Question 1

Solve the problem.
-Use Newton's method to find the positive fourth root of 5 by solving the equation $x ^ { 4 } - 5 = 0$. Start with $x _ { 1 } = 1$ and find $x _ { 2 }$.

Accepted Answer

\[\begin{array} { l } 
f ( x ) = x ^ { 4

Question 2

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.
-$$\text { If } f ( x ) = ( x - 3 ) ^ { 2 } \text { and } g ( x ) = \frac { 1 } { ( x - 3 ) ^ { 2 } } \text {, show that } \lim _ { x \rightarrow 8 } f ( x ) g ( x ) = 0$$

Accepted Answer

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

Question 3

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.
-Given that $x > 0$, find the maximum value, if any, of $x ^ { 1 / x ^ { 6 } }$.

Accepted Answer

A)  $e ^ { 6 e }$ 
B)  $\mathrm { e } ^ { 1 / ( 6 \mathrm { e } ) }$ 
C)  1 
D)  $\infty$ 
A)  $e ^ { 6 e }$ 
B)  $\mathrm { e } ^ { 1 / ( 6 \mathrm { e } ) }$ 
C)  1 
D)  $\infty$

Question 4

Solve the problem.
-A company is constructing an open-top, square-based, rectangular metal tank that will have a volume of $33 \mathrm { ft } 3$. What dimensions yield the minimum surface area? Round to the nearest tenth, if necessary.

Accepted Answer

A)  3.2 ft × 3.2 ft × 3.2 ft 
B)  8.1 ft × 8.1 ft × 0.5 ft 
C)  4 ft × 4 ft × 2 ft 
D)  4.6 ft × 4.6 ft × 1.5 ft 
A)  3.2 ft × 3.2 ft × 3.2 ft 
B)  8.1 ft × 8.1 ft × 0.5 ft 
C)  4 ft × 4 ft × 2 ft 
D)  4.6 ft × 4.6 ft × 1.5 ft

Question 5

Identify the function's extreme values in the given domain, and say where they are assumed. Tell which of the extreme
values, if any, are absolute.
-$$f ( x ) = \sqrt { 64 - x ^ { 2 } } , - 8 \leq x < 8$$

Accepted Answer

A)  no local extrema; no absolute extrema 
B)  local and absolute minimum: 0 at x = -8; local and absolute maximum: 8 at x = 0 
C)  local and absolute minimum: 0 at x = -8 and x = 8; local and absolute maximum: 8 at x = 0 
D)  local and absolute maximum: 0 at x = -8; local and absolute minimum: 8 at x = 0 
A)  no local extrema; no absolute extrema 
B)  local and absolute minimum: 0 at x = -8; local and absolute maximum: 8 at x = 0 
C)  local and absolute minimum: 0 at x = -8 and x = 8; local and absolute maximum: 8 at x = 0 
D)  local and absolute maximum: 0 at x = -8; local and absolute minimum: 8 at x = 0

Question 6

Find the value or values of $c$ that satisfy the equation $\frac { f ( b ) - f ( a ) } { b - a } = f ^ { \prime } ( c )$ in the conclusion of the Mean Value Theorem for the function and interval.
-$f ( x ) = x ^ { 2 } + 3 x + 1 , [ - 1,2 ]$

Accepted Answer

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

Question 7

Find the most general antiderivative.
-$\int \left( \frac { \sqrt { y } } { 8 } + \frac { 8 } { \sqrt { y } } \right) d y$

Accepted Answer

A)  $\frac { 1 } { 12 } y ^ { 3 / 2 } + 16 \sqrt { y } + C$ 
B)  $\frac { 3 } { 16 } \mathrm { y } ^ { 3 / 2 } + \frac { 1 } { 16 } \sqrt { \mathrm { y } } + \mathrm { C }$ 
C)  $\frac { 1 } { 12 } y ^ { 3 / 2 } - 16 \sqrt { y } + C$ 
D)  $\frac { 1 } { 16 } \sqrt { y } - \frac { 1 } { 16 \sqrt { y } } + C$ 
A)  $\frac { 1 } { 12 } y ^ { 3 / 2 } + 16 \sqrt { y } + C$ 
B)  $\frac { 3 } { 16 } \mathrm { y } ^ { 3 / 2 } + \frac { 1 } { 16 } \sqrt { \mathrm { y } } + \mathrm { C }$ 
C)  $\frac { 1 } { 12 } y ^ { 3 / 2 } - 16 \sqrt { y } + C$ 
D)  $\frac { 1 } { 16 } \sqrt { y } - \frac { 1 } { 16 \sqrt { y } } + C$

Question 8

Use l'H^opital's rule to find the limit.
-$\lim _ { x \rightarrow 0 } \frac { \sin 4 x } { \tan 3 x }$

Accepted Answer

A)  $- \frac { 4 } { 3 }$ 
B)  $\frac { 4 } { 3 }$ 
C)  0 
D)  $\frac { 3 } { 4 }$ 
A)  $- \frac { 4 } { 3 }$ 
B)  $\frac { 4 } { 3 }$ 
C)  0 
D)  $\frac { 3 } { 4 }$

Question 9

Solve the problem.
-$$\text { How many solutions does the equation } \cos 4 x = 0.95 - x ^ { 2 } \text { have? }$$

Accepted Answer

The answer of Solve the problem.
-\[\text { How many solutions...

