Question 1

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

Accepted Answer

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

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. Select the correct answer.

Accepted Answer

A)  $0 \mathrm {~m}$ 
B)  $6.09 \mathrm {~m}$ 
C)  $8.09 \mathrm {~m}$ 
D)  $10.09 \mathrm {~m}$ 
E)  $9.09 \mathrm {~m}$ 
A)  $0 \mathrm {~m}$ 
B)  $6.09 \mathrm {~m}$ 
C)  $8.09 \mathrm {~m}$ 
D)  $10.09 \mathrm {~m}$ 
E)  $9.09 \mathrm {~m}$

Question 3

Use Newton's method to obtain an approximation to the root of $\cos ^ { - 1 } x - 2 x = 0$ to within $0.00001$.

Accepted Answer

A)  $0.62821$ 
B)  $0.48918$ 
C)  $0.56127$ 
D)  $0.51493$ 
A)  $0.62821$ 
B)  $0.48918$ 
C)  $0.56127$ 
D)  $0.51493$

Question 4

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

A)  $c = \pm \sqrt { 33 }$ 
B)  $c = \pm 4 \sqrt { 3 }$ 
C)  $- 4 \sqrt { 3 }  4 \sqrt { 3 }$ 
E)  $c \langle - \sqrt { 33 } , c \rangle \sqrt { 33 }$ 
A)  $c = \pm \sqrt { 33 }$ 
B)  $c = \pm 4 \sqrt { 3 }$ 
C)  $- 4 \sqrt { 3 }  4 \sqrt { 3 }$ 
E)  $c \langle - \sqrt { 33 } , c \rangle \sqrt { 33 }$

Question 5

Estimate the value of $\sqrt [ 3 ] { 5 }$ by using three iterations of Newton's method to solve the equation $x ^ { 3 } - 5 = 0$ with initial estimate $x _ { 0 } = 2$. Round your final estimate to four decimal places.

Accepted Answer

A)  $2.2361$ 
B)  $2.4134$ 
C)  $1.71$ 
D)  $1.8532$ 
A)  $2.2361$ 
B)  $2.4134$ 
C)  $1.71$ 
D)  $1.8532$

Question 6

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

Accepted Answer

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

Question 7

$$\text { What is the function of the graph? }$$

Accepted Answer

The answer of \[\text { What is the function of...

Question 8

A fence $10 \mathrm { ft }$ tall runs parallel to a tall building at a distance of $5 \mathrm { ft }$ from the building. What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building? Round the result to the nearest hundredth.

Accepted Answer

The answer of A fence $10 \mathrm { ft }$...

Question 9

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) Increa

Question 10

For what values of $c$ does the curve have maximum and minimum points for the given function $f ( x ) = c x ^ { 4 } - 5 x ^ { 2 } + 1 ?$
Select the correct answer.

Accepted Answer

A) For $c = 0 , f ( x ) = - 5 x ^ { 2 } + 1$, a parabola whose vertex $( 0,2 )$, is the absolute maximum. For $c > 0 , f ( x ) = c x ^ { 4 } - 5 x ^ { 2 } + 1$, opens downward with one minimum points. For $c 0 , f ( x ) = c x ^ { 4 } + 5 x ^ { 2 } + 1$, opens upward with two minimum points. For $c 0 , f ( x ) = c x ^ { 4 } - 5 x ^ { 2 } + 1$, opens upward with two minimum points. For $c 0 , f ( x ) = c x ^ { 4 } - 5 x ^ { 2 } + 1$, opens upward with two minimum points. For $c 0 , f ( x ) = c x ^ { 4 } + 5 x ^ { 2 } - 1$, opens downward with two maximum points. For $c < 0$, the graph opens upward, and has an absolute minimum at $x = 0$. A) For $c = 0 , f ( x ) = - 5 x ^ { 2 } + 1$, a parabola whose vertex $( 0,2 )$, is the absolute maximum. For $c > 0 , f ( x ) = c x ^ { 4 } - 5 x ^ { 2 } + 1$, opens downward with one minimum points. For $c 0 , f ( x ) = c x ^ { 4 } + 5 x ^ { 2 } + 1$, opens upward with two minimum points. For $c 0 , f ( x ) = c x ^ { 4 } - 5 x ^ { 2 } + 1$, opens upward with two minimum points. For $c 0 , f ( x ) = c x ^ { 4 } - 5 x ^ { 2 } + 1$, opens upward with two minimum points. For $c 0 , f ( x ) = c x ^ { 4 } + 5 x ^ { 2 } - 1$, opens downward with two maximum points. For $c < 0$, the graph opens upward, and has an absolute minimum at $x = 0$.

Question 11

Find the critical numbers of $f ( x ) = x ^ { 4 } ( x - 3 ) ^ { 3 }$.

