Question 1

Find the maximum and minimum points of the function. $F(x)=\left(1-x^{2}\right)^{5}+6 x^{6}$

Accepted Answer

0

Question 2

Find the critical number(s), if any, of the function $h(u)=\frac{7 u}{u^{2}+36}$ .

Accepted Answer

A)  $\frac{7}{36}$ 
B)  $\pm 6$ 
C)  $\frac{7}{6}$ 
D)  None of these 
A)  $\frac{7}{36}$ 
B)  $\pm 6$ 
C)  $\frac{7}{6}$ 
D)  None of these B

Question 3

Find the value of the limit. $\lim _{x \rightarrow 0} 3 \frac{\tan 5 x-5 x}{x^{3}}$

Accepted Answer

A)  $125$ 
B)  $185$ 
C)  $175$ 
D)  $195$ 
E)  $115$ 
A)  $125$ 
B)  $185$ 
C)  $175$ 
D)  $195$ 
E)  $115$ A

Question 4

Sketch the graph of the function $f(x)=x^{3}-3 x$ using the curve-sketching guidelines.

Accepted Answer

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

Question 5

The function $f(x)=x^{2}+10 x+27$ satisfies the hypotheses of Rolle's Theorem on the interval $[-10,0]$ . Find all values of c that satisfy the conclusion of the theorem.

Accepted Answer

A)  -5, -4 
B)  -5 
C)  -6, -4 
D)  -6 
A)  -5, -4 
B)  -5 
C)  -6, -4 
D)  -6

Question 6

Evaluate the limit. $\lim _{x \rightarrow \infty}\left(x-\sqrt{x^{2}+2}\right)$

Accepted Answer

A)  -2 
B)  1 
C)  0 
D)  2 
A)  -2 
B)  1 
C)  0 
D)  2

Question 7

An efficiency study showed that the total number of cell phones assembled by the average worker at a manufacturing company t hours after starting work at 8
a.m. is given by
 $N(t)=-0.5 t^{3}+3 t^{2}+7 t, \quad 0 \leq t \leq 4$ 
Sketch the graph of the function N, and interpret your result.

Accepted Answer

The answer of An efficiency study showed that the total...

Question 8

Sketch the curve. Find the equation of the slant asymptote. $y=\frac{x^{2}}{x-1}$

Accepted Answer

The answer of Sketch the curve. Find the equation of...

Question 9

Find the most general antiderivative of the function. $f(x)=5 \cos x-7 \sin x$

Accepted Answer

The answer of Find the most general antiderivative of the...

Question 10

The graph of the derivative $f^{\prime}(x)$ of a continuous function f is shown. On what intervals is f decreasing?   .

Accepted Answer

A)  $(2,6)$ 
B)  $(8,9)$ 
C)  $(-1,1)$ 
D)  $(4,6) \cup(7,9)$ 
E)  None of these 
A)  $(2,6)$ 
B)  $(8,9)$ 
C)  $(-1,1)$ 
D)  $(4,6) \cup(7,9)$ 
E)  None of these

Question 11

A rectangular beam will be cut from a cylindrical log of radius $r=40$ inches. Suppose that the strength of a rectangular beam is proportional to the product of its width and the square of its depth. Find the dimensions of the strongest beam that can be cut from the cylindrical log.

Accepted Answer

A)  $\sqrt{3}$ in, $80$ in 
B)  $\frac{80}{\sqrt{3}}$ in, $80 \sqrt{\frac{2}{3}}$ in 
C)  $\frac{80}{\sqrt{3}}$ in, $\frac{80}{\sqrt{3}}$ in 
D)  $80 \sqrt{\frac{2}{3}}$ in, $80 \sqrt{\frac{2}{3}}$ in 
E)  $80$ in, $\sqrt{\frac{2}{3}}$ in 
A)  $\sqrt{3}$ in, $80$ in 
B)  $\frac{80}{\sqrt{3}}$ in, $80 \sqrt{\frac{2}{3}}$ in 
C)  $\frac{80}{\sqrt{3}}$ in, $\frac{80}{\sqrt{3}}$ in 
D)  $80 \sqrt{\frac{2}{3}}$ in, $80 \sqrt{\frac{2}{3}}$ in 
E)  $80$ in, $\sqrt{\frac{2}{3}}$ in

Question 12

Find all the maximum and minimum values of the function. $F(x)=\frac{x-5}{(x-4)^{2}}$

Accepted Answer

The answer of Find all the maximum and minimum values...

