Question 1

Find an equation of the line through the point $(8,16)$ that cuts off the least area from the first quadrant.

Accepted Answer

The answer of Find an equation of the line through...

Question 2

Determine where the graph of the function $f(x)=3+4 x^{1 / 3}$ is concave upward and where it is concave downward. Also, find all inflection points of the function.

Accepted Answer

A)  CU on $(-\infty, 0)$ , CD on $(0, \infty)$ ,
IP $(0,7)$ 
B)  CU on $(-\infty, 0)$ , CD on $(0, \infty)$ ,
IP $(0,3)$ 
C)  CU on $(0, \infty)$ , CD on $(-\infty, 0)$ ,
IP $(0,7)$ 
D)  CU on $(0, \infty)$ , CD on $(-\infty, 0)$ ,
IP $(0,3)$ 
A)  CU on $(-\infty, 0)$ , CD on $(0, \infty)$ ,
IP $(0,7)$ 
B)  CU on $(-\infty, 0)$ , CD on $(0, \infty)$ ,
IP $(0,3)$ 
C)  CU on $(0, \infty)$ , CD on $(-\infty, 0)$ ,
IP $(0,7)$ 
D)  CU on $(0, \infty)$ , CD on $(-\infty, 0)$ ,
IP $(0,3)$

Question 3

Determine where the graph of the function $f(u)=\frac{3 u}{u^{2}-16}$ is concave upward and where it is concave downward. Also, find all inflection points of the function.

Accepted Answer

CU on @#LAT-DLM& and

Question 4

Estimate the absolute maximum value of the function $y=x \sqrt{3 x-x^{2}}$ to two decimal places on the interval $[0,3]$ .

Accepted Answer

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

Question 5

Determine where the graph of $f(x)=x^{2} e^{-4 x}$ is concave upward and where it is concave downward. Also, find all inflection points of the function.

Accepted Answer

concave upward:
\[
\left(-\infty, \frac{

Question 6

Find the limit. $\lim _{x \rightarrow 0} \frac{x}{\tan ^{-1}(2 x)}$

Accepted Answer

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

Question 7

Given $f(x)=x^{2}+6 x$ . (a) Find the intervals on which f is increasing or decreasing.
(b) Find the relative maxima and relative minima of f.

Accepted Answer

A)  (a) Increasing on $(-3, \infty)$ ,
Decreasing on $(-\infty,-3)$
(b) Rel. min. $f(-3)=-9$ 
B)  (a) Increasing on $(-\infty,-3)$ ,
Decreasing on $(-3, \infty)$
(b) Rel. max. $f(-3)=-9$ 
C)  (a) Increasing on $(-6, \infty)$ ,
Decreasing on $(-\infty,-6)$
(b) Rel. min. $f(-6)=0$ 
D)  (a) Increasing on $(-\infty,-6)$ ,
Decreasing on $(-6, \infty)$
(b) Rel. max. $f(-6)=0$ 
A)  (a) Increasing on $(-3, \infty)$ ,
Decreasing on $(-\infty,-3)$
(b) Rel. min. $f(-3)=-9$ 
B)  (a) Increasing on $(-\infty,-3)$ ,
Decreasing on $(-3, \infty)$
(b) Rel. max. $f(-3)=-9$ 
C)  (a) Increasing on $(-6, \infty)$ ,
Decreasing on $(-\infty,-6)$
(b) Rel. min. $f(-6)=0$ 
D)  (a) Increasing on $(-\infty,-6)$ ,
Decreasing on $(-6, \infty)$
(b) Rel. max. $f(-6)=0$

Question 8

Determine where the graph of $f(x)=\ln |x-4|$ is concave upward and where it is concave downward. Also, find all inflection points of the function.

