Question 1

$\lim _{x \rightarrow-4} \frac{x^{3}+64}{x+4}$
(a) Explain why direct substitution cannot be used to find the limit
(b) Find the limit algebraically, if it exists.

Accepted Answer

(a) $\frac{x^{3}+64}{x+4}$ is not defined at  x=-4 .
(b) 48

Question 2

The following table lists the population statistics for a certain city.
$\begin{array}{|c|c|}
\hline \text { Year } & \text { Population } \
\hline 1975 & 995,855 \
\hline 1985 & 990,928 \
\hline 1995 & 942,547 \
\hline 2005 & 889,370 \
\hline
\end{array}$
What is the average rate of change in the population with respect to time between 1975 and  2005 ?

Accepted Answer

-3,549.2

Question 3

Use a calculator to find the LRAM area approximation for the area under the graph $f(x)=-x^{3}+8$
from  x=0  to  x=2  with 20 approximating rectangles.

Accepted Answer

12.39

Question 4

Use a calculator to find the RRAM area approximation for the area under the graph $f(x)=-x^{3}+8$ from  x=0  to  x=2  with 20 approximating rectangles.

Accepted Answer

The answer of Use a calculator to find the RRAM...

Question 5

Write the integral that would be used to find the shaded area on the graph shown below.

Accepted Answer

The answer of Write the integral that would be used...

Question 6

At what points  c  in the domain of  f(x)  does $\lim _{x \rightarrow c} f(x)$ exist?
$f(x)=\left\{\begin{array}{l}
3 x & -5 \leq x<0  \
1 & x=0\
x^{2}+1& 0 < x \leq 2\
\end{array}\right.$

Accepted Answer

The answer of At what points  c  in...

Question 7

Find the derivative of  f(x)=7 x+10 .

Accepted Answer

The answer of Find the derivative of  f(x)=7 x+10...

Question 8

Use the graph below to find the limits or explain why the limits do not exist.
(a) $\lim _{x \rightarrow 4^{-}} f(x)$
(b) $\lim _{x \rightarrow 4^{+}} f(x)$
(c) $\lim _{x \rightarrow 4} f(x)$

Accepted Answer

The answer of Use the graph below to find the...

Question 9

Draw the graph of $ f(x)=-x(x-3) $ over the interval $ [0,3] $. On the graph, show and shade the rectangles that would be used to approximate the area under the curve $ f(x) $ over $ [0,3] $ by the right rectangle approximation method using 6 subintervals. Compute the estimation.

Accepted Answer

The answer of Draw the graph of $ f(x)=-x(x-3) $...

Question 10

Determine $\lim _{x \rightarrow 0} \frac{\sin x}{x^{2}-x}$
if the limit exists.

Accepted Answer

A)    -1 
B)    −12-\frac{1}{2}−21​ 
C)    12\frac{1}{2}21​ 
D)   1
E)   The limit does not exist.

Question 11

Find the average rate of change of  f(x)=5 \tan x  over the interval $\left[\frac{\pi}{4}, \frac{3 \pi}{4}\right]$

Accepted Answer

A)    −5π-\frac{5}{\pi}−π5​ 
B)    −20π-\frac{20}{\pi}−π20​ 
C)    -10 
D)   0

Question 12

Write the integral that would be used to find the shaded area on the graph shown below.

Accepted Answer

The answer of Write the integral that would be used...

Question 13

Explain how to find the area under the graph of $f(x)=|x-2|$ from  x=0  to  x=4  by computing the geometric area.

Accepted Answer

The answer of Explain how to find the area under...

Question 14

Compute the integral $\int_{2}^{5} 4 x d x$ by computing a geometric area. (Hint: Graph the function first.)

Accepted Answer

The answer of Compute the integral \(\int_{2}^{5} 4 x d...

Question 15

Find the average rate of change of  f(x)=3 \cos x  over the interval $\left[\frac{\pi}{3}, \frac{2 \pi}{3}\right]$

Accepted Answer

A)    -3 
B)   −9π-9 \pi−9π 
C)   −9π-\frac{9}{\pi}−π9​ 
D)   π3\frac{\pi}{3}3π​

Question 16

What is the derivative of $f(x)=4 x^{2} ?$

Accepted Answer

The answer of What is the derivative of \(f(x)=4 x^{2}...