Question 10

Find all possible functions with the given derivative.
-$y ^ { \prime } = 3 t - \frac { 4 } { \sqrt { t } }$

Accepted Answer

A)  $3 t ^ { 2 } - \frac { 4 } { \sqrt { t } } + C$ 
B)  $t ^ { 2 } - 4 \sqrt { t } + C$ 
C)  $3 t ^ { 2 } - 8 \sqrt { t } + C$ 
D)  $\frac { 3 } { 2 } t ^ { 2 } - 8 \sqrt { t } + C$ 
A)  $3 t ^ { 2 } - \frac { 4 } { \sqrt { t } } + C$ 
B)  $t ^ { 2 } - 4 \sqrt { t } + C$ 
C)  $3 t ^ { 2 } - 8 \sqrt { t } + C$ 
D)  $\frac { 3 } { 2 } t ^ { 2 } - 8 \sqrt { t } + C$

Question 11

Solve the problem.
-The stiffness of a rectangular beam is proportional to its width times the cube of its depth. Find the dimensions of the stiffest beam than can be cut from a 13-in.-diameter cylindrical log. (Round answers to the nearest tenth.)

Accepted Answer

A)  w = 7.5 in.; d = 12.3 in. 
B)  w = 7.5 in.; d = 10.3 in. 
C)  w = 5.5 in.; d = 12.3 in. 
D)  w = 6.5 in.; d = 11.3 in. 
A)  w = 7.5 in.; d = 12.3 in. 
B)  w = 7.5 in.; d = 10.3 in. 
C)  w = 5.5 in.; d = 12.3 in. 
D)  w = 6.5 in.; d = 11.3 in.

Question 12

Find the limit.
-$\lim _ { x \rightarrow 0 ^ { + } } x ^ { 8 } \ln x$

Accepted Answer

A)  8 
B)  -1 
C)  1 
D)  0 
A)  8 
B)  -1 
C)  1 
D)  0

Question 13

Solve the problem.
-Use Newton's method to find the value of $x$ for which $e ^ { - x } = x ^ { 2 } - 2 x + 5$. Find the answer correct to five decimal places.

Accepted Answer

A)  -2.99294 
B)  -2.99107 
C)  -2.99263 
D)  -2.99288 
A)  -2.99294 
B)  -2.99107 
C)  -2.99263 
D)  -2.99288

Question 14

L'Hopital's rule does not help with the given limit. Find the limit some other way.
-$\lim _ { x \rightarrow 0 } \frac { \sec x } { \csc x }$

Accepted Answer

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

Question 15

Use the maximum/minimum finder on a graphing calculator to determine the approximate location of all local extrema.
-$$f ( x ) = x ^ { 4 } - 3 x ^ { 3 } - 21 x ^ { 2 } + 74 x + 30$$

Accepted Answer

A)  Approximate local maximum at 1.694; approximate local minima at -3.044 and 3.722 
B)  Approximate local maximum at 1.702; approximate local minima at -3.155 and 3.676 
C)  Approximate local maximum at 1.604; approximate local minima at -3.089 and 3.735 
D)  Approximate local maximum at 1.576; approximate local minima at -3.004 and 3.752 
A)  Approximate local maximum at 1.694; approximate local minima at -3.044 and 3.722 
B)  Approximate local maximum at 1.702; approximate local minima at -3.155 and 3.676 
C)  Approximate local maximum at 1.604; approximate local minima at -3.089 and 3.735 
D)  Approximate local maximum at 1.576; approximate local minima at -3.004 and 3.752

Question 16

Find an antiderivative of the given function.
-$\frac { 4 } { 5 } \csc ^ { 2 } \frac { x } { 5 }$

