Question 1

How many real roots does the equation $x ^ { 5 } - 7 x + c = 0$ have in the interval $[ - 1,1 ]$ ?

Accepted Answer

A)  at most five real roots 
B)  no real roots 
C)  at most one real root 
D)  at most two real roots 
E)  at most three real roots 
A)  at most five real roots 
B)  no real roots 
C)  at most one real root 
D)  at most two real roots 
E)  at most three real roots C

Question 2

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

$$f \left( \frac { \pi } { 6 } \right) = \frac { 3 } { 2 } \sqrt { 3 }$$

Question 3

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)  $\mathrm { CU }$ 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)  $\mathrm { CU }$ on $( - \sqrt { 2 } , \infty ) , \mathrm { CD }$ on $( - \infty , - \sqrt { 2 } )$,
IP $( - \sqrt { 2 } , 4 \sqrt { 2 } )$ C

Question 4

Find the critical number(s), if any, of the function $f ( x ) = e ^ { x ^ { 2 } - 10 x }$.

Accepted Answer

A)  $\sqrt { 10 }$ 
B)  5 
C)  $\sqrt { 10 }$ and $- \sqrt { 10 }$ 
D)  5 and $- 5$ 
A)  $\sqrt { 10 }$ 
B)  5 
C)  $\sqrt { 10 }$ and $- \sqrt { 10 }$ 
D)  5 and $- 5$

Question 5

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 6

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 7

An aircraft manufacturer wants to determine the best selling price for a new airplane. The company estimates that the initial cost of designing the airplane and setting up the factories in which to build it will be 700 million dollars. The additional cost of manufacturing each plane can be modeled by the function
$m ( x ) = 1,600 x + 40 x ^ { 4 / 5 } + 0.15 x ^ { 2 }$
where $x$ is the number of aircraft produced and $m$ is the manufacturing cost, in millions of dollars. The company estimates that if it charges a price $p$ (in millions of dollars) for each plane, it will be able to sell $x ( p ) = 390 - 5.8 p$
Find the cost function.

Accepted Answer

The answer of An aircraft manufacturer wants to determine the...

Question 8

At 4:00 P.M. a car's speedometer reads $29 \mathrm { mi } ^ { 2 }$. At 4:15 it reads $72 \mathrm { mi } / \mathrm { h } ^ { 2 }$. At some time between $4 : 00$ and $4 : 15$ the acceleration is exactly $x \mathrm { mi } ^ { 2 }$. Find $x$.

Accepted Answer

The answer of At 4:00 P.M. a car's speedometer reads...

Question 9

The function $f ( x ) = x ^ { 2 } - 6 x + 13$ satisfies the hypotheses of Rolle's Theorem on the interval $[ 0,6 ]$. Find all values of $c$ that satisfy the conclusion of the theorem.

Accepted Answer

A)  2,4 
B)  2 
C)  3,4 
D)  3 
A)  2,4 
B)  2 
C)  3,4 
D)  3

Question 10

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 11

Find the smallest possible area of an isosceles triangle that is circumscribed about a circle of radius $r = 4$.

Accepted Answer

A)  $4 \pi \sqrt { 3 }$ 
B)  $4 \sqrt { 3 }$ 
C)  $48 \sqrt { 3 }$ 
D)  $\sqrt { 3 }$ 
E)  $\sqrt { 48 } r ^ { 2 }$ 
A)  $4 \pi \sqrt { 3 }$ 
B)  $4 \sqrt { 3 }$ 
C)  $48 \sqrt { 3 }$ 
D)  $\sqrt { 3 }$ 
E)  $\sqrt { 48 } r ^ { 2 }$

Question 12

$\text { Find the absolute minimum value(s) of } y = 2 x ^ { 2 } - 20 x + 9 \text { on the interval } [ 0,6 ]$

Accepted Answer

The answer of \(\text { Find the absolute minimum value(s)...

Question 13

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

Accepted Answer

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

Question 14

Find the most general antiderivative of the function. $f ( x ) = 9 x ^ { 1 / 8 } - 11 x ^ { 1 / 10 }$

Accepted Answer

The answer of Find the most general antiderivative of the...

Question 15

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 16

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 17

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)  $\sqrt { 3 }$ in, 80 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)  $\sqrt { 3 }$ in, 80 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 18

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 19

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 20

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) 
$$c _ { 1 } = 2 + \frac { 2 \sqrt { 3 } } { 3 } , c _ { 2 } = \frac { 2 \sqrt { 3 } } { 3 }$$

How many real roots does the equation $x ^ { 5 } - 7 x + c = 0$ have in the interval $[ - 1,1 ]$ ?

