Question 1

Let $f ( x ) = ( x - 4 ) ^ { 2 } - 1$ on $[ 3,6 ]$ Then the set of all c in (3,6) guaranteed by the Mean Value Theorem is

Accepted Answer

A)  $\{ 4 \}$ 
B)  $\left\{ \frac { 7 } { 2 } \right\}$ 
C)  $\{ 2 \}$ 
D)  $\left\{ \frac { 9 } { 2 } \right\}$ 
E)  $\{ 5 \}$ 
A)  $\{ 4 \}$ 
B)  $\left\{ \frac { 7 } { 2 } \right\}$ 
C)  $\{ 2 \}$ 
D)  $\left\{ \frac { 9 } { 2 } \right\}$ 
E)  $\{ 5 \}$

Question 2

Let $f ( x ) = x ^ { 3 } - x ^ { 2 } - x + 8$ on $[ - 2,2 ]$ The absolute minimum and maximum of ƒ are located respectively at x =

Accepted Answer

A) 1,4 
B) 1,2 
C) 1,3 
D) 2,3 
E) 3,4 
A) 1,4 
B) 1,2 
C) 1,3 
D) 2,3 
E) 3,4

Question 3

Determine which of the following is not true for the graph of $f ( x ) = \frac { 1 } { 4 } x ^ { 3 } - \frac { 3 } { 2 } x ^ { 2 } + 2$

Accepted Answer

A) ƒ is continuous on $( - \infty , \infty )$ 
B) ƒ has a relative maximum at x = 0. 
C) ƒ has relative minima at x = ±1. 
D) ƒ has points of inflection at x = 2. 
E) ƒ is concave downward on $( - \infty , 2 )$ 
A) ƒ is continuous on $( - \infty , \infty )$ 
B) ƒ has a relative maximum at x = 0. 
C) ƒ has relative minima at x = ±1. 
D) ƒ has points of inflection at x = 2. 
E) ƒ is concave downward on $( - \infty , 2 )$

Question 4

Let $z ^ { 2 } = x ^ { 2 } + y ^ { 2 }$ and assume that $z > 0$ If $\frac { d x } { d t } = - 25$ and $\frac { d y } { d t } = - \frac { 50 } { 3 }$ when $x = 200$ and $y = 150 \text {, }$ then $\frac { d z } { d t }$ is

Accepted Answer

A) -40 
B) -30 
C) 30 
D) 40 
E) 50 
A) -40 
B) -30 
C) 30 
D) 40 
E) 50

Question 5

Determine the value of the limit $\lim _ { x \rightarrow 0 ^ { + } } ( x + 1 ) ^ { \cot x }$ ?

Accepted Answer

A)  $- \infty$ 
B)  $- e$ 
C)  $e$ 
D) 0 
E)  $\infty$ 
A)  $- \infty$ 
B)  $- e$ 
C)  $e$ 
D) 0 
E)  $\infty$

Question 6

Assume that $h ( t ) = - 16 t ^ { 2 } + 80 t + 100$ is the height, in feet, of an object above ground at time tmeasured in seconds. The time that this object reaches its maximum height is

Accepted Answer

A) 3.5s 
B) 3s 
C) 2.5s 
D) 2s 
E) 1.5s 
A) 3.5s 
B) 3s 
C) 2.5s 
D) 2s 
E) 1.5s

Question 7

If $P = 4 x + 3 y$ and xy = 3, the minimum value of P for x and y > 0 is

Accepted Answer

A) 6 
B) 8 
C) 10 
D) 12 
E) 14 
A) 6 
B) 8 
C) 10 
D) 12 
E) 14

Question 8

A rectangular field having an area of 2700 square meters is enclosed by a fence. An additional fence is to be used to divide the field down the middle. The cost of the fence down the middle is $12 per meter, and the fence along the sides costs $18 per meter. The dimensions of the field such that the cost of the fencing will be the least are
​

Accepted Answer

A) 40 m by 67.5 m 
B) 45 m by 60 m 
C) 30 m by 90 m 
D) 54 m by 50 m 
E) 27 m by 100 m 
A) 40 m by 67.5 m 
B) 45 m by 60 m 
C) 30 m by 90 m 
D) 54 m by 50 m 
E) 27 m by 100 m

