ 
D)  $x=0, \frac{4}{3}$ 
E)  None of these

Question 9

Find the limit $\lim _{x \rightarrow 0^{+}} \frac{2+\sqrt{x}}{\sqrt{x+49}}$ .

Accepted Answer

The answer of Find the limit \(\lim _{x \rightarrow 0^{+}}...

Question 10

Use the graph of the function to state the value of $\lim _{x \rightarrow 0} f(x)$ if it exists. $f(x)=\frac{1}{1+5^{2 / x}}$

Accepted Answer

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

Question 11

If the tangent line to $y=f(x) \text { at }(8,4)$ passes through the point $(4,-32) \text {, find } f^{\prime}(8)$

Accepted Answer

A)  $f^{\prime}(8)=29$ 
B)  $f^{\prime}(8)=19$ 
C)  $f^{\prime}(8)=9$ 
D)  $f^{\prime}(8)=34$ 
E)  $f^{\prime}(8)=-9$ 
A)  $f^{\prime}(8)=29$ 
B)  $f^{\prime}(8)=19$ 
C)  $f^{\prime}(8)=9$ 
D)  $f^{\prime}(8)=34$ 
E)  $f^{\prime}(8)=-9$

Question 12

Find the limit. $\lim _{t \rightarrow 5} \frac{t^{2}-25}{t^{3}-125}$

Accepted Answer

The answer of Find the limit. \(\lim _{t \rightarrow 5}...

Question 13

If $\lim _{x \rightarrow 3^{-}} f(x)=4.1$ , then if $\lim _{x \rightarrow 3} f(x)$ exists, find the value where it exists.

Accepted Answer

A)  $4.1$ 
B)  1 
C)  $4 .4$ 
D)  2 
E)  3 
A)  $4.1$ 
B)  1 
C)  $4 .4$ 
D)  2 
E)  3

Question 14

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

Accepted Answer

A)  $f(3)=1$ 
B)  $f(3)=3$ 
C)  $f(3)=-3$ 
D)  $f(3)=0$ 
E)  None of these 
A)  $f(3)=1$ 
B)  $f(3)=3$ 
C)  $f(3)=-3$ 
D)  $f(3)=0$ 
E)  None of these

Question 15

Find an equation of the tangent line to the parabola $y=5 x^{3}$ at the point $(-5,-145)$ .

Accepted Answer

A)  $y=375 x-1730$ 
B)  $y=395 x-1730$. 
C)  $y=375 x+1730$ 
D)  $y=395 x+1730$. 
E)  $y=395 x-1730$ 
A)  $y=375 x-1730$ 
B)  $y=395 x-1730$. 
C)  $y=375 x+1730$ 
D)  $y=395 x+1730$. 
E)  $y=395 x-1730$

Question 16

Evaluate $\lim _{h \rightarrow 0} \frac{\cot \left(\frac{\pi}{4}+h\right)-1}{h}$ .

Accepted Answer

A)  0 
B)  -2 
C)  2 
D)  Undefined 
A)  0 
B)  -2 
C)  2 
D)  Undefined

Question 17

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

A)  $1.79$ 
B)  $1.49$ 
C)  $1.69$ 
D)  3.18 
E)  $1.59$ 
A)  $1.79$ 
B)  $1.49$ 
C)  $1.69$ 
D)  3.18 
E)  $1.59$

Question 18

Evaluate the limit. $\lim _{x \rightarrow \infty} \frac{10 x^{2}-3 x+1}{7 x^{2}+3 x-3}$

Accepted Answer

The answer of Evaluate the limit. \(\lim _{x \rightarrow \infty}...

Question 19

Determine where f is discontinuous. $f(x)=\left\{\begin{array}{ccc}
\sqrt{-x} & \text { if } & x11
\end{array}\right.$

Accepted Answer

A)  $0 \text { and } 11$ 
B)  0 only 
C)  $-11$ only 
D)  11 only 
E)  $0 \text { and }-11$ 
A)  $0 \text { and } 11$ 
B)  0 only 
C)  $-11$ only 
D)  11 only 
E)  $0 \text { and }-11$

Question 20

Find the point at which the given function is discontinuous. $f(x)=\left\{\begin{array}{ll}
\frac{1}{x-8}, & x \neq 8 \
8, & x=8
\end{array}\right.$

Accepted Answer

The answer of Find the point at which the given...

