Question 1

For what values of $c$ does the curve have maximum and minimum points for the given function
$$f ( x ) = 2 x ^ { 3 } + c x ^ { 2 } + 8 x ?$$

Accepted Answer

The answer of For what values of $c$ does the...

Question 2

A piece of wire $14 \mathrm {~m}$ long is cut into two pieces. One piece is bent into a square and the other is bent into an equilateral triangle. How should the wire be cut for the square so that the total area enclosed is a minimum?
Round your answer to the nearest hundredth.

Accepted Answer

The answer of A piece of wire $14 \mathrm {~m}$...

Question 3

The function $f ( x ) = \frac { 1 } { 3 } x ^ { 3 } - 25 x$ satisfies the hypotheses of Rolle's Theorem on the interval $[ 0,5 \sqrt { 3 } ]$. Find all values of $c$ that satisfy the conclusion of the theorem.

Accepted Answer

A)  $\pm 5$ 
B)  5 
C)  $\pm 4$ 
D)  4 
A)  $\pm 5$ 
B)  5 
C)  $\pm 4$ 
D)  4

Question 4

$$\text { Find the slant asymptote of the function } f ( x ) = \frac { x ^ { 2 } + 7 } { x } \text {. }$$

Accepted Answer

The answer of \[\text { Find the slant asymptote of...

Question 5

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 6

$$\text { For what values of } c \text { does the curve have maximum and minimum points? }$$ Select the correct answer.
$$F ( x ) = 4 x ^ { 3 } + c x ^ { 2 } + 10 x$$

Accepted Answer

A)  $| c | > 40$ 
B)  $| c | > \sqrt { 1200 }$ 
C)  $| c | > \sqrt { 120 }$ 
D)  $| c | > 12$ 
E)  $| c | > \sqrt { 30 }$ 
A)  $| c | > 40$ 
B)  $| c | > \sqrt { 1200 }$ 
C)  $| c | > \sqrt { 120 }$ 
D)  $| c | > 12$ 
E)  $| c | > \sqrt { 30 }$

Question 7

$$\text { Approximate the zero of } f ( x ) = \cos ( x ) + 3 x \text { in } [ - 1,1 ] \text { to within } 0.00001 \text {. }$$

Accepted Answer

The answer of \[\text { Approximate the zero of }...

Question 8

Use Newton's method to approximate the indicated root of $x ^ { 4 } + x - 4 = 0$ in the interval $[ 1,2 ]$, correct to six decimal places.
Use $x _ { 1 } = 1.5$ as the initial approximation.

Accepted Answer

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

Question 9

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)  $\sqrt { 3 } \mathrm { in } , 80 \mathrm { in }$ 
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)  $\sqrt { 3 } \mathrm { in } , 80 \mathrm { in }$ 
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 10

Find the critical number(s) of the function.
$$y = x ^ { 2 } + 10 x$$

Accepted Answer

The answer of Find the critical number(s) of the function.
\[y...

Question 11

Find the absolute maximum of the function. $$f ( x ) = \sin ( 2 x ) + 2 \cos ( x ) \text { on the interval } \left[ 0 , \frac { \pi } { 2 } \right]$$

Accepted Answer

The answer of Find the absolute maximum of the function....

Question 12

$$\text { Find the slant asymptote of the graph of } f ( x ) = \frac { x ^ { 2 } + 4 x + 3 } { 2 x - 6 } \text { using the curve-sketching guidelines. }$$

Accepted Answer

The answer of \[\text { Find the slant asymptote of...

Question 13

What can you say about point of inflation for $f ( x ) = x e ^ { - 3 x }$ ?

Accepted Answer

A)  point of inflation gets closer to $\pm 6$ 
B)  point of inflation goes away from the origin 
C)  point of inflation gets closer to the origin 
D)  point of inflation goes away from $\pm 3$ 
A)  point of inflation gets closer to $\pm 6$ 
B)  point of inflation goes away from the origin 
C)  point of inflation gets closer to the origin 
D)  point of inflation goes away from $\pm 3$

Question 14

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

Accepted Answer

The answer of \[\text { Find the point(s) of inflection...

