Question 1

Find the most general antiderivative of the function. $f(x)=15 x^{2}-16 x+9$

Accepted Answer

A)  $F(x)=5 x^{3}-8 x^{2}+9 x+C$ 
B)  $F(x)=6 x^{2}-14 x+C$ 
C)  $F(x)=36 x-14+C$ 
D)  $F(x)=30 x^{5}-28 x^{4}+9 x+C$ 
E)  $F(x)=18 x^{3}-14 x^{2}+9 x+C$ 
A)  $F(x)=5 x^{3}-8 x^{2}+9 x+C$ 
B)  $F(x)=6 x^{2}-14 x+C$ 
C)  $F(x)=36 x-14+C$ 
D)  $F(x)=30 x^{5}-28 x^{4}+9 x+C$ 
E)  $F(x)=18 x^{3}-14 x^{2}+9 x+C$

Question 2

Find the maximum area of a rectangle that can be circumscribed about a given rectangle with length L = 8 and width W = 3.

Accepted Answer

The answer of Find the maximum area of a rectangle...

Question 3

What is the shortest possible length of the line segment that is cut off by the first quadrant and is tangent to the curve $y=\frac{5}{x}$ at some point?

Accepted Answer

A)  $\frac{\sqrt{2}}{5}$ 
B)  $\sqrt{2}$ 
C)  $\frac{1}{5}$ 
D)  None of these 
E)  $2 \sqrt{10}$ 
A)  $\frac{\sqrt{2}}{5}$ 
B)  $\sqrt{2}$ 
C)  $\frac{1}{5}$ 
D)  None of these 
E)  $2 \sqrt{10}$

Question 4

Find the point(s) of inflection of the graph of $f(x)=\cos ^{-1} 7 x$

Accepted Answer

A)  $(1,0)$ , $\left(0, \frac{\pi}{2}\right)$ , and $\left(0, \frac{\pi}{7}\right)$ 
B)  $(1,0)$ 
C)  $\left(0, \frac{\pi}{7}\right)$ 
D)  $\left(0, \frac{\pi}{2}\right)$ 
A)  $(1,0)$ , $\left(0, \frac{\pi}{2}\right)$ , and $\left(0, \frac{\pi}{7}\right)$ 
B)  $(1,0)$ 
C)  $\left(0, \frac{\pi}{7}\right)$ 
D)  $\left(0, \frac{\pi}{2}\right)$

Question 5

Find the critical number(s), if any, of the function $g(t)=t^{2} \ln 3 t$

Accepted Answer

A)  $e^{\frac{-1}{3}}$ 
B)  $e^{\frac{1}{3}}$ 
C)  $\frac{1}{3} e^{\frac{1}{3}}$ 
D)  $\frac{1}{3} e^{\frac{-1}{2}}$ 
A)  $e^{\frac{-1}{3}}$ 
B)  $e^{\frac{1}{3}}$ 
C)  $\frac{1}{3} e^{\frac{1}{3}}$ 
D)  $\frac{1}{3} e^{\frac{-1}{2}}$

Question 6

Determine where the graph of the function $f(x)=x^{3}-30 x$ is concave upward and where it is concave downward. Also, find all inflection points of the function.

Accepted Answer

A)  CU on $(-\sqrt{10}, \sqrt{10})$ , CD on $(-\infty,-\sqrt{10})$ and $(\sqrt{10}, \infty)$
IP $(-\sqrt{10}, 20 \sqrt{10})$ and $(\sqrt{10},-20 \sqrt{10})$ 
B)  CU on $(0, \infty)$ , CD on $(-\infty, 0)$ ,
IP $(0,0)$ 
C)  CU on $(-\sqrt{10}, \infty)$ , CD on $(-\infty,-\sqrt{10})$ ,
IP $(-\sqrt{10}, 20 \sqrt{10})$ 
D)  CU on $(\sqrt{10}, \infty)$ , CD on $(-\infty, \sqrt{10})$ ,
IP $(\sqrt{10},-20 \sqrt{10})$ 
A)  CU on $(-\sqrt{10}, \sqrt{10})$ , CD on $(-\infty,-\sqrt{10})$ and $(\sqrt{10}, \infty)$
IP $(-\sqrt{10}, 20 \sqrt{10})$ and $(\sqrt{10},-20 \sqrt{10})$ 
B)  CU on $(0, \infty)$ , CD on $(-\infty, 0)$ ,
IP $(0,0)$ 
C)  CU on $(-\sqrt{10}, \infty)$ , CD on $(-\infty,-\sqrt{10})$ ,
IP $(-\sqrt{10}, 20 \sqrt{10})$ 
D)  CU on $(\sqrt{10}, \infty)$ , CD on $(-\infty, \sqrt{10})$ ,
IP $(\sqrt{10},-20 \sqrt{10})$

Question 7

Find the relative extrema, if any, of $f(t)=e^{t}-7 t-3$ . Use the Second Derivative Test, if possible.