Question 9

The largest set on which the function $f ( x ) = x e ^ { x }$ is increasing is

Accepted Answer

A)  $( - \infty , \infty )$ 
B)  $( - \infty , \infty )$. 
C)  $( 0 , \infty )$ 
D)  $( - \infty , 1 )$ 
E)  $( - 1 , \infty )$ 
A)  $( - \infty , \infty )$ 
B)  $( - \infty , \infty )$. 
C)  $( 0 , \infty )$ 
D)  $( - \infty , 1 )$ 
E)  $( - 1 , \infty )$

Question 10

Let $\frac { y } { x } = 12$ If $\frac { d x } { d t } = - \frac { 1 } { 2 },$ then $\frac { d y } { d t }$ is

Accepted Answer

A) -6 
B) -4 
C) -2 
D) 2 
E) 6 
A) -6 
B) -4 
C) -2 
D) 2 
E) 6

Question 11

Let $f ( x ) = x ^ { 4 } - 4 x ^ { 3 } + 16 x$ Then ƒ has a point of inflection at x =

Accepted Answer

A)  $- \frac { 1 } { 3 } , 0$ 
B)  $1,0$ 
C)  $\frac { 1 } { 3 } , 0$ 
D)  $- 1,0$ 
E)  $2,0$ 
A)  $- \frac { 1 } { 3 } , 0$ 
B)  $1,0$ 
C)  $\frac { 1 } { 3 } , 0$ 
D)  $- 1,0$ 
E)  $2,0$

Question 12

Water is flowing into a vertical cylindrical tank at the rate of 5 m3/min. If the radius of the tank is 3 m, then the rate at which the height of the water is rising is

Accepted Answer

A)  $\frac { 9 } { 5 \pi }$ 
B)  $\frac { 3 } { 5 \pi }$ 
C)  $\frac { 5 } { 6 \pi }$ 
D)  $\frac { 5 } { 9 \pi }$ 
E)  $\frac { 5 } { 3 \pi }$ 
A)  $\frac { 9 } { 5 \pi }$ 
B)  $\frac { 3 } { 5 \pi }$ 
C)  $\frac { 5 } { 6 \pi }$ 
D)  $\frac { 5 } { 9 \pi }$ 
E)  $\frac { 5 } { 3 \pi }$

Question 13

Let $2 x + 3 y = 8$ If $\frac { d y } { d t } = 4,$ then $\frac { d x } { d t }$ is

Accepted Answer

A) -6 
B) -4 
C) -2 
D) 2 
E) 6 
A) -6 
B) -4 
C) -2 
D) 2 
E) 6

Question 14

Let $y = f ( x )$ be a differentiable function for which the graph of its derivative, ƒ ', is given below:
At what x-value(s), if any, does the graph of f have a local minimum?

Accepted Answer

A)  x = 3 only 
B) x = -3 only 
C) x =0 only 
D) x = -3 and x = 3 
E) None 
A)  x = 3 only 
B) x = -3 only 
C) x =0 only 
D) x = -3 and x = 3 
E) None

Question 15

Let $A = \frac { 1 } { 2 } b h .$ If $\frac { d h } { d t } = - 4 , \frac { d b } { d t } = 6,$ when and $h = 6,$ then $\frac { d A } { d t }$ is

Accepted Answer

A) 12 
B) 10 
C) 8 
D) 6 
E) 4 
A) 12 
B) 10 
C) 8 
D) 6 
E) 4

Question 16

Determine which of the following is not true for the graph of $f ( x ) = x ^ { \frac { 2 } { 3 } }$

Accepted Answer

A) ƒ is continuous on $( - \infty , \infty )$ . 
B) ƒ has no relative maximum. 
C) ƒ has a relative minimum at x = 0. 
D) ƒ has a point of inflection at x = 0. 
E) ƒ is concave downward on $( 0 , \infty )$ . 
A) ƒ is continuous on $( - \infty , \infty )$ . 
B) ƒ has no relative maximum. 
C) ƒ has a relative minimum at x = 0. 
D) ƒ has a point of inflection at x = 0. 
E) ƒ is concave downward on $( 0 , \infty )$ .