Accepted Answer

A)  concave downward: $(-\infty,-4)$ and $(4, \infty)$ , concave upward: $(-4,0)$ and $(0,9)$ ; inflection points: $(-4,0)$ and $(4,0)$ 
B)  concave downward: $(-4,0)$ and $(0,9)$ , concave upward: $(-\infty,-4)$ and $(4, \infty)$ ; inflection points: $(-4,0)$ and $(4,0)$ 
C)  concave downward: $(-\infty, 4)$ and $(4, \infty)$ , concave upward: nowhere; inflection points: none 
D)  concave downward: nowhere, concave upward: $(-\infty, 4)$ and $(4, \infty)$ ; inflection points: none 
A)  concave downward: $(-\infty,-4)$ and $(4, \infty)$ , concave upward: $(-4,0)$ and $(0,9)$ ; inflection points: $(-4,0)$ and $(4,0)$ 
B)  concave downward: $(-4,0)$ and $(0,9)$ , concave upward: $(-\infty,-4)$ and $(4, \infty)$ ; inflection points: $(-4,0)$ and $(4,0)$ 
C)  concave downward: $(-\infty, 4)$ and $(4, \infty)$ , concave upward: nowhere; inflection points: none 
D)  concave downward: nowhere, concave upward: $(-\infty, 4)$ and $(4, \infty)$ ; inflection points: none

Question 9

The function $f(t)=\sin ^{2} 2 t$ satisfies the hypotheses of Rolle's Theorem on the interval $\left[0, \frac{\pi}{2}\right]$ . Find all values of c that satisfy the conclusion of the theorem.

Accepted Answer

A)  0 
B)  $\frac{\pi}{4}$ 
C)  $0, \pm \frac{\pi}{4}$ 
D)  $0, \frac{\pi}{4}$ 
A)  0 
B)  $\frac{\pi}{4}$ 
C)  $0, \pm \frac{\pi}{4}$ 
D)  $0, \frac{\pi}{4}$

Question 10

Evaluate the limit $\lim _{x \rightarrow 0} \frac{e^{6 x}-e^{-8 x}}{\ln (1+x)}$ using l'Hôpital's Rule.

Accepted Answer

A)  2 
B)  48 
C)  -2 
D)  14 
A)  2 
B)  48 
C)  -2 
D)  14

Question 11

a) Graph the funtion $f(x)=x^{2} \ln x$ . b) Use l'Hospitals' rule to explain the behavior as $x \rightarrow 0$

Accepted Answer

A)  a)   b) $\lim _{x \rightarrow 0_{+}} x^{2} \ln x=0$
There is a hole at (0, 0) 
B)    b) $\lim _{x \rightarrow 0_{+}} x^{2} \ln x=1$
There is a hole at (0, 1) 
C)    b) $\lim _{x \rightarrow 0_{+}} x^{2} \ln x=1$
There is a hole at (1, 0) 
D)    b) $\lim _{x \rightarrow 0_{+}} x^{2} \ln x=0$
There is a hole at (0, 0) 
A)  a)   b) $\lim _{x \rightarrow 0_{+}} x^{2} \ln x=0$
There is a hole at (0, 0) 
B)    b) $\lim _{x \rightarrow 0_{+}} x^{2} \ln x=1$
There is a hole at (0, 1) 
C)    b) $\lim _{x \rightarrow 0_{+}} x^{2} \ln x=1$
There is a hole at (1, 0) 
D)    b) $\lim _{x \rightarrow 0_{+}} x^{2} \ln x=0$
There is a hole at (0, 0)

Question 12

The graph of the second derivative $f^{\prime t}(x)$ of a function f is shown. State the x-coordinates of the inflection points of 
f.

Accepted Answer

The answer of The graph of the second derivative \(f^{\prime...

Question 13

The sum of two positive numbers is $16$ . What is the smallest possible value of the sum of their squares?

Accepted Answer

The answer of The sum of two positive numbers is...

Question 14

A city's population (in thousands) t years from now is estimated by $P(t)=\frac{26 t^{2}+120 t+225}{t^{2}+5 t+41}$ What will the population of the city be in the long run? Hint: Find $\lim _{t \rightarrow \infty} P(t)$

Accepted Answer

The answer of A city's population (in thousands) t years...

Question 15

Sketch the graph of $f(x)=x^{2}-1$ on $(-2,1]$ and find its absolute maximum and absolute minimum values, if any.

Accepted Answer

@#IMG-DLM&
Abs. min.

Question 16

Use Newton's method to solve the equation $320\left(t+10 e^{-0.1 t}\right)-13,200=0$ to within 0.00001.

Accepted Answer

The answer of Use Newton's method to solve the equation...