Question 15

$$\text { Find the critical number(s), if any of the function } f ( t ) = 3 t ^ { \frac { 3 } { 4 } } - 3 t ^ { \frac { 1 } { 4 } }$$

Accepted Answer

The answer of \[\text { Find the critical number(s), if...

Question 16

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.
$$f ( x ) = x ^ { 3 } - 5 x ^ { 2 } + 6 x + 1 , [ 0,4 ]$$

Accepted Answer

A) 
$$c = 2 + \frac { 2 \sqrt { 3 } } { 3 }$$ 
B) 
$$c _ { 1 } = \frac { 5 } { 3 } + \frac { \sqrt { 7 } } { 3 } , c _ { 2 } = \frac { 5 } { 3 } - \frac { \sqrt { 7 } } { 3 }$$ 
C) 
$$c _ { 1 } = 12 + \frac { \sqrt { 2 } } { 3 } , c _ { 2 } = 12 - \frac { \sqrt { 2 } } { 3 }$$ 
D) 
$$c = 2 - \frac { 2 \sqrt { 3 } } { 3 }$$ 
E) 
$$c _ { 1 } = 2 + \frac { 2 \sqrt { 3 } } { 3 } , c _ { 2 } = \frac { 2 \sqrt { 3 } } { 3 }$$ 
A) 
$$c = 2 + \frac { 2 \sqrt { 3 } } { 3 }$$ 
B) 
$$c _ { 1 } = \frac { 5 } { 3 } + \frac { \sqrt { 7 } } { 3 } , c _ { 2 } = \frac { 5 } { 3 } - \frac { \sqrt { 7 } } { 3 }$$ 
C) 
$$c _ { 1 } = 12 + \frac { \sqrt { 2 } } { 3 } , c _ { 2 } = 12 - \frac { \sqrt { 2 } } { 3 }$$ 
D) 
$$c = 2 - \frac { 2 \sqrt { 3 } } { 3 }$$ 
E) 
$$c _ { 1 } = 2 + \frac { 2 \sqrt { 3 } } { 3 } , c _ { 2 } = \frac { 2 \sqrt { 3 } } { 3 }$$

Question 17

Consider the function $f ( x ) = x ^ { 2 } e ^ { - 9 x }$. (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 18

A right circular cylinder is inscribed in a sphere of radius $r = 5 \mathrm {~cm}$. Find the largest possible surface area of such a cylinder. Round the result to the nearest hundredth.

Accepted Answer

A)  $157.08 \mathrm {~cm} ^ { 2 }$ 
B)  $177.08 \mathrm {~cm} ^ { 2 }$ 
C)  $187.08 \mathrm {~cm} ^ { 2 }$ 
D)  $167.08 \mathrm {~cm} ^ { 2 }$ 
E)  $197.08 \mathrm {~cm} ^ { 2 }$ 
A)  $157.08 \mathrm {~cm} ^ { 2 }$ 
B)  $177.08 \mathrm {~cm} ^ { 2 }$ 
C)  $187.08 \mathrm {~cm} ^ { 2 }$ 
D)  $167.08 \mathrm {~cm} ^ { 2 }$ 
E)  $197.08 \mathrm {~cm} ^ { 2 }$

Question 19

$$\text { Sketch the graph of the function } y = ( x + 1 ) ^ { 3 / 2 } - 2 \text { using the curve-sketching guidelines. }$$

Accepted Answer

The answer of \[\text { Sketch the graph of the...

Question 20

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)  $\left( 0 , \frac { \pi } { 2 } \right)$

For what values of $c$ does the curve have maximum and minimum points for the given function $f ( x ) = 2 x ^ { 3 } + c x ^ { 2 } + 8 x ?$

A piece of wire $14 \mathrm {~m}$ long is cut into two pieces. One piece is bent into a square and the other is bent into an equilateral triangle. How should the wire be cut for the square so that the total area enclosed is a minimum? Round your answer to the nearest hundredth.

The function $f ( x ) = \frac { 1 } { 3 } x ^ { 3 } - 25 x$ satisfies the hypotheses of Rolle's Theorem on the interval $[ 0,5 \sqrt { 3 } ]$ . Find all values of $c$ that satisfy the conclusion of the theorem.

$\text { Find the slant asymptote of the function } f ( x ) = \frac { x ^ { 2 } + 7 } { x } \text {. }$

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.