Question 17

Determine which of the following is not true for the graph of $f ( x ) = x ^ { 4 } - 2 x ^ { 3 } + x$

Accepted Answer

A) ƒ is continuous on $( - \infty , \infty )$ . 
B) ƒ has a relative maximum at $x = \frac { 1 } { 2 }$ 
C) ƒ has relative minima at x = ±1. 
D) ƒ has points of inflection at x = 0,1. 
E) ƒ is concave downward on (0,1). 
A) ƒ is continuous on $( - \infty , \infty )$ . 
B) ƒ has a relative maximum at $x = \frac { 1 } { 2 }$ 
C) ƒ has relative minima at x = ±1. 
D) ƒ has points of inflection at x = 0,1. 
E) ƒ is concave downward on (0,1).

Question 18

Let $f ( x ) = \sqrt { x - 1 }$ on $[ 1,3 ]$ Then the set of all c in (1,3) guaranteed by the Mean Value Theorem is

Accepted Answer

A)  $\left\{ \frac { 5 } { 4 } \right\}$ 
B)  $\left\{ \frac { 9 } { 8 } \right\}$ 
C)  $\left\{ \frac { 3 } { 2 } \right\}$ 
D)  $\{ 2 \}$ 
E)  $\left\{ \frac { 5 } { 2 } \right\}$ 
A)  $\left\{ \frac { 5 } { 4 } \right\}$ 
B)  $\left\{ \frac { 9 } { 8 } \right\}$ 
C)  $\left\{ \frac { 3 } { 2 } \right\}$ 
D)  $\{ 2 \}$ 
E)  $\left\{ \frac { 5 } { 2 } \right\}$

Question 19

Determine which of the following is not true for the graph of $f ( x ) = - x ^ { 4 } + 6 x ^ { 2 } - 4$

Accepted Answer

A) ƒ is continuous on $( - \infty , \infty )$ 
B) ƒ has a relative minimum at x = 0. 
C) ƒ has relative maxima at $x = \pm \sqrt { 3 }$ 
D) ƒ has points of inflection at x = ±1. 
E) ƒ is concave downward on $( - \infty , 1 )$ 
A) ƒ is continuous on $( - \infty , \infty )$ 
B) ƒ has a relative minimum at x = 0. 
C) ƒ has relative maxima at $x = \pm \sqrt { 3 }$ 
D) ƒ has points of inflection at x = ±1. 
E) ƒ is concave downward on $( - \infty , 1 )$

Question 20

Let $y = f ( x )$ be a differentiable function for which the graph of its derivative, ƒ ', is given below:
At what x-value(s), if any, does the graph of f have a horizontal tangent?

Accepted Answer

A) x = -1 and x = 0 
B) x = -2 
C) x = -2, x = 1, and x = 0 
D) x = -2 and x = 1 
E) None 
A) x = -1 and x = 0 
B) x = -2 
C) x = -2, x = 1, and x = 0 
D) x = -2 and x = 1 
E) None

Let $f ( x ) = ( x - 4 ) ^ { 2 } - 1$ on $[ 3,6 ]$ Then the set of all c in (3,6) guaranteed by the Mean Value Theorem is

Let $f ( x ) = x ^ { 3 } - x ^ { 2 } - x + 8$ on $[ - 2,2 ]$ The absolute minimum and maximum of ƒ are located respectively at x =

Determine which of the following is not true for the graph of $f ( x ) = \frac { 1 } { 4 } x ^ { 3 } - \frac { 3 } { 2 } x ^ { 2 } + 2$

Let $z ^ { 2 } = x ^ { 2 } + y ^ { 2 }$ and assume that $z > 0$ If $\frac { d x } { d t } = - 25$ and $\frac { d y } { d t } = - \frac { 50 } { 3 }$ when $x = 200$ and $y = 150 \text {, }$ then $\frac { d z } { d t }$ is

Determine the value of the limit $\lim _ { x \rightarrow 0 ^ { + } } ( x + 1 ) ^ { \cot x }$ ?

