Question 1

Evaluate the limit, if it exists. $\lim _{h \rightarrow 0} \frac{(x-h)^{7}-x^{7}}{h}$

Accepted Answer

A)  $7 x^{6}$ 
B)  $-7 x^{6}$ 
C)  $-7$ 
D)  1 
E)  does not exist 
A)  $7 x^{6}$ 
B)  $-7 x^{6}$ 
C)  $-7$ 
D)  1 
E)  does not exist

Question 2

The position of a car is given by the values in the following table. $\begin{array}{|c|c|c|c|c|c|c|}
\hline t \text { (seconds) } & 0 & 1 & 2 & 3 & 4 & 5 \
\hline s \text { (feet) } & 0 & 19 & 30 & 74 & 111 & 178 \
\hline
\end{array}$ Estimate the instantaneous velocity when $t=2$ by averaging the velocities for the periods $[1,2] \text { and }[2,3]$ .

Accepted Answer

The answer of The position of a car is given...

Question 3

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)  $-35$

Question 4

Sketch the graph of the function f and evaluate $\lim _{x \rightarrow 3} f(x)$ $f(x)=\left\{\begin{array}{ll}
x-2, & \text { if } x \leq 3 \
-2 x+7, & \text { if } x>3
\end{array}\right.$

Accepted Answer

A)   
1 
B)   
3 
C)   
3 
D)   
1 
A)   
1 
B)   
3 
C)   
3 
D)   
1

Question 5

If f and g are continuous functions with $f(3)=10 \text { and } \lim _{x \rightarrow 3}[2 f(x)-g(x)]=3, \text { find } g(3)$

Accepted Answer

The answer of If f and g are continuous functions...

Question 6

The cost (in dollars) of producing x units of a certain commodity is $C(x)=4,571+17 x+0.02 x^{2}$ Find the instantaneous rate of change with respect to x when $x=103 \text {. }$ (This is called the marginal cost.)

Accepted Answer

A)  $4.12$ 
B)  $21.12$ 
C)  $23.18$ 
D)  $19.06$. 
E)  $19.06$ 
A)  $4.12$ 
B)  $21.12$ 
C)  $23.18$ 
D)  $19.06$. 
E)  $19.06$

Question 7

Choose an equation from the following that expresses the fact that a function f is continuous at the number 6 .

Accepted Answer

A)  $\lim _{x \rightarrow 0} f(x)=f(6)$ 
B)  $\lim _{x \rightarrow 0} f(x)=6$ 
C)  $\lim _{x \rightarrow 6} f(x)=f(6)$ 
D)  $\lim _{x \rightarrow 6} f(x)=\infty $ 
E)  $\lim _{x \rightarrow 6} f(x)=-\infty$ 
A)  $\lim _{x \rightarrow 0} f(x)=f(6)$ 
B)  $\lim _{x \rightarrow 0} f(x)=6$ 
C)  $\lim _{x \rightarrow 6} f(x)=f(6)$ 
D)  $\lim _{x \rightarrow 6} f(x)=\infty $ 
E)  $\lim _{x \rightarrow 6} f(x)=-\infty$

Question 8

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

Accepted Answer

A)  3 
B)  5 
C)  2 
D)  Does not exist 
A)  3 
B)  5 
C)  2 
D)  Does not exist

Question 9

Write an equation that expresses the fact that a function f is continuous at the number 6 .

Accepted Answer

The answer of Write an equation that expresses the fact...

Question 10

Find the limit $\lim _{x \rightarrow 2} \frac{x-1}{x^{2}-4 x+7}$ .

Accepted Answer

A)  $\frac{1}{7}$ 
B)  $\frac{1}{3}$ 
C)  $-1$ 
D)  1 
A)  $\frac{1}{7}$ 
B)  $\frac{1}{3}$ 
C)  $-1$ 
D)  1

Question 11

Use continuity to evaluate the limit. $\lim _{x \rightarrow 9_{x}} \sin (x+10 \sin x)$

Accepted Answer

A)  $\infty$ 
B)  $9 \pi$ 
C)  $-1$ 
D)  0 
E)  1 
A)  $\infty$ 
B)  $9 \pi$ 
C)  $-1$ 
D)  0 
E)  1

Question 12

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. $\lim _{x \rightarrow 0} \frac{x}{\sin 5 x}$ $\begin{array}{|l|l|l|l|l|l|l|}
 \hline  \text {x} &-0.1 & -0.01 & -0.001 & 0.001 & 0.01& 0.1 \
\hline  \text {f(x)} &&\
\hline 
\end{array}$

Accepted Answer

None

Question 13

Evaluate the limit. $\lim _{x \rightarrow 0} \frac{(6+x)^{-1}-6^{-1}}{x}$

Accepted Answer

The answer of Evaluate the limit. \(\lim _{x \rightarrow 0}...