Question 13

Find the critical numbers of $f(x)=x^{4}(x-2)^{3}$ .

Accepted Answer

A)  0, 3, $\frac{7}{8}$ 
B)  0, 2, $\frac{12}{11}$ 
C)  $0,2, \frac{8}{7}$ 
D)  0, 3, $\frac{12}{7}$ 
E)  0, 2, $\frac{7}{8}$ 
A)  0, 3, $\frac{7}{8}$ 
B)  0, 2, $\frac{12}{11}$ 
C)  $0,2, \frac{8}{7}$ 
D)  0, 3, $\frac{12}{7}$ 
E)  0, 2, $\frac{7}{8}$

Question 14

Evaluate the limit. $\lim _{x \rightarrow 5^{+}}\left(\frac{x}{x-5}-\frac{5}{\ln x}\right)$

Accepted Answer

A)  $\frac{1}{2}$ 
B)  -2 
C)  $\infty$ 
D)  0 
E)  2 
A)  $\frac{1}{2}$ 
B)  -2 
C)  $\infty$ 
D)  0 
E)  2

Question 15

Find the critical number(s), if any, of the function $f(x)=x^{3}-6 x-4$

Accepted Answer

A)  $4$ 
B)  -4 
C)  $\pm \sqrt{2}$ 
D)  $\pm 2$ 
A)  $4$ 
B)  -4 
C)  $\pm \sqrt{2}$ 
D)  $\pm 2$

Question 16

A manufacturer has been selling 1,200 television sets a week at $400 each. A market survey indicates that for each $30 rebate offered to the buyer, the number of sets sold will increase by 60 per week. Find the demand function.

Accepted Answer

A)  $p(x)=0.5 x+1,000$ 
B)  $p(x)=-0.5 x+1,000$ 
C)  $p(x)=-0.5 x+400.5$ 
D)  $p(x)=-0.5 x+600$ 
E)  $p(x)=0.5 x$ 
A)  $p(x)=0.5 x+1,000$ 
B)  $p(x)=-0.5 x+1,000$ 
C)  $p(x)=-0.5 x+400.5$ 
D)  $p(x)=-0.5 x+600$ 
E)  $p(x)=0.5 x$

Question 17

Find the limit. $\lim _{t \rightarrow 0} \frac{4^{t}-3^{t}}{t}$

Accepted Answer

A)  0 
B)  $\infty$ 
C)  $\ln 3-\ln 4$ 
D)  $\ln 4-\ln 3$ 
E)  1 
A)  0 
B)  $\infty$ 
C)  $\ln 3-\ln 4$ 
D)  $\ln 4-\ln 3$ 
E)  1

Question 18

Sketch the curve. $y=3 x^{3}+3 x$

Accepted Answer

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

Question 19

An apple orchard has an average yield of 32 bushels of apples per tree if tree density is 30 trees per acre. For each unit increase in tree density, the yield decreases by 2 bushels per tree. How many trees per acre should be planted to maximize yield?

Accepted Answer

A)  27 trees/acre 
B)  30 trees/acre 
C)  37 trees/acre 
D)  23 trees/acre 
A)  27 trees/acre 
B)  30 trees/acre 
C)  37 trees/acre 
D)  23 trees/acre

Question 20

Estimate the value of $\sqrt[3]{11}$ by using three iterations of Newton's method to solve the equation $x^{3}-11=0$ with initial estimate $x_{0}=2 \text {. }$ Round your final estimate to four decimal places.

Accepted Answer

A)  3.3166 
B)  2.253 
C)  3.2605 
D)  2.224 
A)  3.3166 
B)  2.253 
C)  3.2605 
D)  2.224

Find the maximum and minimum points of the function. $F(x)=\left(1-x^{2}\right)^{5}+6 x^{6}$

Find the critical number(s), if any, of the function $h(u)=\frac{7 u}{u^{2}+36}$ .

