Question 1

Graph the function, then find the extreme values of the function on the interval and indicate where they occur.
-$y = | x - 3 | + | x + 4 |$ on the interval $- 5 < x < 5$

Accepted Answer

A)  Absolute minimum is 8
on the interval $[ - 3,4 ]$ 
  
B)  Absolute minimum is 7
 on the interval $[ - 4,3 ]$
  
C)  Absolute minimum is 8
on the interval $[ - 4,3 ]$
  
D)  Absolute minimum is 7 
 on the interval $[ - 3,4 ]$
  
A)  Absolute minimum is 8
on the interval $[ - 3,4 ]$ 
  
B)  Absolute minimum is 7
 on the interval $[ - 4,3 ]$
  
C)  Absolute minimum is 8
on the interval $[ - 4,3 ]$
  
D)  Absolute minimum is 7 
 on the interval $[ - 3,4 ]$

Question 2

Answer the problem.
-If the derivative of an even function f(x) is zero at x = c, can anything be said about the value of f at x = -c? Give reasons for your answer.

Accepted Answer

Yes. The value of f at x = -c

Question 3

Use Newton's method to estimate the requested solution of the equation. Start with given value of x0 and then give x2 as
the estimated solution.
-$$- x ^ { 2 } + 4 x - 1 = 0 ; x _ { 0 } = 0 ; \text { Find the left-hand solution. }$$

Accepted Answer

A)  0.25 
B)  0.14 
C)  -0.33 
D)  0.23 
A)  0.25 
B)  0.14 
C)  -0.33 
D)  0.23

Question 4

Find a value of c that makes the function continuous at the given value of x. If it is impossible, state this.
-$$f ( x ) = \left\{ \begin{array} { c c } 
x ^ { 2 } - 4 , & x  0
\end{array} ; x = 0 \right.$$

Accepted Answer

A)  -2 
B)  -4 
C)  Impossible 
D)  4 
A)  -2 
B)  -4 
C)  Impossible 
D)  4

Question 5

Use differentiation to determine whether the integral formula is correct.
-$$\int x \cos x d x = \frac { x ^ { 2 } } { 2 } \sin x + C$$

Accepted Answer

A) True 
 B)False

Question 6

Estimate the limit by graphing the function for an appropriate domain. Confirm your estimate by using L'Hopital's rule.
Show each step of your calculation.
-$\text { Give an example of two differentiable functions } f \text { and } g \text { with } \lim _ { x \rightarrow \infty } f ( x ) = \lim _ { x \rightarrow\infty  } g ( x ) = 0 \text { that satisfy } \lim _ { x \rightarrow \infty } \frac { f ( x ) } { g ( x ) } = \infty \text {. }$

Accepted Answer

The answer of Estimate the limit by graphing the function...

Question 7

Use l'H^opital's rule to find the limit.
-$\lim _ { \theta  \rightarrow \ 0 } \frac { \sin \theta ^ { 4 } } { \theta }$

Accepted Answer

A)  $\infty$ 
B)  0 
C)  1 
D)  $- \infty$ 
A)  $\infty$ 
B)  0 
C)  1 
D)  $- \infty$

Question 8

Solve the problem.
-$$\text { Let } c ( x ) = t \left( p _ { 0 } - p \right) p ^ { 3 } \text { where } t \text { and } p _ { 0 } \text { are constants. Show that } c ( x ) \text { is greatest when } p = \frac { 3 } { 4 } p _ { 0 } \text {. }$$

Accepted Answer

We calculate @#LAT-DLM&, and find that t

Question 9

Graph the function, then find the extreme values of the function on the interval and indicate where they occur.
-$y = | x + 4 | - | x - 5 |$ on the interval $- \infty < x < \infty$

Accepted Answer

A)  Absolute maximum is 9
on the interval $[ 6 , \infty )$;
Absolute minimum is $- 9$ 
on the interval $( - \infty , 3 ]$
  
B)  Absolute maximum is 10 
 on the interval $[ 5 , \infty )$;
 Absolute minimum is $- 10$  
on the interval $( - \infty , 4 ]$
  