Accepted Answer

A)  $0,3 , \frac { 12 } { 7 }$ 
B)  $0,4 , \frac { 7 } { 12 }$ 
C)  $0,4 , \frac { 12 } { 7 }$ 
D)  $0,3 , \frac { 12 } { 11 }$ 
E)  $0,3 , \frac { 7 } { 12 }$ 
A)  $0,3 , \frac { 12 } { 7 }$ 
B)  $0,4 , \frac { 7 } { 12 }$ 
C)  $0,4 , \frac { 12 } { 7 }$ 
D)  $0,3 , \frac { 12 } { 11 }$ 
E)  $0,3 , \frac { 7 } { 12 }$

Question 12

Find the dimensions of the rectangle enclosed in the semicircle $y = \sqrt { 144 - x ^ { 2 } }$ with the largest possible area.

Accepted Answer

A)  5 in. $\times 7$ in. 
B)  6 in. $\times 6$ in. 
C)  $12 \sqrt { 2 }$ in. $\times 6 \sqrt { 2 }$ in. 
D)  $6 \sqrt { 2 }$ in. $\times 6 \sqrt { 2 }$ in. 
A)  5 in. $\times 7$ in. 
B)  6 in. $\times 6$ in. 
C)  $12 \sqrt { 2 }$ in. $\times 6 \sqrt { 2 }$ in. 
D)  $6 \sqrt { 2 }$ in. $\times 6 \sqrt { 2 }$ in.

Question 13

The graph of the derivative $f ^ { \prime } ( x )$ of a continuous function $f$ is shown. On what intervals is $f$ decreasing?

Accepted Answer

The answer of The graph of the derivative \(f ^...

Question 14

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 15

The size of the monthly repayment $k$ that amortizes a loan of $A$ dollars in $N$ years at an interest rate of $r$ per year, compounded monthly, on the unpaid balance is given by
$$k = \frac { A r } { 12 \left[ 1 - \left( 1 + \frac { r } { 12 } \right) ^ { - 12 N } \right] }$$
The value of $r$ can be found by performing the iteration
$$r _ { n + 1 } = r _ { n } - \frac { A r _ { n } + 12 k \left[ \left( 1 + \frac { r _ { n } } { 12 } \right) ^ { - 12 N } - 1 \right] } { A - 12 N k \left( 1 + \frac { r _ { n } } { 12 } \right) ^ { - 12 N - 1 } } .$$
A family secured a loan of $\$ 360,000$ from a bank to finance the purchase of a house. They have agreed to repay the loan in equal monthly installments of $\$ 2476$ over 25 years. Find the interest rate on this loan. Round the rate to one decimal place.

Accepted Answer

A)  $8.7 \%$ 
B)  $7.7 \%$ 
C)  $6.7 \%$ 
D)  $5.7 \%$ 
A)  $8.7 \%$ 
B)  $7.7 \%$ 
C)  $6.7 \%$ 
D)  $5.7 \%$

Question 16

Sketch the graph of the function $g ( x ) = \frac { x - 2 } { x - 1 }$ using the curve-sketching guidelines.

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 17

Determine where the graph of the function $f ( x ) = x ^ { 3 } - 6 x$ is concave upward and where it is concave downward. Also, find all inflection points of the function.

Accepted Answer

A)  $\mathrm { CU }$ on $( \sqrt { 2 } , \infty ) , \mathrm { CD }$ on $( - \infty , \sqrt { 2 } )$,
IP $( \sqrt { 2 } , - 4 \sqrt { 2 } )$ 
B)  $\mathrm { CU }$ on $( - \sqrt { 2 } , \sqrt { 2 } ) , \mathrm { CD }$ on $( - \infty , - \sqrt { 2 } )$ and $( \sqrt { 2 } , \infty )$
IP $( - \sqrt { 2 } , 4 \sqrt { 2 } )$ and $( \sqrt { 2 } , - 4 \sqrt { 2 } )$ 
C)  $\mathrm { CU }$ on $( 0 , \infty ) , \mathrm { CD }$ on $( - \infty , 0 )$,
IP $( 0,0 )$ 
D)  $\operatorname { CU }$ on $( - \sqrt { 2 } , \infty ) , \mathrm { CD }$ on $( - \infty , - \sqrt { 2 } )$,
IP $( - \sqrt { 2 } , 4 \sqrt { 2 } )$ 
A)  $\mathrm { CU }$ on $( \sqrt { 2 } , \infty ) , \mathrm { CD }$ on $( - \infty , \sqrt { 2 } )$,
IP $( \sqrt { 2 } , - 4 \sqrt { 2 } )$ 
B)  $\mathrm { CU }$ on $( - \sqrt { 2 } , \sqrt { 2 } ) , \mathrm { CD }$ on $( - \infty , - \sqrt { 2 } )$ and $( \sqrt { 2 } , \infty )$
IP $( - \sqrt { 2 } , 4 \sqrt { 2 } )$ and $( \sqrt { 2 } , - 4 \sqrt { 2 } )$ 
C)  $\mathrm { CU }$ on $( 0 , \infty ) , \mathrm { CD }$ on $( - \infty , 0 )$,
IP $( 0,0 )$ 
D)  $\operatorname { CU }$ on $( - \sqrt { 2 } , \infty ) , \mathrm { CD }$ on $( - \infty , - \sqrt { 2 } )$,
IP $( - \sqrt { 2 } , 4 \sqrt { 2 } )$

