Question 1

Find the intervals where $f(x)=\frac{\log 9 x}{x}$ is increasing and where it is decreasing.

Accepted Answer

A)  increasing: $(9 e, \infty)$ , decreasing: $(0,9 e)$ 
B)  increasing: $\left(0, \frac{e}{9}\right)$ , decreasing: $\left(\frac{e}{9}, \infty\right)$ 
C)  increasing: $(90, \infty)$ , decreasing: $(0,90)$ 
D)  increasing: $\left(0, \frac{9}{10}\right)$ , decreasing: $\left(\frac{9}{10}, \infty\right)$ 
A)  increasing: $(9 e, \infty)$ , decreasing: $(0,9 e)$ 
B)  increasing: $\left(0, \frac{e}{9}\right)$ , decreasing: $\left(\frac{e}{9}, \infty\right)$ 
C)  increasing: $(90, \infty)$ , decreasing: $(0,90)$ 
D)  increasing: $\left(0, \frac{9}{10}\right)$ , decreasing: $\left(\frac{9}{10}, \infty\right)$

Question 2

A production editor decided that a promotional flyer should have a 1-in. margin at the top and the bottom, and a $\frac{1}{2}$ -in. margin on each side. The editor further stipulated that the flyer should have an area of 392 $\text { in. }^{2}$ . Determine the dimensions of the flyer that will result in the maximum printed area on the flyer.

Accepted Answer

A)  7 in. $x$ 56 in. 
B)  $7 \sqrt{2}$ in. $x$ $7 \sqrt{2}$ in. 
C)  14 in. $x$ 28 in. 
D)  $14 \sqrt{2}$ in. $x$ $14 \sqrt{2}$ in. 
A)  7 in. $x$ 56 in. 
B)  $7 \sqrt{2}$ in. $x$ $7 \sqrt{2}$ in. 
C)  14 in. $x$ 28 in. 
D)  $14 \sqrt{2}$ in. $x$ $14 \sqrt{2}$ in.

Question 3

What can you say about point of inflation for $f(x)=x e^{-3 x}$ ?

Accepted Answer

A)  point of inflation gets closer to $\pm$ 6 
B)  point of inflation goes away from the origin 
C)  point of inflation gets closer to the origin 
D)  point of inflation goes away from $\pm$ 3 
A)  point of inflation gets closer to $\pm$ 6 
B)  point of inflation goes away from the origin 
C)  point of inflation gets closer to the origin 
D)  point of inflation goes away from $\pm$ 3

Question 4

If an open box is made from a metal sheet 9 in. square by cutting out identical squares from each corner an bending up the resulting flaps, determine the dimensions of the box with the largest volume that can be made.

Accepted Answer

The answer of If an open box is made from...

Question 5

Find the absolute maximum of the function. $f(x)=\sin (2 x)+2 \cos (x)$ on the interval $\left[0, \frac{\pi}{2}\right]$

Accepted Answer

The answer of Find the absolute maximum of the function....

Question 6

Find the limit. $\lim _{y \rightarrow-\infty}\left(\frac{5 y+\sqrt{y^{2}+2 y}}{y}\right)$

Accepted Answer

The answer of Find the limit. \(\lim _{y \rightarrow-\infty}\left(\frac{5 y+\sqrt{y^{2}+2...

Question 7

Sketch the graph of the function $f(x)=x-\ln 6 x$ using the curve-sketching guidelines.

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Question 8

Find two numbers whose difference is 170 and whose product is a minimum.

Accepted Answer

The answer of Find two numbers whose difference is 170...

Question 9

The average cost of producing x units of a commodity is given by the equation $C(x)=20.4-0.0007 x$ .
Find the marginal cost at a production level of 1,255 units.

Accepted Answer

The answer of The average cost of producing x units...

Question 10

How many real roots does the equation $x^{5}-7 x+c=0$ have in the interval $[-1,1]$ ?