Accepted Answer

A)  $\frac { 4 } { 25 } \cot \frac { x } { 5 }$ 
B)  $\frac { 8 } { 5 } \csc ^ { 2 } \frac { x } { 5 } \cot \frac { x } { 5 }$ 
C)  $4 \cot \frac { x } { 5 }$ 
D)  $- 4 \cot \frac { x } { 5 }$ 
A)  $\frac { 4 } { 25 } \cot \frac { x } { 5 }$ 
B)  $\frac { 8 } { 5 } \csc ^ { 2 } \frac { x } { 5 } \cot \frac { x } { 5 }$ 
C)  $4 \cot \frac { x } { 5 }$ 
D)  $- 4 \cot \frac { x } { 5 }$

Question 17

Provide an appropriate response.
-$\text { Find the absolute maximum and minimum values of } f ( x ) = 2 x - e ^ { x } \text { on } [ 0,1 ] \text {. }$

Accepted Answer

A)  Maximum = ln 4 - 2 at x = ln 2, minimum = 1 at x = 0 
B)  Maximum = ln 2 - 2 at x = ln 2, minimum = -1 at x = 0 
C)  Maximum = 0 at x = ln 2, minimum = 1 at x = 0 
D)  Maximum = ln 4 - 2 at x = ln 2, minimum = -1 at x = 0 
A)  Maximum = ln 4 - 2 at x = ln 2, minimum = 1 at x = 0 
B)  Maximum = ln 2 - 2 at x = ln 2, minimum = -1 at x = 0 
C)  Maximum = 0 at x = ln 2, minimum = 1 at x = 0 
D)  Maximum = ln 4 - 2 at x = ln 2, minimum = -1 at x = 0

Question 18

Find the absolute extreme values of the function on the interval.
-$$F ( x ) = \sqrt [ 3 ] { x } , - 3 \leq x \leq 8$$

Accepted Answer

A)  absolute maximum is 2 at $x = 8$; absolute minimum is $- 2$ at $x = - 8$ 
B)  absolute maximum is 2 at $x = - 8$; absolute minimum is 0 at $x = 0$ 
C)  absolute maximum is 0 at $x = 0$; absolute minimum is 2 at $x = 8$ 
D)  absolute maximum is 2 at $x = 8$; absolute minimum is 0 at $x = 0$ 
A)  absolute maximum is 2 at $x = 8$; absolute minimum is $- 2$ at $x = - 8$ 
B)  absolute maximum is 2 at $x = - 8$; absolute minimum is 0 at $x = 0$ 
C)  absolute maximum is 0 at $x = 0$; absolute minimum is 2 at $x = 8$ 
D)  absolute maximum is 2 at $x = 8$; absolute minimum is 0 at $x = 0$

Question 19

Sketch the graph of the function and determine whether it has any absolute extreme values on its domain.
-

Accepted Answer

A)  no absolute maximum;
no absolute minimum
  
B)  no absolute maximum;
 absolute minimum at $x = \pi$
  
C)  absolute maximum at $x = 0$;
 absolute minimum at $x = \pi$ 
  
D)  absolute maximum at $x = 0$ and $x = 2 \pi$;
absolute minimum at $x = \pi$
  
A)  no absolute maximum;
no absolute minimum
  
B)  no absolute maximum;
 absolute minimum at $x = \pi$
  
C)  absolute maximum at $x = 0$;
 absolute minimum at $x = \pi$ 
  
D)  absolute maximum at $x = 0$ and $x = 2 \pi$;
absolute minimum at $x = \pi$

Question 20

Solve the initial value problem.
-$\frac { \mathrm { dv } } { \mathrm { dt } } = \frac { 1 } { 4 } \csc t \cot t , v \left( \frac { 3 \pi } { 2 } \right) = \frac { 3 } { 4 }$

Accepted Answer

A)  $v = - \frac { \csc t } { 4 } + \frac { 1 } { 2 }$ 
B)  $v = \csc t - \frac { 1 } { 4 }$ 
C)  $v = - \frac { \sec t } { 4 } + \frac { 3 } { 4 }$ 
D)  $v = - \frac { \operatorname { csct } } { 4 } + 1$ 
A)  $v = - \frac { \csc t } { 4 } + \frac { 1 } { 2 }$ 
B)  $v = \csc t - \frac { 1 } { 4 }$ 
C)  $v = - \frac { \sec t } { 4 } + \frac { 3 } { 4 }$ 
D)  $v = - \frac { \operatorname { csct } } { 4 } + 1$

Solve the problem. -Use Newton's method to find the positive fourth root of 5 by solving the equation $x ^ { 4 } - 5 = 0$ . Start with $x _ { 1 } = 1$ and find $x _ { 2 }$ .

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. -Given that $x > 0$ , find the maximum value, if any, of $x ^ { 1 / x ^ { 6 } }$ .

Solve the problem. -A company is constructing an open-top, square-based, rectangular metal tank that will have a volume of $33 \mathrm { ft } 3$ . What dimensions yield the minimum surface area? Round to the nearest tenth, if necessary.