Accepted Answer

A)  relative minimum: none, relative maximum: $f(\ln 3)=-7 \ln 7$ 
B)  relative minimum: $f(\ln 7)=4-7 \ln 7$ , relative maximum: none 
C)  relative minimum: $f(\ln 3)=-7 \ln 7$ , relative maximum: none 
D)  relative minimum: none, relative maximum: $f(\ln 7)=4-7 \ln 7$ 
A)  relative minimum: none, relative maximum: $f(\ln 3)=-7 \ln 7$ 
B)  relative minimum: $f(\ln 7)=4-7 \ln 7$ , relative maximum: none 
C)  relative minimum: $f(\ln 3)=-7 \ln 7$ , relative maximum: none 
D)  relative minimum: none, relative maximum: $f(\ln 7)=4-7 \ln 7$

Question 8

Use Newton's method with the specified initial approximation $x_{1}$ to find $x_{4}$ , the third approximation to the root of the given equation. (Give your answer to four decimal places.) $x^{4}-12=0, x_{1}=6$

Accepted Answer

The answer of Use Newton's method with the specified initial...

Question 9

Use Newton's method to obtain an approximation to the root of $\cos ^{-1} x-6 x=0$ to within 0.00001.

Accepted Answer

A)  0.20844 
B)  0.2443 
C)  0.22413 
D)  0.2712 
A)  0.20844 
B)  0.2443 
C)  0.22413 
D)  0.2712

Question 10

Estimate the absolute minimum value of the function $y=x^{3}-6 x+2$ to two decimal places on the interval $[0,10]$ .   __________

Accepted Answer

The answer of Estimate the absolute minimum value of the...

Question 11

Find the absolute maximum and absolute minimum values, if any, of the function $f(x)=x-\sqrt{x}$ on [0, 25]

Accepted Answer

A)  Abs. max. $f(25)=20$
Abs) min. $f\left(\frac{1}{4}\right)=-\frac{1}{4}$ 
B)  Abs. max. $f(25)=20$
Abs) min. $f(0)=0$ 
C)  Abs. max. $f(25)=20$
Abs) min. $f(0)=f(1)=0$ 
D)  Abs. max. $f(0)=0$
Abs) min. $f\left(\frac{1}{4}\right)=-\frac{1}{4}$ 
A)  Abs. max. $f(25)=20$
Abs) min. $f\left(\frac{1}{4}\right)=-\frac{1}{4}$ 
B)  Abs. max. $f(25)=20$
Abs) min. $f(0)=0$ 
C)  Abs. max. $f(25)=20$
Abs) min. $f(0)=f(1)=0$ 
D)  Abs. max. $f(0)=0$
Abs) min. $f\left(\frac{1}{4}\right)=-\frac{1}{4}$

Question 12

A cylindrical tank of height 16 feet is filled with water. Suppose a jet of water flows through an orifice on the tank. It can be shown that the range R (in feet) of the jet of water is given by $R=2 \sqrt{x(16-x)}$ .
Where should the orifice be located so that the jet of water will have the maximum range?

Accepted Answer

Halfway up the side

Question 13

Find the critical number(s), if any, of the function $f(x)=e^{x^{2}-x}$ .

Accepted Answer

A)  $\frac{1}{2}$ 
B)  $\frac{1}{2}$ and   $\frac{1}{2}$ 
C)  1 
D)  1 and   1 
A)  $\frac{1}{2}$ 
B)  $\frac{1}{2}$ and   $\frac{1}{2}$ 
C)  1 
D)  1 and   1

Question 14

Use Newton's method to approximate the zero of $f(x)=x^{3}+3 x+3$ between $x=0$ and $x=2$ using $x_{0}=1$ . Continue until two successive approximations differ by less than 0.00001.

Accepted Answer

The answer of Use Newton's method to approximate the zero...

Question 15

Find the position function of a particle moving along a coordinate line that satisfies the given condition. $ \mathcal{v}(t)=3 t^{2}-4 t+4 $ , s(1) = -1

Accepted Answer

The answer of Find the position function of a particle...

Question 16

Find a cubic function $f(x)=a x^{3}-b x^{2}+c x-d$ that has a local maximum value of $115$ at 1 and a local minimum value of -1,184 at 7.

Accepted Answer

The answer of Find a cubic function \(f(x)=a x^{3}-b x^{2}+c...