Question 17

Find the limit. $\lim _{x \rightarrow(x / 2)^{-}} \frac{-3}{5 \cos (x)}$

Accepted Answer

A)  -$\infty$ 
B)  $\infty$ 
C)  $-\frac{3}{5}$ 
D)  $\frac{3}{5}$ 
A)  -$\infty$ 
B)  $\infty$ 
C)  $-\frac{3}{5}$ 
D)  $\frac{3}{5}$

Question 18

Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle's Theorem. $f(x)=\sin 5 \pi x$ , $\left[-\frac{2}{5}, \frac{2}{5}\right]$

Accepted Answer

A)  None of these 
B)  $c_{1}=\pm \frac{1}{10}$ , $c_{2}=\frac{3}{10}$ 
C)  $c_{1}=-\frac{1}{10}$ , $c_{2}=\pm \frac{1}{10}$ 
D)  $c_{1}=\frac{1}{10}$ , $c_{2}=\pm \frac{3}{10}$ 
E)  $c_{1}=\pm \frac{1}{10}$ , $c_{2}=\pm \frac{3}{10}$ 
A)  None of these 
B)  $c_{1}=\pm \frac{1}{10}$ , $c_{2}=\frac{3}{10}$ 
C)  $c_{1}=-\frac{1}{10}$ , $c_{2}=\pm \frac{1}{10}$ 
D)  $c_{1}=\frac{1}{10}$ , $c_{2}=\pm \frac{3}{10}$ 
E)  $c_{1}=\pm \frac{1}{10}$ , $c_{2}=\pm \frac{3}{10}$

Question 19

Consider the function $f(x)=\frac{8 x}{\ln x}$ . (a) Find the intervals on which f is increasing or decreasing. (b) Find the relative maxima and relative minima of f.

Accepted Answer

(a) decreasing on @#LAT-DLM& a

Question 20

Find the relative extrema, if any, of $f(t)=t-7 \ln t$ . Use the Second Derivative Test, if possible.

Accepted Answer

A)  relative minimum: none, relative maximum: $f\left(\frac{1}{7}\right)=\frac{1}{7}+7 \ln 7$ 
B)  relative minimum: $f(7)=7-7 \ln 7$ , relative maximum: none 
C)  relative minimum: none, relative maximum: $f(7)=7-7 \ln 7$ 
D)  relative minimum: $f\left(\frac{1}{7}\right)=\frac{1}{7}+7 \ln 7$ , relative maximum: none 
A)  relative minimum: none, relative maximum: $f\left(\frac{1}{7}\right)=\frac{1}{7}+7 \ln 7$ 
B)  relative minimum: $f(7)=7-7 \ln 7$ , relative maximum: none 
C)  relative minimum: none, relative maximum: $f(7)=7-7 \ln 7$ 
D)  relative minimum: $f\left(\frac{1}{7}\right)=\frac{1}{7}+7 \ln 7$ , relative maximum: none

Find an equation of the line through the point $(8,16)$ that cuts off the least area from the first quadrant.

Determine where the graph of the function $f(x)=3+4 x^{1 / 3}$ is concave upward and where it is concave downward. Also, find all inflection points of the function.

Determine where the graph of the function $f(u)=\frac{3 u}{u^{2}-16}$ is concave upward and where it is concave downward. Also, find all inflection points of the function.

Estimate the absolute maximum value of the function $y=x \sqrt{3 x-x^{2}}$ to two decimal places on the interval $[0,3]$ . $Estimate the absolute maximum value of the function y=x \sqrt{3 x-x^{2}} to two decimal places on the interval [0,3] .$

Determine where the graph of $f(x)=x^{2} e^{-4 x}$ is concave upward and where it is concave downward. Also, find all inflection points of the function.

Find the limit. $\lim _{x \rightarrow 0} \frac{x}{\tan ^{-1}(2 x)}$

Given $f(x)=x^{2}+6 x$ . (a) Find the intervals on which f is increasing or decreasing. (b) Find the relative maxima and relative minima of f.

Determine where the graph of $f(x)=\ln |x-4|$ is concave upward and where it is concave downward. Also, find all inflection points of the function.

The function $f(t)=\sin ^{2} 2 t$ satisfies the hypotheses of Rolle's Theorem on the interval $\left[0, \frac{\pi}{2}\right]$ . Find all values of c that satisfy the conclusion of the theorem.