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

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 the critical number(s), if any, of the function $f ( x ) = e ^ { x ^ { 2 } - 10 x }$ .

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.

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

At 4:00 P.M. a car's speedometer reads $29 \mathrm { mi } ^ { 2 }$ . At 4:15 it reads $72 \mathrm { mi } / \mathrm { h } ^ { 2 }$ . At some time between $4 : 00$ and $4 : 15$ the acceleration is exactly $x \mathrm { mi } ^ { 2 }$ . Find $x$ .

The function $f ( x ) = x ^ { 2 } - 6 x + 13$ satisfies the hypotheses of Rolle's Theorem on the interval $[ 0,6 ]$ . Find all values of $c$ that satisfy the conclusion of the theorem.

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

Find the smallest possible area of an isosceles triangle that is circumscribed about a circle of radius $r = 4$ .

$\text { Find the absolute minimum value(s) of } y = 2 x ^ { 2 } - 20 x + 9 \text { on the interval } [ 0,6 ]$

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

Find the most general antiderivative of the function. $f ( x ) = 9 x ^ { 1 / 8 } - 11 x ^ { 1 / 10 }$

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

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

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.

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

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.

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

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

How many real roots does the equation x5−7x+c=0x ^ { 5 } - 7 x + c = 0x5−7x+c=0 have in the interval [−1,1][ - 1,1 ][−1,1] ?

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​]

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 the critical number(s), if any, of the function f(x)=ex2−10xf ( x ) = e ^ { x ^ { 2 } - 10 x }f(x)=ex2−10x .

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.

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.

At 4:00 P.M. a car's speedometer reads 29mi229 \mathrm { mi } ^ { 2 }29mi2 . At 4:15 it reads 72mi/h272 \mathrm { mi } / \mathrm { h } ^ { 2 }72mi/h2 . At some time between 4:004 : 004:00 and 4:154 : 154:15 the acceleration is exactly xmi2x \mathrm { mi } ^ { 2 }xmi2 . Find xxx .

The function f(x)=x2−6x+13f ( x ) = x ^ { 2 } - 6 x + 13f(x)=x2−6x+13 satisfies the hypotheses of Rolle's Theorem on the interval [0,6][ 0,6 ][0,6] . Find all values of ccc that satisfy the conclusion of the theorem.

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 .

Find the smallest possible area of an isosceles triangle that is circumscribed about a circle of radius r=4r = 4r=4 .

Find the absolute minimum value(s) of y=2x2−20x+9 on the interval [0,6]\text { Find the absolute minimum value(s) of } y = 2 x ^ { 2 } - 20 x + 9 \text { on the interval } [ 0,6 ] Find the absolute minimum value(s) of y=2x2−20x+9 on the interval [0,6]

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.

Find the most general antiderivative of the function. f(x)=9x1/8−11x1/10f ( x ) = 9 x ^ { 1 / 8 } - 11 x ^ { 1 / 10 }f(x)=9x1/8−11x1/10

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] .

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.

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.

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?

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.

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]

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

How many real roots does the equation $x ^ { 5 } - 7 x + c = 0$ have in the interval $[ - 1,1 ]$ ?

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

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 the critical number(s), if any, of the function $f ( x ) = e ^ { x ^ { 2 } - 10 x }$ .

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.

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

At 4:00 P.M. a car's speedometer reads $29 \mathrm { mi } ^ { 2 }$ . At 4:15 it reads $72 \mathrm { mi } / \mathrm { h } ^ { 2 }$ . At some time between $4 : 00$ and $4 : 15$ the acceleration is exactly $x \mathrm { mi } ^ { 2 }$ . Find $x$ .

The function $f ( x ) = x ^ { 2 } - 6 x + 13$ satisfies the hypotheses of Rolle's Theorem on the interval $[ 0,6 ]$ . Find all values of $c$ that satisfy the conclusion of the theorem.

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

Find the smallest possible area of an isosceles triangle that is circumscribed about a circle of radius $r = 4$ .

$\text { Find the absolute minimum value(s) of } y = 2 x ^ { 2 } - 20 x + 9 \text { on the interval } [ 0,6 ]$

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

Find the most general antiderivative of the function. $f ( x ) = 9 x ^ { 1 / 8 } - 11 x ^ { 1 / 10 }$

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

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

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.

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

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.

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