Accepted Answer

A)  at most five real roots 
B)  no real roots 
C)  at most one real root 
D)  at most two real roots 
E)  at most three real roots 
A)  at most five real roots 
B)  no real roots 
C)  at most one real root 
D)  at most two real roots 
E)  at most three real roots

Question 11

Evaluate the limit using l'Hôpital's Rule. $\lim _{x \rightarrow 0^{+}} \frac{2 e^{x^{2}}+x-2}{2-\sqrt{4-x^{2}}}$

Accepted Answer

A)  ($\infty$) 
B)  1 
C)  e 
D)  0 
A)  ($\infty$) 
B)  1 
C)  e 
D)  0

Question 12

Evaluate the limit using l'Hôpital's Rule. $\lim _{x \rightarrow 0} \frac{(7 \sin x)^{2}}{1-\sec x}$

Accepted Answer

A)  -7 
B)  -98 
C)  -49 
D)  7 
A)  -7 
B)  -98 
C)  -49 
D)  7

Question 13

Suppose the line $y=7 x-1$ is tangent to the curve $y=f(x)$ when $x=-8$ . If Newton's method is used to locate a root of the equation $f(x)=0$ and the initial approximation is $x_{1}=-8$ , find the second approximation $x_{2}$ .

Accepted Answer

The answer of Suppose the line $y=7 x-1$ is tangent...

Question 14

Find the absolute maximum and absolute minimum values, if any, of the function $f(x)=x \ln x-x$ on $\left[\frac{1}{2}, 7\right]$ .

Accepted Answer

None

Question 15

Find the dimensions of a rectangle of area 64 $\mathrm{t}^{2}$ that has the smallest possible perimeter.

Accepted Answer

A)  4 ft by 16 ft 
B)  2 ft by 32 ft 
C)  1 ft by 64 ft 
D)  8 ft by 8 ft 
A)  4 ft by 16 ft 
B)  2 ft by 32 ft 
C)  1 ft by 64 ft 
D)  8 ft by 8 ft

Question 16

You are given the graph of the second derivative $f^{\prime\prime}$ of a function f.
   
(a) Determine the intervals where the graph of f is concave upward and where it is concave downward. (b) Find the x-coordinates all inflection points of f.

Accepted Answer

CU on @#LAT-DLM& and

Question 17

Find the number c that satisfies the conclusion of the Mean Value Theorem on the given interval. $f(x)=2 \sqrt{x}$ , $[0,25]$

Accepted Answer

A)  $c=\frac{1}{5}$ 
B)  $c=0$ 
C)  $c=5$ 
D)  $c=$ $\frac{25}{4}$ 
E)  None of these 
A)  $c=\frac{1}{5}$ 
B)  $c=0$ 
C)  $c=5$ 
D)  $c=$ $\frac{25}{4}$ 
E)  None of these

Question 18

A rectangular box having a top and a square base is to be constructed at a cost of $1. If the material for the bottom costs $0.35 per square foot, the material for the top costs $0.15 per square foot, and the material for the sides costs $0.20 per square foot, find the dimensions and volume of the box of maximum volume that can be constructed.

Accepted Answer

The answer of A rectangular box having a top and...

Question 19

For what values of c does the curve have maximum and minimum points? $F(x)=8 x^{3}+c x^{2}+10 x$

Accepted Answer

A)  $|c|>\sqrt{30}$ 
B)  $|c|>\sqrt{160}$ 
C)  $|c|>480$ 
D)  $|c|>\sqrt{240}$ 
E)  $|c|>15$ 
A)  $|c|>\sqrt{30}$ 
B)  $|c|>\sqrt{160}$ 
C)  $|c|>480$ 
D)  $|c|>\sqrt{240}$ 
E)  $|c|>15$

Question 20

Evaluate the limit $\lim _{x \rightarrow 6}\left(\frac{1}{\ln (x-5)}-\frac{1}{x-6}\right)$ using l'Hôpital's Rule.

Accepted Answer

A)  $-\frac{1}{30}$ 
B)  $\frac{1}{30}$ 
C)  $\frac{1}{2}$ 
D)  $\frac{11}{30}$ 
A)  $-\frac{1}{30}$ 
B)  $\frac{1}{30}$ 
C)  $\frac{1}{2}$ 
D)  $\frac{11}{30}$

Find the intervals where $f(x)=\frac{\log 9 x}{x}$ is increasing and where it is decreasing.

What can you say about point of inflation for $f(x)=x e^{-3 x}$ ?