C)  Absolute maximum is 8
 on the interval $[ 5 , \infty)$;
Absolute minimum is $\mathbf { - 8 }$.
 on the interval $( - \infty , 4 ]$ 
  
D)  Absolute maximum is 9 
on the interval $[ 5 , \infty )$; 
Absolute minimum is $- 9$
on the interval $( - \infty , 4 ]$
  
A)  Absolute maximum is 9
on the interval $[ 6 , \infty )$;
Absolute minimum is $- 9$ 
on the interval $( - \infty , 3 ]$
  
B)  Absolute maximum is 10 
 on the interval $[ 5 , \infty )$;
 Absolute minimum is $- 10$  
on the interval $( - \infty , 4 ]$
  
C)  Absolute maximum is 8
 on the interval $[ 5 , \infty)$;
Absolute minimum is $\mathbf { - 8 }$.
 on the interval $( - \infty , 4 ]$ 
  
D)  Absolute maximum is 9 
on the interval $[ 5 , \infty )$; 
Absolute minimum is $- 9$
on the interval $( - \infty , 4 ]$

Question 10

Find the absolute extreme values of the function on the interval.
-$$f ( x ) = \ln ( x + 2 ) + \frac { 1 } { x } , 1 \leq x \leq 6$$

Accepted Answer

A)  Minimum value is $- 1$ at $x = - 1$; maximum value is $\ln 8 + \frac { 1 } { 6 }$ at $x = 6$ 
B)  Minimum value is $\ln 3 + 1$ at $x = 1$; maximum value is $\ln 8 + \frac { 1 } { 6 }$ at $x = 6$ 
C)  Minimum value is $\ln 4 + \frac { 1 } { 2 }$ at $x = 2$; maximum value is $\ln 3 + 1$ at $x = 1$ 
D)  Minimum value is $\ln 4 + \frac { 1 } { 2 }$ at $x = 2$; maximum value is $\ln 8 + \frac { 1 } { 6 }$ at $x = 6$ 
A)  Minimum value is $- 1$ at $x = - 1$; maximum value is $\ln 8 + \frac { 1 } { 6 }$ at $x = 6$ 
B)  Minimum value is $\ln 3 + 1$ at $x = 1$; maximum value is $\ln 8 + \frac { 1 } { 6 }$ at $x = 6$ 
C)  Minimum value is $\ln 4 + \frac { 1 } { 2 }$ at $x = 2$; maximum value is $\ln 3 + 1$ at $x = 1$ 
D)  Minimum value is $\ln 4 + \frac { 1 } { 2 }$ at $x = 2$; maximum value is $\ln 8 + \frac { 1 } { 6 }$ at $x = 6$

Question 11

Find the extreme values of the function and where they occur.
-$$y = \frac { x + 1 } { x ^ { 2 } + 2 x + 2 }$$

Accepted Answer

A)  The maximum is $- \frac { 1 } { 2 }$ at $x = 0$; the minimum is $\frac { 1 } { 2 }$ at $x = - 2$. 
B)  The maximum is $\frac { 1 } { 2 }$ at $x = 0$; the minimum is $- \frac { 1 } { 2 }$ at $x = - 2$. 
C)  None 
D)  The maximum is 2 at $x = 0$; the minimum is $\frac { 1 } { 2 }$ at $x = - 2$. 
A)  The maximum is $- \frac { 1 } { 2 }$ at $x = 0$; the minimum is $\frac { 1 } { 2 }$ at $x = - 2$. 
B)  The maximum is $\frac { 1 } { 2 }$ at $x = 0$; the minimum is $- \frac { 1 } { 2 }$ at $x = - 2$. 
C)  None 
D)  The maximum is 2 at $x = 0$; the minimum is $\frac { 1 } { 2 }$ at $x = - 2$.