Find the value of the limit. $\lim _{x \rightarrow 0} 3 \frac{\tan 5 x-5 x}{x^{3}}$

Sketch the graph of the function $f(x)=x^{3}-3 x$ using the curve-sketching guidelines.

The function $f(x)=x^{2}+10 x+27$ satisfies the hypotheses of Rolle's Theorem on the interval $[-10,0]$ . Find all values of c that satisfy the conclusion of the theorem.

Evaluate the limit. $\lim _{x \rightarrow \infty}\left(x-\sqrt{x^{2}+2}\right)$

An efficiency study showed that the total number of cell phones assembled by the average worker at a manufacturing company t hours after starting work at 8 a.m. is given by $N(t)=-0.5 t^{3}+3 t^{2}+7 t, \quad 0 \leq t \leq 4$ Sketch the graph of the function N, and interpret your result.

Sketch the curve. Find the equation of the slant asymptote. $y=\frac{x^{2}}{x-1}$

Find the most general antiderivative of the function. $f(x)=5 \cos x-7 \sin x$

The graph of the derivative $f^{\prime}(x)$ of a continuous function f is shown. On what intervals is f decreasing? $The graph of the derivative f^{\prime}(x) of a continuous function f is shown. On what intervals is f decreasing? .$ .

A rectangular beam will be cut from a cylindrical log of radius $r=40$ inches. Suppose that the strength of a rectangular beam is proportional to the product of its width and the square of its depth. Find the dimensions of the strongest beam that can be cut from the cylindrical log.

Find all the maximum and minimum values of the function. $F(x)=\frac{x-5}{(x-4)^{2}}$

Find the critical numbers of $f(x)=x^{4}(x-2)^{3}$ .

Evaluate the limit. $\lim _{x \rightarrow 5^{+}}\left(\frac{x}{x-5}-\frac{5}{\ln x}\right)$

Find the critical number(s), if any, of the function $f(x)=x^{3}-6 x-4$

A manufacturer has been selling 1,200 television sets a week at $400 each. A market survey indicates that for each $30 rebate offered to the buyer, the number of sets sold will increase by 60 per week. Find the demand function.

Find the limit. $\lim _{t \rightarrow 0} \frac{4^{t}-3^{t}}{t}$

Sketch the curve. $y=3 x^{3}+3 x$

An apple orchard has an average yield of 32 bushels of apples per tree if tree density is 30 trees per acre. For each unit increase in tree density, the yield decreases by 2 bushels per tree. How many trees per acre should be planted to maximize yield?

Estimate the value of $\sqrt[3]{11}$ by using three iterations of Newton's method to solve the equation $x^{3}-11=0$ with initial estimate $x_{0}=2 \text {. }$ Round your final estimate to four decimal places.

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 maximum and minimum points of the function. F(x)=(1−x2)5+6x6F(x)=\left(1-x^{2}\right)^{5}+6 x^{6}F(x)=(1−x2)5+6x6

Find the critical number(s), if any, of the function h(u)=7uu2+36h(u)=\frac{7 u}{u^{2}+36}h(u)=u2+367u​ .

Find the value of the limit. lim⁡x→03tan⁡5x−5xx3\lim _{x \rightarrow 0} 3 \frac{\tan 5 x-5 x}{x^{3}}limx→0​3x3tan5x−5x​

Sketch the graph of the function f(x)=x3−3xf(x)=x^{3}-3 xf(x)=x3−3x using the curve-sketching guidelines.

The function f(x)=x2+10x+27f(x)=x^{2}+10 x+27f(x)=x2+10x+27 satisfies the hypotheses of Rolle's Theorem on the interval [−10,0][-10,0][−10,0] . Find all values of c that satisfy the conclusion of the theorem.

Evaluate the limit. lim⁡x→∞(x−x2+2)\lim _{x \rightarrow \infty}\left(x-\sqrt{x^{2}+2}\right)limx→∞​(x−x2+2​)

Sketch the curve. Find the equation of the slant asymptote. y=x2x−1y=\frac{x^{2}}{x-1}y=x−1x2​

Find the most general antiderivative of the function. f(x)=5cos⁡x−7sin⁡xf(x)=5 \cos x-7 \sin xf(x)=5cosx−7sinx

The graph of the derivative f′(x)f^{\prime}(x)f′(x) of a continuous function f is shown. On what intervals is f decreasing? .