Assume that $h ( t ) = - 16 t ^ { 2 } + 80 t + 100$ is the height, in feet, of an object above ground at time tmeasured in seconds. The time that this object reaches its maximum height is

If $P = 4 x + 3 y$ and xy = 3, the minimum value of P for x and y > 0 is

The largest set on which the function $f ( x ) = x e ^ { x }$ is increasing is

Let $\frac { y } { x } = 12$ If $\frac { d x } { d t } = - \frac { 1 } { 2 },$ then $\frac { d y } { d t }$ is

Let $f ( x ) = x ^ { 4 } - 4 x ^ { 3 } + 16 x$ Then ƒ has a point of inflection at x =

Water is flowing into a vertical cylindrical tank at the rate of 5 m³/min. If the radius of the tank is 3 m, then the rate at which the height of the water is rising is

Let $2 x + 3 y = 8$ If $\frac { d y } { d t } = 4,$ then $\frac { d x } { d t }$ is

Let $y = f ( x )$ be a differentiable function for which the graph of its derivative, ƒ ', is given below: At what x-value(s), if any, does the graph of f have a local minimum?

Let $A = \frac { 1 } { 2 } b h .$ If $\frac { d h } { d t } = - 4 , \frac { d b } { d t } = 6,$ when and $h = 6,$ then $\frac { d A } { d t }$ is

Determine which of the following is not true for the graph of $f ( x ) = x ^ { \frac { 2 } { 3 } }$

Determine which of the following is not true for the graph of $f ( x ) = x ^ { 4 } - 2 x ^ { 3 } + x$

Let $f ( x ) = \sqrt { x - 1 }$ on $[ 1,3 ]$ Then the set of all c in (1,3) guaranteed by the Mean Value Theorem is

Determine which of the following is not true for the graph of $f ( x ) = - x ^ { 4 } + 6 x ^ { 2 } - 4$

Let $y = f ( x )$ be a differentiable function for which the graph of its derivative, ƒ ', is given below: At what x-value(s), if any, does the graph of f have a horizontal tangent?

Preparing for Calculus

Limits and Continuity

The Derivative

More About Derivatives

The Integral

Applications of the Integral

Techniques of Integration

Infinite Series

Parametric Equations; Polar Equations

Vectors; Lines, Planes, and Quadric Surfaces in Space

Vector Functions

Functions of Several Variables

Directional Derivatives, Gradients, and Extrema

Multiple Integrals

Vector Calculus

Differential Equations

Filters

Exam 5: Applications of the Derivative

Let f(x)=(x−4)2−1f ( x ) = ( x - 4 ) ^ { 2 } - 1f(x)=(x−4)2−1 on [3,6][ 3,6 ][3,6] Then the set of all c in (3,6) guaranteed by the Mean Value Theorem is

Let f(x)=x3−x2−x+8f ( x ) = x ^ { 3 } - x ^ { 2 } - x + 8f(x)=x3−x2−x+8 on [−2,2][ - 2,2 ][−2,2] The absolute minimum and maximum of ƒ are located respectively at x =

Determine which of the following is not true for the graph of f(x)=14x3−32x2+2f ( x ) = \frac { 1 } { 4 } x ^ { 3 } - \frac { 3 } { 2 } x ^ { 2 } + 2f(x)=41​x3−23​x2+2

Determine the value of the limit lim⁡x→0+(x+1)cot⁡x\lim _ { x \rightarrow 0 ^ { + } } ( x + 1 ) ^ { \cot x }limx→0+​(x+1)cotx ?