Question 12

Find the absolute extreme values of the function on the interval.
-$$f ( x ) = 5 x ^ { 4 / 3 } , - 27 \leq x \leq 8$$

Accepted Answer

A)  absolute maximum is 405 at x = -27 ; absolute minimum is 80 at x = 8 
B)  absolute maximum is 80 at x = 8 ; absolute minimum is 0 at x = 0 
C)  absolute maximum is 81 at x = -27 ; absolute minimum is 0 at x = 0 
D)  absolute maximum is 405 at x = -27 ; absolute minimum is 0 at x = 0 
A)  absolute maximum is 405 at x = -27 ; absolute minimum is 80 at x = 8 
B)  absolute maximum is 80 at x = 8 ; absolute minimum is 0 at x = 0 
C)  absolute maximum is 81 at x = -27 ; absolute minimum is 0 at x = 0 
D)  absolute maximum is 405 at x = -27 ; absolute minimum is 0 at x = 0

Question 13

Find all possible functions with the given derivative.
-$y ^ { \prime } = 5 x ^ { 7 }$

Accepted Answer

A)  $\frac { 5 } { 7 } x ^ { 8 } + C$ 
B)  $\frac { 7 } { 5 } x ^ { 8 } + C$ 
C)  $\frac { 1 } { 4 } x ^ { 8 } + C$ 
D)  $\frac { 5 } { 8 } x ^ { 8 } + C$ 
A)  $\frac { 5 } { 7 } x ^ { 8 } + C$ 
B)  $\frac { 7 } { 5 } x ^ { 8 } + C$ 
C)  $\frac { 1 } { 4 } x ^ { 8 } + C$ 
D)  $\frac { 5 } { 8 } x ^ { 8 } + C$

Question 14

Using the derivative of f(x) given below, determine the critical points of f(x).
-$$f ^ { \prime } ( x ) = ( x - 5 ) e ^ { - x }$$

Accepted Answer

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

Question 15

Use a computer algebra system (CAS) to solve the given initial value problem.
-$$y ^ { \prime } = \frac { 2 \left( 1 - x ^ { 2 } \right) } { 1 + x ^ { 2 } } , y ( 0 ) = 1$$

Accepted Answer

A)  $y = 4 \tan ^ { - 1 } x - 2 x$ 
B)  $y = 2 \ln \left( \frac { x + 1 } { | x - 1 | } \right) - 2 x + 1$ 
C)  $y = 3 \tan ^ { - 1 } x - x + 1$ 
D)  $y = 4 \tan ^ { - 1 } x - 2 x + 1$ 
A)  $y = 4 \tan ^ { - 1 } x - 2 x$ 
B)  $y = 2 \ln \left( \frac { x + 1 } { | x - 1 | } \right) - 2 x + 1$ 
C)  $y = 3 \tan ^ { - 1 } x - x + 1$ 
D)  $y = 4 \tan ^ { - 1 } x - 2 x + 1$

Question 16

Find the most general antiderivative.
-$$\int \frac { x \sqrt { x } + \sqrt { x } } { x ^ { 2 } } d x$$

Accepted Answer

A)  $- \frac { \sqrt { x } } { 2 } - \frac { 3 \sqrt { x } } { 2 } + C$ 
B)  $2 \sqrt { x } - \frac { 2 } { \sqrt { x } } + C$ 
C)  $\mathrm { C }$ 
D)  $\frac { 2 } { \sqrt { x } } - 2 \sqrt { x } + C$ 
A)  $- \frac { \sqrt { x } } { 2 } - \frac { 3 \sqrt { x } } { 2 } + C$ 
B)  $2 \sqrt { x } - \frac { 2 } { \sqrt { x } } + C$ 
C)  $\mathrm { C }$ 
D)  $\frac { 2 } { \sqrt { x } } - 2 \sqrt { x } + C$

Question 17

Find the extreme values of the function and where they occur.
-$$y = x ^ { 3 } - 3 x ^ { 2 } + 6 x - 8$$

Accepted Answer

A)  The minimum is 4 at x = -1. 
B)  None 
C)  The maximum is 4 at x = 1. 
D)  The maximum is 4 at x = 2. 
A)  The minimum is 4 at x = -1. 
B)  None 
C)  The maximum is 4 at x = 1. 
D)  The maximum is 4 at x = 2.