Question 17

Use the graph below to find the limits or explain why the limits do not exist.
(a) $\lim _{x \rightarrow 2^{-}} f(x)$
(b) $\lim _{x \rightarrow 2^{+}} f(x)$
(c) $\lim _{x \rightarrow 2} f(x)$

Accepted Answer

The answer of Use the graph below to find the...

Question 18

The following table lists the population statistics for a certain city.
$\begin{array}{|c|c|}
\hline \text { Year } & \text { Population } \
\hline 1975 & 695,854 \
\hline 1985 & 720,928 \
\hline 1995 & 742,547 \
\hline 2005 & 750,370 \
\hline
\end{array}$
What is the average rate of change in the population with respect to time between 1975 and 2005 ?

Accepted Answer

The answer of The following table lists the population statistics...

Question 19

Find the derivative of  f(x)=-4 x-8 .

Accepted Answer

The answer of Find the derivative of  f(x)=-4 x-8...

Question 20

What is the average rate of change of $f(x)=\sqrt{3 x-1}$ over the interval  [1.9,2.1]  ?

Accepted Answer

The answer of What is the average rate of change...

Exam 10: An Introduction to Calculus: Limits, Derivatives, and Integrals

$\lim _{x \rightarrow-4} \frac{x^{3}+64}{x+4}$ (a) Explain why direct substitution cannot be used to find the limit (b) Find the limit algebraically, if it exists.

Use a calculator to find the LRAM area approximation for the area under the graph $f(x)=-x^{3}+8$ from x=0 to x=2 with 20 approximating rectangles.

Use a calculator to find the RRAM area approximation for the area under the graph $f(x)=-x^{3}+8$ from x=0 to x=2 with 20 approximating rectangles.

Write the integral that would be used to find the shaded area on the graph shown below.

At what points c in the domain of f(x) does $\lim _{x \rightarrow c} f(x)$ exist? $f(x)=\left\{\begin{array}{l}3 x & -5 \leq x<0 \\1 & x=0\\x^{2}+1& 0 < x \leq 2\\\end{array}\right.$

Find the derivative of f(x)=7 x+10 .

Draw the graph of $f(x)=-x(x-3)$ over the interval $[0,3]$ . On the graph, show and shade the rectangles that would be used to approximate the area under the curve $f(x)$ over $[0,3]$ by the right rectangle approximation method using 6 subintervals. Compute the estimation.

Determine $\lim _{x \rightarrow 0} \frac{\sin x}{x^{2}-x}$ if the limit exists.

Find the average rate of change of f(x) =5 \tan x over the interval $\left[\frac{\pi}{4}, \frac{3 \pi}{4}\right]$

Write the integral that would be used to find the shaded area on the graph shown below.

Explain how to find the area under the graph of $f(x)=|x-2|$ from x=0 to x=4 by computing the geometric area.

Compute the integral $\int_{2}^{5} 4 x d x$ by computing a geometric area. (Hint: Graph the function first.)

Find the average rate of change of f(x) =3 \cos x over the interval $\left[\frac{\pi}{3}, \frac{2 \pi}{3}\right]$

What is the derivative of $f(x)=4 x^{2} ?$

Find the derivative of f(x)=-4 x-8 .

What is the average rate of change of $f(x)=\sqrt{3 x-1}$ over the interval [1.9,2.1] ?

Exam 10: An Introduction to Calculus: Limits, Derivatives, and Integrals

lim⁡x→−4x3+64x+4\lim _{x \rightarrow-4} \frac{x^{3}+64}{x+4}limx→−4​x+4x3+64​ (a) Explain why direct substitution cannot be used to find the limit (b) Find the limit algebraically, if it exists.

Use a calculator to find the LRAM area approximation for the area under the graph f(x)=−x3+8f(x)=-x^{3}+8f(x)=−x3+8 from x=0 to x=2 with 20 approximating rectangles.

Use a calculator to find the RRAM area approximation for the area under the graph f(x)=−x3+8f(x)=-x^{3}+8f(x)=−x3+8 from x=0 to x=2 with 20 approximating rectangles.

Write the integral that would be used to find the shaded area on the graph shown below.