Find the derivative of the function. $f(x)=14-4 x+5 x^{2}$

Complete the table by computing $f(x)$ at the given values of x, accurate to five decimal places. Use the results to guess at the indicated limit, if it exists, to three decimal places. $\lim _{x \rightarrow 5} \frac{\sqrt{x+20}-5}{x-5}$ x 4.9 4.99 4.999 5.001 5.01 5.1 f(x)

Use the definition of the derivative to find $f^{\prime}(3) \text {, where } f(x)=x^{3}-2 x \text {. }$

The position of a car is given by the values in the following table. $t$ t 0 1 2 3 4 (seconds) s 0 23.7 27 58.3 90.5 (meters) Find the average velocity for the time period beginning when $t=2$ and lasting 2 seconds.

Find the vertical asymptotes of the function. $y=\frac{4 x^{2}+1}{3 x-4 x^{2}}$

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

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

Find the vertical asymptotes of the function. $y=\frac{4 x^{2}+1}{3 x-4 x^{2}}$

Find the limit $\lim _{x \rightarrow 0^{+}} \frac{2+\sqrt{x}}{\sqrt{x+49}}$ .

Use the graph of the function to state the value of $\lim _{x \rightarrow 0} f(x)$ if it exists. $f(x)=\frac{1}{1+5^{2 / x}}$

If the tangent line to $y=f(x) \text { at }(8,4)$ passes through the point $(4,-32) \text {, find } f^{\prime}(8)$

Find the limit. $\lim _{t \rightarrow 5} \frac{t^{2}-25}{t^{3}-125}$

If $\lim _{x \rightarrow 3^{-}} f(x)=4.1$ , then if $\lim _{x \rightarrow 3} f(x)$ exists, find the value where it exists.

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

Find an equation of the tangent line to the parabola $y=5 x^{3}$ at the point $(-5,-145)$ .

Evaluate $\lim _{h \rightarrow 0} \frac{\cot \left(\frac{\pi}{4}+h\right)-1}{h}$ .

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.

Evaluate the limit. $\lim _{x \rightarrow \infty} \frac{10 x^{2}-3 x+1}{7 x^{2}+3 x-3}$

Determine where f is discontinuous. $f(x)=\left\{\begin{array}{ccc}\sqrt{-x} & \text { if } & x<0 \\11-x & \text { if } & 0 \leq x<11 \\(11-x)^{2} & \text { if } & x>11\end{array}\right.$

Find the point at which the given function is discontinuous. $f(x)=\left\{\begin{array}{ll}\frac{1}{x-8}, & x \neq 8 \\8, & x=8\end{array}\right.$

Functions and Limits

Applications of Differentiation

Integrals

Applications of Integration

Inverse Functions

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

Final Exam

Filters

Exam 2: Derivatives

Find the derivative of the function. f(x)=14−4x+5x2f(x)=14-4 x+5 x^{2}f(x)=14−4x+5x2

Use the definition of the derivative to find f′(3), where f(x)=x3−2x. f^{\prime}(3) \text {, where } f(x)=x^{3}-2 x \text {. }f′(3), where f(x)=x3−2x.

The position of a car is given by the values in the following table. ttt t 0 1 2 3 4 (seconds) s 0 23.7 27 58.3 90.5 (meters) Find the average velocity for the time period beginning when t=2t=2t=2 and lasting 2 seconds.

Find the vertical asymptotes of the function. y=4x2+13x−4x2y=\frac{4 x^{2}+1}{3 x-4 x^{2}}y=3x−4x24x2+1​

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

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 vertical asymptotes of the function. y=4x2+13x−4x2y=\frac{4 x^{2}+1}{3 x-4 x^{2}}y=3x−4x24x2+1​

Find the limit lim⁡x→0+2+xx+49\lim _{x \rightarrow 0^{+}} \frac{2+\sqrt{x}}{\sqrt{x+49}}limx→0+​x+49​2+x​​ .