$\text { For what values of } c \text { does the curve have maximum and minimum points? }$ Select the correct answer. $F ( x ) = 4 x ^ { 3 } + c x ^ { 2 } + 10 x$

$\text { Approximate the zero of } f ( x ) = \cos ( x ) + 3 x \text { in } [ - 1,1 ] \text { to within } 0.00001 \text {. }$

Use Newton's method to approximate the indicated root of $x ^ { 4 } + x - 4 = 0$ in the interval $[ 1,2 ]$ , correct to six decimal places. Use $x _ { 1 } = 1.5$ as the initial approximation.

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 the critical number(s) of the function. $y = x ^ { 2 } + 10 x$

Find the absolute maximum of the function. $f ( x ) = \sin ( 2 x ) + 2 \cos ( x ) \text { on the interval } \left[ 0 , \frac { \pi } { 2 } \right]$

$\text { Find the slant asymptote of the graph of } f ( x ) = \frac { x ^ { 2 } + 4 x + 3 } { 2 x - 6 } \text { using the curve-sketching guidelines. }$

What can you say about point of inflation for $f ( x ) = x e ^ { - 3 x }$ ?

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

$\text { Find the critical number(s), if any of the function } f ( t ) = 3 t ^ { \frac { 3 } { 4 } } - 3 t ^ { \frac { 1 } { 4 } }$

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. $f ( x ) = x ^ { 3 } - 5 x ^ { 2 } + 6 x + 1 , [ 0,4 ]$

Consider the function $f ( x ) = x ^ { 2 } e ^ { - 9 x }$ . (a) Find the intervals on which $f$ is increasing or decreasing. (b) Find the relative maxima and relative minima of $f$ .

A right circular cylinder is inscribed in a sphere of radius $r = 5 \mathrm {~cm}$ . Find the largest possible surface area of such a cylinder. Round the result to the nearest hundredth.

$\text { Sketch the graph of the function } y = ( x + 1 ) ^ { 3 / 2 } - 2 \text { using the curve-sketching guidelines. }$

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

Functions and Models

Limits and Derivatives

Applications of Differentiation

Integrals

Applications of Integration

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

Filters

Exam 3: Differentiation Rules

For what values of ccc does the curve have maximum and minimum points for the given function f(x)=2x3+cx2+8x?f ( x ) = 2 x ^ { 3 } + c x ^ { 2 } + 8 x ?f(x)=2x3+cx2+8x?

A piece of wire 14 m14 \mathrm {~m}14 m long is cut into two pieces. One piece is bent into a square and the other is bent into an equilateral triangle. How should the wire be cut for the square so that the total area enclosed is a minimum? Round your answer to the nearest hundredth.

The function f(x)=13x3−25xf ( x ) = \frac { 1 } { 3 } x ^ { 3 } - 25 xf(x)=31​x3−25x satisfies the hypotheses of Rolle's Theorem on the interval [0,53][ 0,5 \sqrt { 3 } ][0,53​] . Find all values of ccc that satisfy the conclusion of the theorem.

Find the slant asymptote of the function f(x)=x2+7x. \text { Find the slant asymptote of the function } f ( x ) = \frac { x ^ { 2 } + 7 } { x } \text {. } Find the slant asymptote of the function f(x)=xx2+7​.

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 ccc that satisfy the conclusion of the theorem.

Approximate the zero of f(x)=cos⁡(x)+3x in [−1,1] to within 0.00001. \text { Approximate the zero of } f ( x ) = \cos ( x ) + 3 x \text { in } [ - 1,1 ] \text { to within } 0.00001 \text {. } Approximate the zero of f(x)=cos(x)+3x in [−1,1] to within 0.00001.

Use Newton's method to approximate the indicated root of x4+x−4=0x ^ { 4 } + x - 4 = 0x4+x−4=0 in the interval [1,2][ 1,2 ][1,2] , correct to six decimal places. Use x1=1.5x _ { 1 } = 1.5x1​=1.5 as the initial approximation.