Question 17

Find the critical numbers of the function. $y=\frac{x}{x^{2}+36}$

Accepted Answer

The answer of Find the critical numbers of the function....

Question 18

How many points of inflection are on the graph of the function? $f(x)=18 x^{3}+5 x^{2}-12 x-20$

Accepted Answer

A)  5 
B)  2 
C)  3 
D)  4 
E)  1 
A)  5 
B)  2 
C)  3 
D)  4 
E)  1

Question 19

Find the absolute maximum and absolute minimum values, if any, of the function $f(x)=x e^{-4 x}$ on $[-1,1]$

Accepted Answer

None

Question 20

Find the absolute maximum value of $y=\sqrt{16-x^{2}}$ on the interval $[-6,6]$ .

Accepted Answer

A)  8 
B)  5 
C)  7 
D)  6 
E)  4 
A)  8 
B)  5 
C)  7 
D)  6 
E)  4

Find the most general antiderivative of the function. $f(x)=15 x^{2}-16 x+9$

Find the maximum area of a rectangle that can be circumscribed about a given rectangle with length L = 8 and width W = 3.

What is the shortest possible length of the line segment that is cut off by the first quadrant and is tangent to the curve $y=\frac{5}{x}$ at some point?

Find the point(s) of inflection of the graph of $f(x)=\cos ^{-1} 7 x$

Find the critical number(s), if any, of the function $g(t)=t^{2} \ln 3 t$

Determine where the graph of the function $f(x)=x^{3}-30 x$ is concave upward and where it is concave downward. Also, find all inflection points of the function.

Find the relative extrema, if any, of $f(t)=e^{t}-7 t-3$ . Use the Second Derivative Test, if possible.

Use Newton's method with the specified initial approximation $x_{1}$ to find $x_{4}$ , the third approximation to the root of the given equation. (Give your answer to four decimal places.) $x^{4}-12=0, x_{1}=6$

Use Newton's method to obtain an approximation to the root of $\cos ^{-1} x-6 x=0$ to within 0.00001.

Estimate the absolute minimum value of the function $y=x^{3}-6 x+2$ to two decimal places on the interval $[0,10]$ . $Estimate the absolute minimum value of the function y=x^{3}-6 x+2 to two decimal places on the interval [0,10] . $

Find the absolute maximum and absolute minimum values, if any, of the function $f(x)=x-\sqrt{x}$ on [0, 25]

Find the critical number(s), if any, of the function $f(x)=e^{x^{2}-x}$ .

Use Newton's method to approximate the zero of $f(x)=x^{3}+3 x+3$ between $x=0$ and $x=2$ using $x_{0}=1$ . Continue until two successive approximations differ by less than 0.00001.

Find the position function of a particle moving along a coordinate line that satisfies the given condition. $\mathcal{v}(t)=3 t^{2}-4 t+4$ , s(1) = -1

Find a cubic function $f(x)=a x^{3}-b x^{2}+c x-d$ that has a local maximum value of $115$ at 1 and a local minimum value of -1,184 at 7.

Find the critical numbers of the function. $y=\frac{x}{x^{2}+36}$

How many points of inflection are on the graph of the function? $f(x)=18 x^{3}+5 x^{2}-12 x-20$

Find the absolute maximum and absolute minimum values, if any, of the function $f(x)=x e^{-4 x}$ on $[-1,1]$

Find the absolute maximum value of $y=\sqrt{16-x^{2}}$ on the interval $[-6,6]$ .

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 most general antiderivative of the function. f(x)=15x2−16x+9f(x)=15 x^{2}-16 x+9f(x)=15x2−16x+9

Find the maximum area of a rectangle that can be circumscribed about a given rectangle with length L = 8 and width W = 3.

What is the shortest possible length of the line segment that is cut off by the first quadrant and is tangent to the curve y=5xy=\frac{5}{x}y=x5​ at some point?

Find the point(s) of inflection of the graph of f(x)=cos⁡−17xf(x)=\cos ^{-1} 7 xf(x)=cos−17x

Find the critical number(s), if any, of the function g(t)=t2ln⁡3tg(t)=t^{2} \ln 3 tg(t)=t2ln3t

Determine where the graph of the function f(x)=x3−30xf(x)=x^{3}-30 xf(x)=x3−30x is concave upward and where it is concave downward. Also, find all inflection points of the function.

Find the relative extrema, if any, of f(t)=et−7t−3f(t)=e^{t}-7 t-3f(t)=et−7t−3 . Use the Second Derivative Test, if possible.