Question 18

Solve the initial value problem.
-$$\frac { \mathrm { ds } } { \mathrm { dt } } = \cos \mathrm { t } - \sin t , s \left( \frac { \pi } { 2 } \right) = 4$$

Accepted Answer

A)  $s = 2 \sin t + 2$ 
B)  $s = \sin t - \cos t + 3$ 
C)  $s = \sin t + \cos t + 5$ 
D)  $s = \sin t + \cos t + 3$ 
A)  $s = 2 \sin t + 2$ 
B)  $s = \sin t - \cos t + 3$ 
C)  $s = \sin t + \cos t + 5$ 
D)  $s = \sin t + \cos t + 3$

Question 19

Find all possible functions with the given derivative.
-$y ^ { \prime } = 3 - \frac { 1 } { x ^ { 2 } }$

Accepted Answer

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

Question 20

Answer the problem.
-The function $P ( x ) = 2 x + \frac { 200 } { x } , 0 < x < \infty$ models the perimeter of a rectangle of dimensions $x$ by $\frac { 100 } { x }$.
(a) Find the extreme values for P.
(b) Give an interpretation in terms of perimeter of the rectangle for any values found in part (a).

Accepted Answer

(a) The only critical point in the inter

Graph the function, then find the extreme values of the function on the interval and indicate where they occur. - $y = | x - 3 | + | x + 4 |$ on the interval $- 5 < x < 5$

Answer the problem. -If the derivative of an even function f(x) is zero at x = c, can anything be said about the value of f at x = -c? Give reasons for your answer.

Use Newton's method to estimate the requested solution of the equation. Start with given value of x0 and then give x2 as the estimated solution. - $- x ^ { 2 } + 4 x - 1 = 0 ; x _ { 0 } = 0 ; \text { Find the left-hand solution. }$

Find a value of c that makes the function continuous at the given value of x. If it is impossible, state this. - $f ( x ) = \left\{ \begin{array} { c c } x ^ { 2 } - 4 , & x < 0 \\c , & x = 0 \\- 2 ( x - 1 ) - 6 , & x > 0\end{array} ; x = 0 \right.$

Use differentiation to determine whether the integral formula is correct. - $\int x \cos x d x = \frac { x ^ { 2 } } { 2 } \sin x + C$

Use l'H^opital's rule to find the limit. - $\lim _ { \theta \rightarrow \ 0 } \frac { \sin \theta ^ { 4 } } { \theta }$

Solve the problem. - $\text { Let } c ( x ) = t \left( p _ { 0 } - p \right) p ^ { 3 } \text { where } t \text { and } p _ { 0 } \text { are constants. Show that } c ( x ) \text { is greatest when } p = \frac { 3 } { 4 } p _ { 0 } \text {. }$

Find the absolute extreme values of the function on the interval. - $f ( x ) = \ln ( x + 2 ) + \frac { 1 } { x } , 1 \leq x \leq 6$

Find the extreme values of the function and where they occur. - $y = \frac { x + 1 } { x ^ { 2 } + 2 x + 2 }$

Find the absolute extreme values of the function on the interval. - $f ( x ) = 5 x ^ { 4 / 3 } , - 27 \leq x \leq 8$

Find all possible functions with the given derivative. - $y ^ { \prime } = 5 x ^ { 7 }$

Using the derivative of f(x) given below, determine the critical points of f(x). - $f ^ { \prime } ( x ) = ( x - 5 ) e ^ { - x }$

Use a computer algebra system (CAS) to solve the given initial value problem. - $y ^ { \prime } = \frac { 2 \left( 1 - x ^ { 2 } \right) } { 1 + x ^ { 2 } } , y ( 0 ) = 1$

Find the most general antiderivative. - $\int \frac { x \sqrt { x } + \sqrt { x } } { x ^ { 2 } } d x$

Find the extreme values of the function and where they occur. - $y = x ^ { 3 } - 3 x ^ { 2 } + 6 x - 8$

Solve the initial value problem. - $\frac { \mathrm { ds } } { \mathrm { dt } } = \cos \mathrm { t } - \sin t , s \left( \frac { \pi } { 2 } \right) = 4$