Evaluate the limit $\lim _{x \rightarrow 0} \frac{e^{6 x}-e^{-8 x}}{\ln (1+x)}$ using l'Hôpital's Rule.

a) Graph the funtion $f(x)=x^{2} \ln x$ . b) Use l'Hospitals' rule to explain the behavior as $x \rightarrow 0$

The graph of the second derivative $f^{\prime t}(x)$ of a function f is shown. State the x-coordinates of the inflection points of f. $The graph of the second derivative f^{\prime t}(x) of a function f is shown. State the x-coordinates of the inflection points of f.$

The sum of two positive numbers is $16$ . What is the smallest possible value of the sum of their squares?

A city's population (in thousands) t years from now is estimated by $P(t)=\frac{26 t^{2}+120 t+225}{t^{2}+5 t+41}$ What will the population of the city be in the long run? Hint: Find $\lim _{t \rightarrow \infty} P(t)$

Sketch the graph of $f(x)=x^{2}-1$ on $(-2,1]$ and find its absolute maximum and absolute minimum values, if any.

Use Newton's method to solve the equation $320\left(t+10 e^{-0.1 t}\right)-13,200=0$ to within 0.00001.

Find the limit. $\lim _{x \rightarrow(x / 2)^{-}} \frac{-3}{5 \cos (x)}$

Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle's Theorem. $f(x)=\sin 5 \pi x$ , $\left[-\frac{2}{5}, \frac{2}{5}\right]$

Consider the function $f(x)=\frac{8 x}{\ln x}$ . (a) Find the intervals on which f is increasing or decreasing. (b) Find the relative maxima and relative minima of f.

Find the relative extrema, if any, of $f(t)=t-7 \ln t$ . Use the Second Derivative Test, if possible.

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 an equation of the line through the point (8,16)(8,16)(8,16) that cuts off the least area from the first quadrant.

Determine where the graph of the function f(x)=3+4x1/3f(x)=3+4 x^{1 / 3}f(x)=3+4x1/3 is concave upward and where it is concave downward. Also, find all inflection points of the function.

Determine where the graph of the function f(u)=3uu2−16f(u)=\frac{3 u}{u^{2}-16}f(u)=u2−163u​ is concave upward and where it is concave downward. Also, find all inflection points of the function.

Estimate the absolute maximum value of the function y=x3x−x2y=x \sqrt{3 x-x^{2}}y=x3x−x2​ to two decimal places on the interval [0,3][0,3][0,3] .

Determine where the graph of f(x)=x2e−4xf(x)=x^{2} e^{-4 x}f(x)=x2e−4x is concave upward and where it is concave downward. Also, find all inflection points of the function.

Find the limit. lim⁡x→0xtan⁡−1(2x)\lim _{x \rightarrow 0} \frac{x}{\tan ^{-1}(2 x)}limx→0​tan−1(2x)x​

Given f(x)=x2+6xf(x)=x^{2}+6 xf(x)=x2+6x . (a) Find the intervals on which f is increasing or decreasing. (b) Find the relative maxima and relative minima of f.

Determine where the graph of f(x)=ln⁡∣x−4∣f(x)=\ln |x-4|f(x)=ln∣x−4∣ is concave upward and where it is concave downward. Also, find all inflection points of the function.

The function f(t)=sin⁡22tf(t)=\sin ^{2} 2 tf(t)=sin22t satisfies the hypotheses of Rolle's Theorem on the interval [0,π2]\left[0, \frac{\pi}{2}\right][0,2π​] . Find all values of c that satisfy the conclusion of the theorem.

Evaluate the limit lim⁡x→0e6x−e−8xln⁡(1+x)\lim _{x \rightarrow 0} \frac{e^{6 x}-e^{-8 x}}{\ln (1+x)}limx→0​ln(1+x)e6x−e−8x​ using l'Hôpital's Rule.

a) Graph the funtion f(x)=x2ln⁡xf(x)=x^{2} \ln xf(x)=x2lnx . b) Use l'Hospitals' rule to explain the behavior as x→0x \rightarrow 0x→0

The graph of the second derivative f′t(x)f^{\prime t}(x)f′t(x) of a function f is shown. State the x-coordinates of the inflection points of f.