At what points c in the domain of f(x) does lim⁡x→cf(x)\lim _{x \rightarrow c} f(x)limx→c​f(x) exist? f(x)={3x−5≤x<01x=0x2+10<x≤2f(x)=\left\{\begin{array}{l}3 x & -5 \leq x<0 \\1 & x=0\\x^{2}+1& 0 < x \leq 2\\\end{array}\right.f(x)=⎩⎨⎧​3x1x2+1​−5≤x<0x=00<x≤2​

Find the derivative of f(x)=7 x+10 .

Use the graph below to find the limits or explain why the limits do not exist. (a) lim⁡x→4−f(x)\lim _{x \rightarrow 4^{-}} f(x)limx→4−​f(x) (b) lim⁡x→4+f(x)\lim _{x \rightarrow 4^{+}} f(x)limx→4+​f(x) (c) lim⁡x→4f(x)\lim _{x \rightarrow 4} f(x)limx→4​f(x)

Determine lim⁡x→0sin⁡xx2−x\lim _{x \rightarrow 0} \frac{\sin x}{x^{2}-x}limx→0​x2−xsinx​ if the limit exists.

Find the average rate of change of f(x) =5 \tan x over the interval [π4,3π4]\left[\frac{\pi}{4}, \frac{3 \pi}{4}\right][4π​,43π​]

Write the integral that would be used to find the shaded area on the graph shown below.

Explain how to find the area under the graph of f(x)=∣x−2∣f(x)=|x-2|f(x)=∣x−2∣ from x=0 to x=4 by computing the geometric area.

Compute the integral ∫254xdx\int_{2}^{5} 4 x d x∫25​4xdx by computing a geometric area. (Hint: Graph the function first.)

Find the average rate of change of f(x) =3 \cos x over the interval [π3,2π3]\left[\frac{\pi}{3}, \frac{2 \pi}{3}\right][3π​,32π​]

What is the derivative of f(x)=4x2?f(x)=4 x^{2} ?f(x)=4x2?

Use the graph below to find the limits or explain why the limits do not exist. (a) lim⁡x→2−f(x)\lim _{x \rightarrow 2^{-}} f(x)limx→2−​f(x) (b) lim⁡x→2+f(x)\lim _{x \rightarrow 2^{+}} f(x)limx→2+​f(x) (c) lim⁡x→2f(x)\lim _{x \rightarrow 2} f(x)limx→2​f(x)

Find the derivative of f(x)=-4 x-8 .

What is the average rate of change of f(x)=3x−1f(x)=\sqrt{3 x-1}f(x)=3x−1​ over the interval [1.9,2.1] ?

$\lim _{x \rightarrow-4} \frac{x^{3}+64}{x+4}$ (a) Explain why direct substitution cannot be used to find the limit (b) Find the limit algebraically, if it exists.

Use a calculator to find the LRAM area approximation for the area under the graph $f(x)=-x^{3}+8$ from x=0 to x=2 with 20 approximating rectangles.

Use a calculator to find the RRAM area approximation for the area under the graph $f(x)=-x^{3}+8$ from x=0 to x=2 with 20 approximating rectangles.

At what points c in the domain of f(x) does $\lim _{x \rightarrow c} f(x)$ exist? $f(x)=\left\{\begin{array}{l}3 x & -5 \leq x<0 \\1 & x=0\\x^{2}+1& 0 < x \leq 2\\\end{array}\right.$

Determine $\lim _{x \rightarrow 0} \frac{\sin x}{x^{2}-x}$ if the limit exists.

Find the average rate of change of f(x) =5 \tan x over the interval $\left[\frac{\pi}{4}, \frac{3 \pi}{4}\right]$

Explain how to find the area under the graph of $f(x)=|x-2|$ from x=0 to x=4 by computing the geometric area.

Compute the integral $\int_{2}^{5} 4 x d x$ by computing a geometric area. (Hint: Graph the function first.)

Find the average rate of change of f(x) =3 \cos x over the interval $\left[\frac{\pi}{3}, \frac{2 \pi}{3}\right]$

What is the derivative of $f(x)=4 x^{2} ?$

What is the average rate of change of $f(x)=\sqrt{3 x-1}$ over the interval [1.9,2.1] ?