Identify the function's extreme values in the given domain, and say where they are assumed. Tell which of the extreme values, if any, are absolute. - $f ( x ) = \sqrt { 64 - x ^ { 2 } } , - 8 \leq x < 8$

Find the value or values of $c$ that satisfy the equation $\frac { f ( b ) - f ( a ) } { b - a } = f ^ { \prime } ( c )$ in the conclusion of the Mean Value Theorem for the function and interval. - $f ( x ) = x ^ { 2 } + 3 x + 1 , [ - 1,2 ]$

Find the most general antiderivative. - $\int \left( \frac { \sqrt { y } } { 8 } + \frac { 8 } { \sqrt { y } } \right) d y$

Use l'H^opital's rule to find the limit. - $\lim _ { x \rightarrow 0 } \frac { \sin 4 x } { \tan 3 x }$

Solve the problem. - $\text { How many solutions does the equation } \cos 4 x = 0.95 - x ^ { 2 } \text { have? }$

Find all possible functions with the given derivative. - $y ^ { \prime } = 3 t - \frac { 4 } { \sqrt { t } }$

Solve the problem. -The stiffness of a rectangular beam is proportional to its width times the cube of its depth. Find the dimensions of the stiffest beam than can be cut from a 13-in.-diameter cylindrical log. (Round answers to the nearest tenth.)

Find the limit. - $\lim _ { x \rightarrow 0 ^ { + } } x ^ { 8 } \ln x$

Solve the problem. -Use Newton's method to find the value of $x$ for which $e ^ { - x } = x ^ { 2 } - 2 x + 5$ . Find the answer correct to five decimal places.

L'Hopital's rule does not help with the given limit. Find the limit some other way. - $\lim _ { x \rightarrow 0 } \frac { \sec x } { \csc x }$

Use the maximum/minimum finder on a graphing calculator to determine the approximate location of all local extrema. - $f ( x ) = x ^ { 4 } - 3 x ^ { 3 } - 21 x ^ { 2 } + 74 x + 30$

Find an antiderivative of the given function. - $\frac { 4 } { 5 } \csc ^ { 2 } \frac { x } { 5 }$

Provide an appropriate response. - $\text { Find the absolute maximum and minimum values of } f ( x ) = 2 x - e ^ { x } \text { on } [ 0,1 ] \text {. }$

Find the absolute extreme values of the function on the interval. - $F ( x ) = \sqrt [ 3 ] { x } , - 3 \leq x \leq 8$

Sketch the graph of the function and determine whether it has any absolute extreme values on its domain. -

Solve the initial value problem. - $\frac { \mathrm { dv } } { \mathrm { dt } } = \frac { 1 } { 4 } \csc t \cot t , v \left( \frac { 3 \pi } { 2 } \right) = \frac { 3 } { 4 }$

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Exam 5: Applications of Derivatives

Solve the problem. -Use Newton's method to find the positive fourth root of 5 by solving the equation x4−5=0x ^ { 4 } - 5 = 0x4−5=0 . Start with x1=1x _ { 1 } = 1x1​=1 and find x2x _ { 2 }x2​ .

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. -Given that x>0x > 0x>0 , find the maximum value, if any, of x1/x6x ^ { 1 / x ^ { 6 } }x1/x6 .

Solve the problem. -A company is constructing an open-top, square-based, rectangular metal tank that will have a volume of 33ft333 \mathrm { ft } 333ft3 . What dimensions yield the minimum surface area? Round to the nearest tenth, if necessary.

Identify the function's extreme values in the given domain, and say where they are assumed. Tell which of the extreme values, if any, are absolute. - f(x)=64−x2,−8≤x<8f ( x ) = \sqrt { 64 - x ^ { 2 } } , - 8 \leq x < 8f(x)=64−x2​,−8≤x<8

Find the most general antiderivative. - ∫(y8+8y)dy\int \left( \frac { \sqrt { y } } { 8 } + \frac { 8 } { \sqrt { y } } \right) d y∫(8y​​+y​8​)dy

Use l'H^opital's rule to find the limit. - lim⁡x→0sin⁡4xtan⁡3x\lim _ { x \rightarrow 0 } \frac { \sin 4 x } { \tan 3 x }limx→0​tan3xsin4x​

Solve the problem. - How many solutions does the equation cos⁡4x=0.95−x2 have? \text { How many solutions does the equation } \cos 4 x = 0.95 - x ^ { 2 } \text { have? } How many solutions does the equation cos4x=0.95−x2 have?

Find all possible functions with the given derivative. - y′=3t−4ty ^ { \prime } = 3 t - \frac { 4 } { \sqrt { t } }y′=3t−t​4​