The sum of two positive numbers is 161616 . What is the smallest possible value of the sum of their squares?

Sketch the graph of f(x)=x2−1f(x)=x^{2}-1f(x)=x2−1 on (−2,1](-2,1](−2,1] and find its absolute maximum and absolute minimum values, if any.

Use Newton's method to solve the equation 320(t+10e−0.1t)−13,200=0320\left(t+10 e^{-0.1 t}\right)-13,200=0320(t+10e−0.1t)−13,200=0 to within 0.00001.

Find the limit. lim⁡x→(x/2)−−35cos⁡(x)\lim _{x \rightarrow(x / 2)^{-}} \frac{-3}{5 \cos (x)}limx→(x/2)−​5cos(x)−3​

Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle's Theorem. f(x)=sin⁡5πxf(x)=\sin 5 \pi xf(x)=sin5πx , [−25,25]\left[-\frac{2}{5}, \frac{2}{5}\right][−52​,52​]

Consider the function f(x)=8xln⁡xf(x)=\frac{8 x}{\ln x}f(x)=lnx8x​ . (a) Find the intervals on which f is increasing or decreasing. (b) Find the relative maxima and relative minima of f.

Find the relative extrema, if any, of f(t)=t−7ln⁡tf(t)=t-7 \ln tf(t)=t−7lnt . Use the Second Derivative Test, if possible.

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 an equation of the line through the point $(8,16)$ that cuts off the least area from the first quadrant.

Determine where the graph of the function $f(x)=3+4 x^{1 / 3}$ is concave upward and where it is concave downward. Also, find all inflection points of the function.

Determine where the graph of the function $f(u)=\frac{3 u}{u^{2}-16}$ is concave upward and where it is concave downward. Also, find all inflection points of the function.

Estimate the absolute maximum value of the function $y=x \sqrt{3 x-x^{2}}$ to two decimal places on the interval $[0,3]$ . $Estimate the absolute maximum value of the function y=x \sqrt{3 x-x^{2}} to two decimal places on the interval [0,3] .$

Determine where the graph of $f(x)=x^{2} e^{-4 x}$ is concave upward and where it is concave downward. Also, find all inflection points of the function.

Find the limit. $\lim _{x \rightarrow 0} \frac{x}{\tan ^{-1}(2 x)}$

Given $f(x)=x^{2}+6 x$ . (a) Find the intervals on which f is increasing or decreasing. (b) Find the relative maxima and relative minima of f.

Determine where the graph of $f(x)=\ln |x-4|$ is concave upward and where it is concave downward. Also, find all inflection points of the function.

The function $f(t)=\sin ^{2} 2 t$ satisfies the hypotheses of Rolle's Theorem on the interval $\left[0, \frac{\pi}{2}\right]$ . Find all values of c that satisfy the conclusion of the theorem.

Evaluate the limit $\lim _{x \rightarrow 0} \frac{e^{6 x}-e^{-8 x}}{\ln (1+x)}$ using l'Hôpital's Rule.

a) Graph the funtion $f(x)=x^{2} \ln x$ . b) Use l'Hospitals' rule to explain the behavior as $x \rightarrow 0$

The graph of the second derivative $f^{\prime t}(x)$ of a function f is shown. State the x-coordinates of the inflection points of f. $The graph of the second derivative f^{\prime t}(x) of a function f is shown. State the x-coordinates of the inflection points of f.$

The sum of two positive numbers is $16$ . What is the smallest possible value of the sum of their squares?

Sketch the graph of $f(x)=x^{2}-1$ on $(-2,1]$ and find its absolute maximum and absolute minimum values, if any.

Use Newton's method to solve the equation $320\left(t+10 e^{-0.1 t}\right)-13,200=0$ to within 0.00001.

Find the limit. $\lim _{x \rightarrow(x / 2)^{-}} \frac{-3}{5 \cos (x)}$

Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle's Theorem. $f(x)=\sin 5 \pi x$ , $\left[-\frac{2}{5}, \frac{2}{5}\right]$

Consider the function $f(x)=\frac{8 x}{\ln x}$ . (a) Find the intervals on which f is increasing or decreasing. (b) Find the relative maxima and relative minima of f.

Find the relative extrema, if any, of $f(t)=t-7 \ln t$ . Use the Second Derivative Test, if possible.