Question 14

If a rock is thrown upward on the planet Mars with a velocity of $15 \mathrm{~m} / \mathrm{s}$ , its height in meters t seconds later is given by $y=15 t-1.86 t^{2}$ . Find the average velocity over the time interval $[1,4]$ .

Accepted Answer

A)  $2.7$ 
B)  $6 .7$ 
C)  $7.7$ 
D)  $5.7$ 
E)  $4.7$ 
A)  $2.7$ 
B)  $6 .7$ 
C)  $7.7$ 
D)  $5.7$ 
E)  $4.7$

Question 15

Find the slope of the tangent line to the curve $y=5 x^{2}$ at the point $(-4,22)$ .

Accepted Answer

A)  $-40$ 
B)  40 
C)  $-4$ 
D)  4 
E)  25 
A)  $-40$ 
B)  40 
C)  $-4$ 
D)  4 
E)  25

Question 16

For $x=5$ determine whether f is continuous from the right, from the left, or neither.

Accepted Answer

The answer of For $x=5$ determine whether f is continuous...

Question 17

Use the graph of the function to state the value of $\lim _{x \rightarrow 0} f(x)$ , if it exists. $f(x)=\frac{x^{2}+x}{4 \sqrt{x^{3}+x^{2}}}$

Accepted Answer

A)  -∞ 
B)  $\frac{1}{4}$ 
C)  $-\frac{1}{4}$ 
D)  ∞ 
E)  does not exist 
A)  -∞ 
B)  $\frac{1}{4}$ 
C)  $-\frac{1}{4}$ 
D)  ∞ 
E)  does not exist

Question 18

The point $P(4,2)$ lies on the curve $y=\sqrt{x}$ . If is the point Q $(x, \sqrt{x})$ , use your calculator to find the slope of the secant line PQ (correct to six decimal places) for the value $x=3.89 .$

Accepted Answer

A)  $-0.128743$ 
B)  $-0.251743$ 
C)  $0.461743$ 
D)  $-0.391743$ 
E)  $0.251743$ 
A)  $-0.128743$ 
B)  $-0.251743$ 
C)  $0.461743$ 
D)  $-0.391743$ 
E)  $0.251743$

Question 19

Find the limit $\lim _{x \rightarrow 0} \frac{\sqrt{x+14}-\sqrt{14}}{x}$ , if it exists.

Accepted Answer

A)  Does not exist 
B)  $\frac{\sqrt{14}}{28}$ 
C)  $\frac{\sqrt{14}}{2}$ 
D)  $\frac{\sqrt{14}}{14}$ 
A)  Does not exist 
B)  $\frac{\sqrt{14}}{28}$ 
C)  $\frac{\sqrt{14}}{2}$ 
D)  $\frac{\sqrt{14}}{14}$

Question 20

Find the limit. $\lim _{x \rightarrow 3} \frac{x^{2}+3 x-18}{x-3}$

Accepted Answer

A)  18 
B)  0 
C)  9 
D)  5 
E)  11 
A)  18 
B)  0 
C)  9 
D)  5 
E)  11

Evaluate the limit, if it exists. $\lim _{h \rightarrow 0} \frac{(x-h)^{7}-x^{7}}{h}$

The position of a car is given by the values in the following table. t (seconds) 0 1 2 3 4 5 s (feet) 0 19 30 74 111 178 Estimate the instantaneous velocity when $t=2$ by averaging the velocities for the periods $[1,2] \text { and }[2,3]$ .