Question 18

Find $f$. $f ^ { \prime } ( t ) = 2 t - 4 \sin t , \quad f ( 0 ) = 5$

Accepted Answer

A)  $f ( t ) = t ^ { 2 } - 3 \cos t - 5$ 
B)  $f ( t ) = t ^ { 2 } + 4 \cos t + 1$ 
C)  $f ( t ) = 2 t - 3 \sin t + 5$ 
D)  $f ( t ) = t ^ { 2 } + 3 \cos t$ 
E)  None of these 
A)  $f ( t ) = t ^ { 2 } - 3 \cos t - 5$ 
B)  $f ( t ) = t ^ { 2 } + 4 \cos t + 1$ 
C)  $f ( t ) = 2 t - 3 \sin t + 5$ 
D)  $f ( t ) = t ^ { 2 } + 3 \cos t$ 
E)  None of these

Question 19

$$\text { Find the absolute maximum value of } y = \sqrt { 9 - x ^ { 2 } } \text { on the interval } [ - 9,9 ] \text {. }$$

Accepted Answer

The answer of \[\text { Find the absolute maximum value...

Question 20

Use the Second Derivative Test to find the relative extrema, if any, of the function $f ( x ) = 2 x ^ { 3 } - 3 x ^ { 2 } - 36 x - 5$

Accepted Answer

The answer of Use the Second Derivative Test to find...

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

Use Newton's method to obtain an approximation to the root of $\cos ^ { - 1 } x - 2 x = 0$ to within $0.00001$ .

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

Estimate the value of $\sqrt [ 3 ] { 5 }$ by using three iterations of Newton's method to solve the equation $x ^ { 3 } - 5 = 0$ with initial estimate $x _ { 0 } = 2$ . Round your final estimate to four decimal places.

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

$\text { What is the function of the graph? }$ $\text { What is the function of the graph? }$

A fence $10 \mathrm { ft }$ tall runs parallel to a tall building at a distance of $5 \mathrm { ft }$ from the building. What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building? Round the result to the nearest hundredth.

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 $f$ .

For what values of $c$ does the curve have maximum and minimum points for the given function $f ( x ) = c x ^ { 4 } - 5 x ^ { 2 } + 1 ?$ Select the correct answer.

Find the critical numbers of $f ( x ) = x ^ { 4 } ( x - 3 ) ^ { 3 }$ .

Find the dimensions of the rectangle enclosed in the semicircle $y = \sqrt { 144 - x ^ { 2 } }$ with the largest possible area. $Find the dimensions of the rectangle enclosed in the semicircle y = \sqrt { 144 - x ^ { 2 } } with the largest possible area.$

The graph of the derivative $f ^ { \prime } ( x )$ of a continuous function $f$ is shown. On what intervals is $f$ decreasing? $The graph of the derivative f ^ { \prime } ( x ) of a continuous function f is shown. On what intervals is f decreasing?$

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

Sketch the graph of the function $g ( x ) = \frac { x - 2 } { x - 1 }$ using the curve-sketching guidelines.

Determine where the graph of the function $f ( x ) = x ^ { 3 } - 6 x$ is concave upward and where it is concave downward. Also, find all inflection points of the function.

Find $f$ . $f ^ { \prime } ( t ) = 2 t - 4 \sin t , \quad f ( 0 ) = 5$

$\text { Find the absolute maximum value of } y = \sqrt { 9 - x ^ { 2 } } \text { on the interval } [ - 9,9 ] \text {. }$

Use the Second Derivative Test to find the relative extrema, if any, of the function $f ( x ) = 2 x ^ { 3 } - 3 x ^ { 2 } - 36 x - 5$

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

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Exam 3: Differentiation Rules

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] .

Use Newton's method to obtain an approximation to the root of cos⁡−1x−2x=0\cos ^ { - 1 } x - 2 x = 0cos−1x−2x=0 to within 0.000010.000010.00001 .

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?

Estimate the value of 53\sqrt [ 3 ] { 5 }35​ by using three iterations of Newton's method to solve the equation x3−5=0x ^ { 3 } - 5 = 0x3−5=0 with initial estimate x0=2x _ { 0 } = 2x0​=2 . Round your final estimate to four decimal places.