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 the critical number(s) of the function. y=x2+10xy = x ^ { 2 } + 10 xy=x2+10x

Find the absolute maximum of the function. f(x)=sin⁡(2x)+2cos⁡(x) on the interval [0,π2]f ( x ) = \sin ( 2 x ) + 2 \cos ( x ) \text { on the interval } \left[ 0 , \frac { \pi } { 2 } \right]f(x)=sin(2x)+2cos(x) on the interval [0,2π​]

What can you say about point of inflation for f(x)=xe−3xf ( x ) = x e ^ { - 3 x }f(x)=xe−3x ?

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

Find the critical number(s), if any of the function f(t)=3t34−3t14\text { Find the critical number(s), if any of the function } f ( t ) = 3 t ^ { \frac { 3 } { 4 } } - 3 t ^ { \frac { 1 } { 4 } } Find the critical number(s), if any of the function f(t)=3t43​−3t41​

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)=x3−5x2+6x+1,[0,4]f ( x ) = x ^ { 3 } - 5 x ^ { 2 } + 6 x + 1 , [ 0,4 ]f(x)=x3−5x2+6x+1,[0,4]

Consider the function f(x)=x2e−9xf ( x ) = x ^ { 2 } e ^ { - 9 x }f(x)=x2e−9x . (a) Find the intervals on which fff is increasing or decreasing. (b) Find the relative maxima and relative minima of fff .

A right circular cylinder is inscribed in a sphere of radius r=5 cmr = 5 \mathrm {~cm}r=5 cm . Find the largest possible surface area of such a cylinder. Round the result to the nearest hundredth.

Sketch the graph of the function y=(x+1)3/2−2 using the curve-sketching guidelines. \text { Sketch the graph of the function } y = ( x + 1 ) ^ { 3 / 2 } - 2 \text { using the curve-sketching guidelines. } Sketch the graph of the function y=(x+1)3/2−2 using the curve-sketching guidelines.

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

Functions and Models

Limits and Derivatives

Applications of Differentiation

Integrals

Applications of Integration

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

Filters

For what values of $c$ does the curve have maximum and minimum points for the given function $f ( x ) = 2 x ^ { 3 } + c x ^ { 2 } + 8 x ?$

A piece of wire $14 \mathrm {~m}$ long is cut into two pieces. One piece is bent into a square and the other is bent into an equilateral triangle. How should the wire be cut for the square so that the total area enclosed is a minimum? Round your answer to the nearest hundredth.

The function $f ( x ) = \frac { 1 } { 3 } x ^ { 3 } - 25 x$ satisfies the hypotheses of Rolle's Theorem on the interval $[ 0,5 \sqrt { 3 } ]$ . Find all values of $c$ that satisfy the conclusion of the theorem.

$\text { Find the slant asymptote of the function } f ( x ) = \frac { x ^ { 2 } + 7 } { x } \text {. }$

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.

$\text { Approximate the zero of } f ( x ) = \cos ( x ) + 3 x \text { in } [ - 1,1 ] \text { to within } 0.00001 \text {. }$

Use Newton's method to approximate the indicated root of $x ^ { 4 } + x - 4 = 0$ in the interval $[ 1,2 ]$ , correct to six decimal places. Use $x _ { 1 } = 1.5$ as the initial approximation.

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 the critical number(s) of the function. $y = x ^ { 2 } + 10 x$

Find the absolute maximum of the function. $f ( x ) = \sin ( 2 x ) + 2 \cos ( x ) \text { on the interval } \left[ 0 , \frac { \pi } { 2 } \right]$

What can you say about point of inflation for $f ( x ) = x e ^ { - 3 x }$ ?

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

$\text { Find the critical number(s), if any of the function } f ( t ) = 3 t ^ { \frac { 3 } { 4 } } - 3 t ^ { \frac { 1 } { 4 } }$

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. $f ( x ) = x ^ { 3 } - 5 x ^ { 2 } + 6 x + 1 , [ 0,4 ]$

Consider the function $f ( x ) = x ^ { 2 } e ^ { - 9 x }$ . (a) Find the intervals on which $f$ is increasing or decreasing. (b) Find the relative maxima and relative minima of $f$ .

A right circular cylinder is inscribed in a sphere of radius $r = 5 \mathrm {~cm}$ . Find the largest possible surface area of such a cylinder. Round the result to the nearest hundredth.

$\text { Sketch the graph of the function } y = ( x + 1 ) ^ { 3 / 2 } - 2 \text { using the curve-sketching guidelines. }$

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