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

Sketch the graph of the function f and evaluate $\lim _{x \rightarrow 3} f(x)$ $f(x)=\left\{\begin{array}{ll}x-2, & \text { if } x \leq 3 \\-2 x+7, & \text { if } x>3\end{array}\right.$

If f and g are continuous functions with $f(3)=10 \text { and } \lim _{x \rightarrow 3}[2 f(x)-g(x)]=3, \text { find } g(3)$

The cost (in dollars) of producing x units of a certain commodity is $C(x)=4,571+17 x+0.02 x^{2}$ Find the instantaneous rate of change with respect to x when $x=103 \text {. }$ (This is called the marginal cost.)

Choose an equation from the following that expresses the fact that a function f is continuous at the number 6 .

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

Write an equation that expresses the fact that a function f is continuous at the number 6 .

Find the limit $\lim _{x \rightarrow 2} \frac{x-1}{x^{2}-4 x+7}$ .

Use continuity to evaluate the limit. $\lim _{x \rightarrow 9_{x}} \sin (x+10 \sin x)$

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. $\lim _{x \rightarrow 0} \frac{x}{\sin 5 x}$ x -0.1 -0.01 -0.001 0.001 0.01 0.1 f(x)

Evaluate the limit. $\lim _{x \rightarrow 0} \frac{(6+x)^{-1}-6^{-1}}{x}$

If a rock is thrown upward on the planet Mars with a velocity of $15 \mathrm{~m} / \mathrm{s}$ , its height in meters t seconds later is given by $y=15 t-1.86 t^{2}$ . Find the average velocity over the time interval $[1,4]$ .

Find the slope of the tangent line to the curve $y=5 x^{2}$ at the point $(-4,22)$ .

For $x=5$ determine whether f is continuous from the right, from the left, or neither.

Use the graph of the function to state the value of $\lim _{x \rightarrow 0} f(x)$ , if it exists. $f(x)=\frac{x^{2}+x}{4 \sqrt{x^{3}+x^{2}}}$

The point $P(4,2)$ lies on the curve $y=\sqrt{x}$ . If is the point Q $(x, \sqrt{x})$ , use your calculator to find the slope of the secant line PQ (correct to six decimal places) for the value $x=3.89 .$

Find the limit $\lim _{x \rightarrow 0} \frac{\sqrt{x+14}-\sqrt{14}}{x}$ , if it exists.

Find the limit. $\lim _{x \rightarrow 3} \frac{x^{2}+3 x-18}{x-3}$

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

Evaluate the limit, if it exists. lim⁡h→0(x−h)7−x7h\lim _{h \rightarrow 0} \frac{(x-h)^{7}-x^{7}}{h}limh→0​h(x−h)7−x7​

The position of a car is given by the values in the following table. t (seconds) 0 1 2 3 4 5 s (feet) 0 19 30 74 111 178 Estimate the instantaneous velocity when t=2t=2t=2 by averaging the velocities for the periods [1,2] and [2,3][1,2] \text { and }[2,3][1,2] and [2,3] .

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

Sketch the graph of the function f and evaluate lim⁡x→3f(x)\lim _{x \rightarrow 3} f(x)limx→3​f(x) f(x)={x−2, if x≤3−2x+7, if x>3f(x)=\left\{\begin{array}{ll}x-2, & \text { if } x \leq 3 \\-2 x+7, & \text { if } x>3\end{array}\right.f(x)={x−2,−2x+7,​ if x≤3 if x>3​

If f and g are continuous functions with f(3)=10 and lim⁡x→3[2f(x)−g(x)]=3, find g(3)f(3)=10 \text { and } \lim _{x \rightarrow 3}[2 f(x)-g(x)]=3, \text { find } g(3)f(3)=10 and limx→3​[2f(x)−g(x)]=3, find g(3)

The cost (in dollars) of producing x units of a certain commodity is C(x)=4,571+17x+0.02x2C(x)=4,571+17 x+0.02 x^{2}C(x)=4,571+17x+0.02x2 Find the instantaneous rate of change with respect to x when x=103. x=103 \text {. }x=103. (This is called the marginal cost.)

Choose an equation from the following that expresses the fact that a function f is continuous at the number 6 .

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

Write an equation that expresses the fact that a function f is continuous at the number 6 .

Find the limit lim⁡x→2x−1x2−4x+7\lim _{x \rightarrow 2} \frac{x-1}{x^{2}-4 x+7}limx→2​x2−4x+7x−1​ .