A rectangular beam will be cut from a cylindrical log of radius r=40r=40r=40 inches. Suppose that the strength of a rectangular beam is proportional to the product of its width and the square of its depth. Find the dimensions of the strongest beam that can be cut from the cylindrical log.

Find all the maximum and minimum values of the function. F(x)=x−5(x−4)2F(x)=\frac{x-5}{(x-4)^{2}}F(x)=(x−4)2x−5​

Find the critical numbers of f(x)=x4(x−2)3f(x)=x^{4}(x-2)^{3}f(x)=x4(x−2)3 .

Evaluate the limit. lim⁡x→5+(xx−5−5ln⁡x)\lim _{x \rightarrow 5^{+}}\left(\frac{x}{x-5}-\frac{5}{\ln x}\right)limx→5+​(x−5x​−lnx5​)

Find the critical number(s), if any, of the function f(x)=x3−6x−4f(x)=x^{3}-6 x-4f(x)=x3−6x−4

A manufacturer has been selling 1,200 television sets a week at $400 each. A market survey indicates that for each $30 rebate offered to the buyer, the number of sets sold will increase by 60 per week. Find the demand function.

Find the limit. lim⁡t→04t−3tt\lim _{t \rightarrow 0} \frac{4^{t}-3^{t}}{t}limt→0​t4t−3t​

Sketch the curve. y=3x3+3xy=3 x^{3}+3 xy=3x3+3x

An apple orchard has an average yield of 32 bushels of apples per tree if tree density is 30 trees per acre. For each unit increase in tree density, the yield decreases by 2 bushels per tree. How many trees per acre should be planted to maximize yield?

Estimate the value of 113\sqrt[3]{11}311​ by using three iterations of Newton's method to solve the equation x3−11=0x^{3}-11=0x3−11=0 with initial estimate x0=2. x_{0}=2 \text {. }x0​=2. Round your final estimate to four decimal places.

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 maximum and minimum points of the function. $F(x)=\left(1-x^{2}\right)^{5}+6 x^{6}$

Find the critical number(s), if any, of the function $h(u)=\frac{7 u}{u^{2}+36}$ .

Find the value of the limit. $\lim _{x \rightarrow 0} 3 \frac{\tan 5 x-5 x}{x^{3}}$

Sketch the graph of the function $f(x)=x^{3}-3 x$ using the curve-sketching guidelines.

The function $f(x)=x^{2}+10 x+27$ satisfies the hypotheses of Rolle's Theorem on the interval $[-10,0]$ . Find all values of c that satisfy the conclusion of the theorem.

Evaluate the limit. $\lim _{x \rightarrow \infty}\left(x-\sqrt{x^{2}+2}\right)$

Sketch the curve. Find the equation of the slant asymptote. $y=\frac{x^{2}}{x-1}$

Find the most general antiderivative of the function. $f(x)=5 \cos x-7 \sin x$

The graph of the derivative $f^{\prime}(x)$ of a continuous function f is shown. On what intervals is f decreasing? $The graph of the derivative f^{\prime}(x) of a continuous function f is shown. On what intervals is f decreasing? .$ .

A rectangular beam will be cut from a cylindrical log of radius $r=40$ inches. Suppose that the strength of a rectangular beam is proportional to the product of its width and the square of its depth. Find the dimensions of the strongest beam that can be cut from the cylindrical log.

Find all the maximum and minimum values of the function. $F(x)=\frac{x-5}{(x-4)^{2}}$

Find the critical numbers of $f(x)=x^{4}(x-2)^{3}$ .

Evaluate the limit. $\lim _{x \rightarrow 5^{+}}\left(\frac{x}{x-5}-\frac{5}{\ln x}\right)$

Find the critical number(s), if any, of the function $f(x)=x^{3}-6 x-4$

Find the limit. $\lim _{t \rightarrow 0} \frac{4^{t}-3^{t}}{t}$

Sketch the curve. $y=3 x^{3}+3 x$

Estimate the value of $\sqrt[3]{11}$ by using three iterations of Newton's method to solve the equation $x^{3}-11=0$ with initial estimate $x_{0}=2 \text {. }$ Round your final estimate to four decimal places.