For what values of ccc does the curve have maximum and minimum points? F(x)=4x3+cx2+10xF ( x ) = 4 x ^ { 3 } + c x ^ { 2 } + 10 xF(x)=4x3+cx2+10x

What is the function of the graph? \text { What is the function of the graph? } What is the function of the graph?

A fence 10ft10 \mathrm { ft }10ft tall runs parallel to a tall building at a distance of 5ft5 \mathrm { ft }5ft from the building. What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building? Round the result to the nearest hundredth.

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

For what values of ccc does the curve have maximum and minimum points for the given function f(x)=cx4−5x2+1?f ( x ) = c x ^ { 4 } - 5 x ^ { 2 } + 1 ?f(x)=cx4−5x2+1? Select the correct answer.

Find the critical numbers of f(x)=x4(x−3)3f ( x ) = x ^ { 4 } ( x - 3 ) ^ { 3 }f(x)=x4(x−3)3 .

Find the dimensions of the rectangle enclosed in the semicircle y=144−x2y = \sqrt { 144 - x ^ { 2 } }y=144−x2​ with the largest possible area.

The graph of the derivative f′(x)f ^ { \prime } ( x )f′(x) of a continuous function fff is shown. On what intervals is fff decreasing?

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

Sketch the graph of the function g(x)=x−2x−1g ( x ) = \frac { x - 2 } { x - 1 }g(x)=x−1x−2​ using the curve-sketching guidelines.

Determine where the graph of the function f(x)=x3−6xf ( x ) = x ^ { 3 } - 6 xf(x)=x3−6x is concave upward and where it is concave downward. Also, find all inflection points of the function.

Find fff . f′(t)=2t−4sin⁡t,f(0)=5f ^ { \prime } ( t ) = 2 t - 4 \sin t , \quad f ( 0 ) = 5f′(t)=2t−4sint,f(0)=5

Find the absolute maximum value of y=9−x2 on the interval [−9,9]. \text { Find the absolute maximum value of } y = \sqrt { 9 - x ^ { 2 } } \text { on the interval } [ - 9,9 ] \text {. } Find the absolute maximum value of y=9−x2​ on the interval [−9,9].

Use the Second Derivative Test to find the relative extrema, if any, of the function f(x)=2x3−3x2−36x−5f ( x ) = 2 x ^ { 3 } - 3 x ^ { 2 } - 36 x - 5f(x)=2x3−3x2−36x−5

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

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

Use Newton's method to obtain an approximation to the root of $\cos ^ { - 1 } x - 2 x = 0$ to within $0.00001$ .

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

Estimate the value of $\sqrt [ 3 ] { 5 }$ by using three iterations of Newton's method to solve the equation $x ^ { 3 } - 5 = 0$ with initial estimate $x _ { 0 } = 2$ . Round your final estimate to four decimal places.

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

$\text { What is the function of the graph? }$ $\text { What is the function of the graph? }$

A fence $10 \mathrm { ft }$ tall runs parallel to a tall building at a distance of $5 \mathrm { ft }$ from the building. What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building? Round the result to the nearest hundredth.

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 $f$ .

For what values of $c$ does the curve have maximum and minimum points for the given function $f ( x ) = c x ^ { 4 } - 5 x ^ { 2 } + 1 ?$ Select the correct answer.

Find the critical numbers of $f ( x ) = x ^ { 4 } ( x - 3 ) ^ { 3 }$ .

Find the dimensions of the rectangle enclosed in the semicircle $y = \sqrt { 144 - x ^ { 2 } }$ with the largest possible area. $Find the dimensions of the rectangle enclosed in the semicircle y = \sqrt { 144 - x ^ { 2 } } with the largest possible area.$

The graph of the derivative $f ^ { \prime } ( x )$ of a continuous function $f$ is shown. On what intervals is $f$ decreasing? $The graph of the derivative f ^ { \prime } ( x ) of a continuous function f is shown. On what intervals is f decreasing?$

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

Sketch the graph of the function $g ( x ) = \frac { x - 2 } { x - 1 }$ using the curve-sketching guidelines.

Determine where the graph of the function $f ( x ) = x ^ { 3 } - 6 x$ is concave upward and where it is concave downward. Also, find all inflection points of the function.

Find $f$ . $f ^ { \prime } ( t ) = 2 t - 4 \sin t , \quad f ( 0 ) = 5$

$\text { Find the absolute maximum value of } y = \sqrt { 9 - x ^ { 2 } } \text { on the interval } [ - 9,9 ] \text {. }$

Use the Second Derivative Test to find the relative extrema, if any, of the function $f ( x ) = 2 x ^ { 3 } - 3 x ^ { 2 } - 36 x - 5$