Use continuity to evaluate the limit. lim⁡x→9xsin⁡(x+10sin⁡x)\lim _{x \rightarrow 9_{x}} \sin (x+10 \sin x)limx→9x​​sin(x+10sinx)

Evaluate the limit. lim⁡x→0(6+x)−1−6−1x\lim _{x \rightarrow 0} \frac{(6+x)^{-1}-6^{-1}}{x}limx→0​x(6+x)−1−6−1​

If a rock is thrown upward on the planet Mars with a velocity of 15 m/s15 \mathrm{~m} / \mathrm{s}15 m/s , its height in meters t seconds later is given by y=15t−1.86t2y=15 t-1.86 t^{2}y=15t−1.86t2 . Find the average velocity over the time interval [1,4][1,4][1,4] .

Find the slope of the tangent line to the curve y=5x2y=5 x^{2}y=5x2 at the point (−4,22)(-4,22)(−4,22) .

For x=5x=5x=5 determine whether f is continuous from the right, from the left, or neither.

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)=x2+x4x3+x2f(x)=\frac{x^{2}+x}{4 \sqrt{x^{3}+x^{2}}}f(x)=4x3+x2​x2+x​

The point P(4,2)P(4,2)P(4,2) lies on the curve y=xy=\sqrt{x}y=x​ . If is the point Q (x,x)(x, \sqrt{x})(x,x​) , use your calculator to find the slope of the secant line PQ (correct to six decimal places) for the value x=3.89.x=3.89 .x=3.89.

Find the limit lim⁡x→0x+14−14x\lim _{x \rightarrow 0} \frac{\sqrt{x+14}-\sqrt{14}}{x}limx→0​xx+14​−14​​ , if it exists.

Find the limit. lim⁡x→3x2+3x−18x−3\lim _{x \rightarrow 3} \frac{x^{2}+3 x-18}{x-3}limx→3​x−3x2+3x−18​

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

Evaluate the limit, if it exists. $\lim _{h \rightarrow 0} \frac{(x-h)^{7}-x^{7}}{h}$

The position of a car is given by the values in the following table. t (seconds) 0 1 2 3 4 5 s (feet) 0 19 30 74 111 178 Estimate the instantaneous velocity when $t=2$ by averaging the velocities for the periods $[1,2] \text { and }[2,3]$ .

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

Sketch the graph of the function f and evaluate $\lim _{x \rightarrow 3} f(x)$ $f(x)=\left\{\begin{array}{ll}x-2, & \text { if } x \leq 3 \\-2 x+7, & \text { if } x>3\end{array}\right.$

If f and g are continuous functions with $f(3)=10 \text { and } \lim _{x \rightarrow 3}[2 f(x)-g(x)]=3, \text { find } g(3)$

The cost (in dollars) of producing x units of a certain commodity is $C(x)=4,571+17 x+0.02 x^{2}$ Find the instantaneous rate of change with respect to x when $x=103 \text {. }$ (This is called the marginal cost.)

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

Find the limit $\lim _{x \rightarrow 2} \frac{x-1}{x^{2}-4 x+7}$ .

Use continuity to evaluate the limit. $\lim _{x \rightarrow 9_{x}} \sin (x+10 \sin x)$

Evaluate the limit. $\lim _{x \rightarrow 0} \frac{(6+x)^{-1}-6^{-1}}{x}$

If a rock is thrown upward on the planet Mars with a velocity of $15 \mathrm{~m} / \mathrm{s}$ , its height in meters t seconds later is given by $y=15 t-1.86 t^{2}$ . Find the average velocity over the time interval $[1,4]$ .

Find the slope of the tangent line to the curve $y=5 x^{2}$ at the point $(-4,22)$ .

For $x=5$ determine whether f is continuous from the right, from the left, or neither.

Use the graph of the function to state the value of $\lim _{x \rightarrow 0} f(x)$ , if it exists. $f(x)=\frac{x^{2}+x}{4 \sqrt{x^{3}+x^{2}}}$

The point $P(4,2)$ lies on the curve $y=\sqrt{x}$ . If is the point Q $(x, \sqrt{x})$ , use your calculator to find the slope of the secant line PQ (correct to six decimal places) for the value $x=3.89 .$

Find the limit $\lim _{x \rightarrow 0} \frac{\sqrt{x+14}-\sqrt{14}}{x}$ , if it exists.

Find the limit. $\lim _{x \rightarrow 3} \frac{x^{2}+3 x-18}{x-3}$