If an open box is made from a metal sheet 9 in. square by cutting out identical squares from each corner an bending up the resulting flaps, determine the dimensions of the box with the largest volume that can be made.

Find the absolute maximum of the function. $f(x)=\sin (2 x)+2 \cos (x)$ on the interval $\left[0, \frac{\pi}{2}\right]$

Find the limit. $\lim _{y \rightarrow-\infty}\left(\frac{5 y+\sqrt{y^{2}+2 y}}{y}\right)$

Sketch the graph of the function $f(x)=x-\ln 6 x$ using the curve-sketching guidelines.

Find two numbers whose difference is 170 and whose product is a minimum.

The average cost of producing x units of a commodity is given by the equation $C(x)=20.4-0.0007 x$ . Find the marginal cost at a production level of 1,255 units.

How many real roots does the equation $x^{5}-7 x+c=0$ have in the interval $[-1,1]$ ?

Evaluate the limit using l'Hôpital's Rule. $\lim _{x \rightarrow 0^{+}} \frac{2 e^{x^{2}}+x-2}{2-\sqrt{4-x^{2}}}$

Evaluate the limit using l'Hôpital's Rule. $\lim _{x \rightarrow 0} \frac{(7 \sin x)^{2}}{1-\sec x}$

Suppose the line $y=7 x-1$ is tangent to the curve $y=f(x)$ when $x=-8$ . If Newton's method is used to locate a root of the equation $f(x)=0$ and the initial approximation is $x_{1}=-8$ , find the second approximation $x_{2}$ .

Find the absolute maximum and absolute minimum values, if any, of the function $f(x)=x \ln x-x$ on $\left[\frac{1}{2}, 7\right]$ .

Find the dimensions of a rectangle of area 64 $\mathrm{t}^{2}$ that has the smallest possible perimeter.

Find the number c that satisfies the conclusion of the Mean Value Theorem on the given interval. $f(x)=2 \sqrt{x}$ , $[0,25]$

For what values of c does the curve have maximum and minimum points? $F(x)=8 x^{3}+c x^{2}+10 x$

Evaluate the limit $\lim _{x \rightarrow 6}\left(\frac{1}{\ln (x-5)}-\frac{1}{x-6}\right)$ using l'Hôpital's Rule.

Functions and Limits

Derivatives

Applications of Differentiation

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 4: Integrals

Find the intervals where f(x)=log⁡9xxf(x)=\frac{\log 9 x}{x}f(x)=xlog9x​ is increasing and where it is decreasing.

What can you say about point of inflation for f(x)=xe−3xf(x)=x e^{-3 x}f(x)=xe−3x ?

If an open box is made from a metal sheet 9 in. square by cutting out identical squares from each corner an bending up the resulting flaps, determine the dimensions of the box with the largest volume that can be made.

Find the absolute maximum of the function. f(x)=sin⁡(2x)+2cos⁡(x)f(x)=\sin (2 x)+2 \cos (x)f(x)=sin(2x)+2cos(x) on the interval [0,π2]\left[0, \frac{\pi}{2}\right][0,2π​]

Find the limit. lim⁡y→−∞(5y+y2+2yy)\lim _{y \rightarrow-\infty}\left(\frac{5 y+\sqrt{y^{2}+2 y}}{y}\right)limy→−∞​(y5y+y2+2y​​)

Sketch the graph of the function f(x)=x−ln⁡6xf(x)=x-\ln 6 xf(x)=x−ln6x using the curve-sketching guidelines.

Find two numbers whose difference is 170 and whose product is a minimum.

The average cost of producing x units of a commodity is given by the equation C(x)=20.4−0.0007xC(x)=20.4-0.0007 xC(x)=20.4−0.0007x . Find the marginal cost at a production level of 1,255 units.

How many real roots does the equation x5−7x+c=0x^{5}-7 x+c=0x5−7x+c=0 have in the interval [−1,1][-1,1][−1,1] ?