Find all possible functions with the given derivative. - $y ^ { \prime } = 3 - \frac { 1 } { x ^ { 2 } }$

Functions

Limits and Continuity

Derivatives

Integrals

Applications of Definite Integrals

Integrals and Transcendental Functions

Techniques of Integration

First-Order Differential Equations

Infinite Sequences and Series

Parametric Equations and Polar Coordinates

Vectors and the Geometry of Space

Vector-Valued Functions and Motion in Space

Partial Derivatives

Multiple Integrals

Integrals and Vector Fields

Filters

Exam 5: Applications of Derivatives

Graph the function, then find the extreme values of the function on the interval and indicate where they occur. - y=∣x−3∣+∣x+4∣y = | x - 3 | + | x + 4 |y=∣x−3∣+∣x+4∣ on the interval −5<x<5- 5 < x < 5−5<x<5

Answer the problem. -If the derivative of an even function f(x) is zero at x = c, can anything be said about the value of f at x = -c? Give reasons for your answer.

Use differentiation to determine whether the integral formula is correct. - ∫xcos⁡xdx=x22sin⁡x+C\int x \cos x d x = \frac { x ^ { 2 } } { 2 } \sin x + C∫xcosxdx=2x2​sinx+C

Use l'H^opital's rule to find the limit. - lim⁡θ→ 0sin⁡θ4θ\lim _ { \theta \rightarrow \ 0 } \frac { \sin \theta ^ { 4 } } { \theta }limθ→ 0​θsinθ4​

Graph the function, then find the extreme values of the function on the interval and indicate where they occur. - y=∣x+4∣−∣x−5∣y = | x + 4 | - | x - 5 |y=∣x+4∣−∣x−5∣ on the interval −∞<x<∞- \infty < x < \infty−∞<x<∞

Find the absolute extreme values of the function on the interval. - f(x)=ln⁡(x+2)+1x,1≤x≤6f ( x ) = \ln ( x + 2 ) + \frac { 1 } { x } , 1 \leq x \leq 6f(x)=ln(x+2)+x1​,1≤x≤6

Find the extreme values of the function and where they occur. - y=x+1x2+2x+2y = \frac { x + 1 } { x ^ { 2 } + 2 x + 2 }y=x2+2x+2x+1​

Find the absolute extreme values of the function on the interval. - f(x)=5x4/3,−27≤x≤8f ( x ) = 5 x ^ { 4 / 3 } , - 27 \leq x \leq 8f(x)=5x4/3,−27≤x≤8

Find all possible functions with the given derivative. - y′=5x7y ^ { \prime } = 5 x ^ { 7 }y′=5x7

Using the derivative of f(x) given below, determine the critical points of f(x). - f′(x)=(x−5)e−xf ^ { \prime } ( x ) = ( x - 5 ) e ^ { - x }f′(x)=(x−5)e−x

Use a computer algebra system (CAS) to solve the given initial value problem. - y′=2(1−x2)1+x2,y(0)=1y ^ { \prime } = \frac { 2 \left( 1 - x ^ { 2 } \right) } { 1 + x ^ { 2 } } , y ( 0 ) = 1y′=1+x22(1−x2)​,y(0)=1

Find the most general antiderivative. - ∫xx+xx2dx\int \frac { x \sqrt { x } + \sqrt { x } } { x ^ { 2 } } d x∫x2xx​+x​​dx

Find the extreme values of the function and where they occur. - y=x3−3x2+6x−8y = x ^ { 3 } - 3 x ^ { 2 } + 6 x - 8y=x3−3x2+6x−8

Solve the initial value problem. - dsdt=cos⁡t−sin⁡t,s(π2)=4\frac { \mathrm { ds } } { \mathrm { dt } } = \cos \mathrm { t } - \sin t , s \left( \frac { \pi } { 2 } \right) = 4dtds​=cost−sint,s(2π​)=4

Find all possible functions with the given derivative. - y′=3−1x2y ^ { \prime } = 3 - \frac { 1 } { x ^ { 2 } }y′=3−x21​