Use Newton's method with the specified initial approximation x1x_{1}x1​ to find x4x_{4}x4​ , the third approximation to the root of the given equation. (Give your answer to four decimal places.) x4−12=0,x1=6x^{4}-12=0, x_{1}=6x4−12=0,x1​=6

Use Newton's method to obtain an approximation to the root of cos⁡−1x−6x=0\cos ^{-1} x-6 x=0cos−1x−6x=0 to within 0.00001.

Estimate the absolute minimum value of the function y=x3−6x+2y=x^{3}-6 x+2y=x3−6x+2 to two decimal places on the interval [0,10][0,10][0,10] . __________

Find the absolute maximum and absolute minimum values, if any, of the function f(x)=x−xf(x)=x-\sqrt{x}f(x)=x−x​ on [0, 25]

Find the critical number(s), if any, of the function f(x)=ex2−xf(x)=e^{x^{2}-x}f(x)=ex2−x .

Use Newton's method to approximate the zero of f(x)=x3+3x+3f(x)=x^{3}+3 x+3f(x)=x3+3x+3 between x=0x=0x=0 and x=2x=2x=2 using x0=1x_{0}=1x0​=1 . Continue until two successive approximations differ by less than 0.00001.

Find the position function of a particle moving along a coordinate line that satisfies the given condition. v(t)=3t2−4t+4 \mathcal{v}(t)=3 t^{2}-4 t+4 v(t)=3t2−4t+4 , s(1) = -1

Find a cubic function f(x)=ax3−bx2+cx−df(x)=a x^{3}-b x^{2}+c x-df(x)=ax3−bx2+cx−d that has a local maximum value of 115115115 at 1 and a local minimum value of -1,184 at 7.

Find the critical numbers of the function. y=xx2+36y=\frac{x}{x^{2}+36}y=x2+36x​

How many points of inflection are on the graph of the function? f(x)=18x3+5x2−12x−20f(x)=18 x^{3}+5 x^{2}-12 x-20f(x)=18x3+5x2−12x−20

Find the absolute maximum and absolute minimum values, if any, of the function f(x)=xe−4xf(x)=x e^{-4 x}f(x)=xe−4x on [−1,1][-1,1][−1,1]

Find the absolute maximum value of y=16−x2y=\sqrt{16-x^{2}}y=16−x2​ on the interval [−6,6][-6,6][−6,6] .

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 most general antiderivative of the function. $f(x)=15 x^{2}-16 x+9$

What is the shortest possible length of the line segment that is cut off by the first quadrant and is tangent to the curve $y=\frac{5}{x}$ at some point?

Find the point(s) of inflection of the graph of $f(x)=\cos ^{-1} 7 x$

Find the critical number(s), if any, of the function $g(t)=t^{2} \ln 3 t$

Determine where the graph of the function $f(x)=x^{3}-30 x$ is concave upward and where it is concave downward. Also, find all inflection points of the function.

Find the relative extrema, if any, of $f(t)=e^{t}-7 t-3$ . Use the Second Derivative Test, if possible.

Use Newton's method with the specified initial approximation $x_{1}$ to find $x_{4}$ , the third approximation to the root of the given equation. (Give your answer to four decimal places.) $x^{4}-12=0, x_{1}=6$

Use Newton's method to obtain an approximation to the root of $\cos ^{-1} x-6 x=0$ to within 0.00001.

Estimate the absolute minimum value of the function $y=x^{3}-6 x+2$ to two decimal places on the interval $[0,10]$ . $Estimate the absolute minimum value of the function y=x^{3}-6 x+2 to two decimal places on the interval [0,10] . $

Find the absolute maximum and absolute minimum values, if any, of the function $f(x)=x-\sqrt{x}$ on [0, 25]

Find the critical number(s), if any, of the function $f(x)=e^{x^{2}-x}$ .

Use Newton's method to approximate the zero of $f(x)=x^{3}+3 x+3$ between $x=0$ and $x=2$ using $x_{0}=1$ . Continue until two successive approximations differ by less than 0.00001.

Find the position function of a particle moving along a coordinate line that satisfies the given condition. $\mathcal{v}(t)=3 t^{2}-4 t+4$ , s(1) = -1

Find a cubic function $f(x)=a x^{3}-b x^{2}+c x-d$ that has a local maximum value of $115$ at 1 and a local minimum value of -1,184 at 7.

Find the critical numbers of the function. $y=\frac{x}{x^{2}+36}$

How many points of inflection are on the graph of the function? $f(x)=18 x^{3}+5 x^{2}-12 x-20$

Find the absolute maximum and absolute minimum values, if any, of the function $f(x)=x e^{-4 x}$ on $[-1,1]$

Find the absolute maximum value of $y=\sqrt{16-x^{2}}$ on the interval $[-6,6]$ .