Evaluate the limit using l'Hôpital's Rule. lim⁡x→0+2ex2+x−22−4−x2\lim _{x \rightarrow 0^{+}} \frac{2 e^{x^{2}}+x-2}{2-\sqrt{4-x^{2}}}limx→0+​2−4−x2​2ex2+x−2​

Evaluate the limit using l'Hôpital's Rule. lim⁡x→0(7sin⁡x)21−sec⁡x\lim _{x \rightarrow 0} \frac{(7 \sin x)^{2}}{1-\sec x}limx→0​1−secx(7sinx)2​

Suppose the line y=7x−1y=7 x-1y=7x−1 is tangent to the curve y=f(x)y=f(x)y=f(x) when x=−8x=-8x=−8 . If Newton's method is used to locate a root of the equation f(x)=0f(x)=0f(x)=0 and the initial approximation is x1=−8x_{1}=-8x1​=−8 , find the second approximation x2x_{2}x2​ .

Find the absolute maximum and absolute minimum values, if any, of the function f(x)=xln⁡x−xf(x)=x \ln x-xf(x)=xlnx−x on [12,7]\left[\frac{1}{2}, 7\right][21​,7] .

Find the dimensions of a rectangle of area 64 t2\mathrm{t}^{2}t2 that has the smallest possible perimeter.

You are given the graph of the second derivative f′′f^{\prime\prime}f′′ of a function f. (a) Determine the intervals where the graph of f is concave upward and where it is concave downward. (b) Find the x-coordinates all inflection points of f.

Find the number c that satisfies the conclusion of the Mean Value Theorem on the given interval. f(x)=2xf(x)=2 \sqrt{x}f(x)=2x​ , [0,25][0,25][0,25]

For what values of c does the curve have maximum and minimum points? F(x)=8x3+cx2+10xF(x)=8 x^{3}+c x^{2}+10 xF(x)=8x3+cx2+10x

Evaluate the limit lim⁡x→6(1ln⁡(x−5)−1x−6)\lim _{x \rightarrow 6}\left(\frac{1}{\ln (x-5)}-\frac{1}{x-6}\right)limx→6​(ln(x−5)1​−x−61​) using l'Hôpital's Rule.

Functions and Limits

Derivatives

Applications of Differentiation

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 intervals where $f(x)=\frac{\log 9 x}{x}$ is increasing and where it is decreasing.

What can you say about point of inflation for $f(x)=x e^{-3 x}$ ?

Find the absolute maximum of the function. $f(x)=\sin (2 x)+2 \cos (x)$ on the interval $\left[0, \frac{\pi}{2}\right]$

Find the limit. $\lim _{y \rightarrow-\infty}\left(\frac{5 y+\sqrt{y^{2}+2 y}}{y}\right)$

Sketch the graph of the function $f(x)=x-\ln 6 x$ using the curve-sketching guidelines.

The average cost of producing x units of a commodity is given by the equation $C(x)=20.4-0.0007 x$ . Find the marginal cost at a production level of 1,255 units.

How many real roots does the equation $x^{5}-7 x+c=0$ have in the interval $[-1,1]$ ?

Evaluate the limit using l'Hôpital's Rule. $\lim _{x \rightarrow 0^{+}} \frac{2 e^{x^{2}}+x-2}{2-\sqrt{4-x^{2}}}$

Evaluate the limit using l'Hôpital's Rule. $\lim _{x \rightarrow 0} \frac{(7 \sin x)^{2}}{1-\sec x}$

Suppose the line $y=7 x-1$ is tangent to the curve $y=f(x)$ when $x=-8$ . If Newton's method is used to locate a root of the equation $f(x)=0$ and the initial approximation is $x_{1}=-8$ , find the second approximation $x_{2}$ .

Find the absolute maximum and absolute minimum values, if any, of the function $f(x)=x \ln x-x$ on $\left[\frac{1}{2}, 7\right]$ .

Find the dimensions of a rectangle of area 64 $\mathrm{t}^{2}$ that has the smallest possible perimeter.

Find the number c that satisfies the conclusion of the Mean Value Theorem on the given interval. $f(x)=2 \sqrt{x}$ , $[0,25]$

For what values of c does the curve have maximum and minimum points? $F(x)=8 x^{3}+c x^{2}+10 x$

Evaluate the limit $\lim _{x \rightarrow 6}\left(\frac{1}{\ln (x-5)}-\frac{1}{x-6}\right)$ using l'Hôpital's Rule.