Use the graph of the function to state the value of lim⁡x→0f(x)\lim _{x \rightarrow 0} f(x)limx→0​f(x) if it exists. f(x)=11+52/xf(x)=\frac{1}{1+5^{2 / x}}f(x)=1+52/x1​

If the tangent line to y=f(x) at (8,4)y=f(x) \text { at }(8,4)y=f(x) at (8,4) passes through the point (4,−32), find f′(8)(4,-32) \text {, find } f^{\prime}(8)(4,−32), find f′(8)

Find the limit. lim⁡t→5t2−25t3−125\lim _{t \rightarrow 5} \frac{t^{2}-25}{t^{3}-125}limt→5​t3−125t2−25​

If lim⁡x→3−f(x)=4.1\lim _{x \rightarrow 3^{-}} f(x)=4.1limx→3−​f(x)=4.1 , then if lim⁡x→3f(x)\lim _{x \rightarrow 3} f(x)limx→3​f(x) exists, find the value where it exists.

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

Find an equation of the tangent line to the parabola y=5x3y=5 x^{3}y=5x3 at the point (−5,−145)(-5,-145)(−5,−145) .

Evaluate lim⁡h→0cot⁡(π4+h)−1h\lim _{h \rightarrow 0} \frac{\cot \left(\frac{\pi}{4}+h\right)-1}{h}limh→0​hcot(4π​+h)−1​ .

Evaluate the limit. lim⁡x→∞10x2−3x+17x2+3x−3\lim _{x \rightarrow \infty} \frac{10 x^{2}-3 x+1}{7 x^{2}+3 x-3}limx→∞​7x2+3x−310x2−3x+1​

Find the point at which the given function is discontinuous. f(x)={1x−8,x≠88,x=8f(x)=\left\{\begin{array}{ll}\frac{1}{x-8}, & x \neq 8 \\8, & x=8\end{array}\right.f(x)={x−81​,8,​x=8x=8​

Functions and Limits

Applications of Differentiation

Integrals

Applications of Integration

Inverse Functions

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

Final Exam

Filters

Find the derivative of the function. $f(x)=14-4 x+5 x^{2}$

Use the definition of the derivative to find $f^{\prime}(3) \text {, where } f(x)=x^{3}-2 x \text {. }$

The position of a car is given by the values in the following table. $t$ t 0 1 2 3 4 (seconds) s 0 23.7 27 58.3 90.5 (meters) Find the average velocity for the time period beginning when $t=2$ and lasting 2 seconds.

Find the vertical asymptotes of the function. $y=\frac{4 x^{2}+1}{3 x-4 x^{2}}$

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

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

Find the vertical asymptotes of the function. $y=\frac{4 x^{2}+1}{3 x-4 x^{2}}$

Find the limit $\lim _{x \rightarrow 0^{+}} \frac{2+\sqrt{x}}{\sqrt{x+49}}$ .

Use the graph of the function to state the value of $\lim _{x \rightarrow 0} f(x)$ if it exists. $f(x)=\frac{1}{1+5^{2 / x}}$

If the tangent line to $y=f(x) \text { at }(8,4)$ passes through the point $(4,-32) \text {, find } f^{\prime}(8)$

Find the limit. $\lim _{t \rightarrow 5} \frac{t^{2}-25}{t^{3}-125}$

If $\lim _{x \rightarrow 3^{-}} f(x)=4.1$ , then if $\lim _{x \rightarrow 3} f(x)$ exists, find the value where it exists.

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

Find an equation of the tangent line to the parabola $y=5 x^{3}$ at the point $(-5,-145)$ .

Evaluate $\lim _{h \rightarrow 0} \frac{\cot \left(\frac{\pi}{4}+h\right)-1}{h}$ .

Evaluate the limit. $\lim _{x \rightarrow \infty} \frac{10 x^{2}-3 x+1}{7 x^{2}+3 x-3}$

Find the point at which the given function is discontinuous. $f(x)=\left\{\begin{array}{ll}\frac{1}{x-8}, & x \neq 8 \\8, & x=8\end{array}\right.$