Assume that h(t)=−16t2+80t+100h ( t ) = - 16 t ^ { 2 } + 80 t + 100h(t)=−16t2+80t+100 is the height, in feet, of an object above ground at time tmeasured in seconds. The time that this object reaches its maximum height is

If P=4x+3yP = 4 x + 3 yP=4x+3y and xy = 3, the minimum value of P for x and y > 0 is

The largest set on which the function f(x)=xexf ( x ) = x e ^ { x }f(x)=xex is increasing is

Let yx=12\frac { y } { x } = 12xy​=12 If dxdt=−12,\frac { d x } { d t } = - \frac { 1 } { 2 },dtdx​=−21​, then dydt\frac { d y } { d t }dtdy​ is

Let f(x)=x4−4x3+16xf ( x ) = x ^ { 4 } - 4 x ^ { 3 } + 16 xf(x)=x4−4x3+16x Then ƒ has a point of inflection at x =

Water is flowing into a vertical cylindrical tank at the rate of 5 m3/min. If the radius of the tank is 3 m, then the rate at which the height of the water is rising is

Let 2x+3y=82 x + 3 y = 82x+3y=8 If dydt=4,\frac { d y } { d t } = 4,dtdy​=4, then dxdt\frac { d x } { d t }dtdx​ is

Let y=f(x)y = f ( x )y=f(x) be a differentiable function for which the graph of its derivative, ƒ ', is given below: At what x-value(s), if any, does the graph of f have a local minimum?

Let A=12bh.A = \frac { 1 } { 2 } b h .A=21​bh. If dhdt=−4,dbdt=6,\frac { d h } { d t } = - 4 , \frac { d b } { d t } = 6,dtdh​=−4,dtdb​=6, when and h=6,h = 6,h=6, then dAdt\frac { d A } { d t }dtdA​ is

Determine which of the following is not true for the graph of f(x)=x23f ( x ) = x ^ { \frac { 2 } { 3 } }f(x)=x32​

Determine which of the following is not true for the graph of f(x)=x4−2x3+xf ( x ) = x ^ { 4 } - 2 x ^ { 3 } + xf(x)=x4−2x3+x

Let f(x)=x−1f ( x ) = \sqrt { x - 1 }f(x)=x−1​ on [1,3][ 1,3 ][1,3] Then the set of all c in (1,3) guaranteed by the Mean Value Theorem is

Determine which of the following is not true for the graph of f(x)=−x4+6x2−4f ( x ) = - x ^ { 4 } + 6 x ^ { 2 } - 4f(x)=−x4+6x2−4

Let y=f(x)y = f ( x )y=f(x) be a differentiable function for which the graph of its derivative, ƒ ', is given below: At what x-value(s), if any, does the graph of f have a horizontal tangent?

Preparing for Calculus

Limits and Continuity

The Derivative

More About Derivatives

The Integral

Applications of the Integral

Techniques of Integration

Infinite Series

Parametric Equations; Polar Equations

Vectors; Lines, Planes, and Quadric Surfaces in Space

Vector Functions

Functions of Several Variables

Directional Derivatives, Gradients, and Extrema

Multiple Integrals

Vector Calculus

Differential Equations

Filters

Let $f ( x ) = ( x - 4 ) ^ { 2 } - 1$ on $[ 3,6 ]$ Then the set of all c in (3,6) guaranteed by the Mean Value Theorem is

Let $f ( x ) = x ^ { 3 } - x ^ { 2 } - x + 8$ on $[ - 2,2 ]$ The absolute minimum and maximum of ƒ are located respectively at x =

Determine which of the following is not true for the graph of $f ( x ) = \frac { 1 } { 4 } x ^ { 3 } - \frac { 3 } { 2 } x ^ { 2 } + 2$

Determine the value of the limit $\lim _ { x \rightarrow 0 ^ { + } } ( x + 1 ) ^ { \cot x }$ ?

Assume that $h ( t ) = - 16 t ^ { 2 } + 80 t + 100$ is the height, in feet, of an object above ground at time tmeasured in seconds. The time that this object reaches its maximum height is

If $P = 4 x + 3 y$ and xy = 3, the minimum value of P for x and y > 0 is

The largest set on which the function $f ( x ) = x e ^ { x }$ is increasing is

Let $\frac { y } { x } = 12$ If $\frac { d x } { d t } = - \frac { 1 } { 2 },$ then $\frac { d y } { d t }$ is

Let $f ( x ) = x ^ { 4 } - 4 x ^ { 3 } + 16 x$ Then ƒ has a point of inflection at x =

Water is flowing into a vertical cylindrical tank at the rate of 5 m³/min. If the radius of the tank is 3 m, then the rate at which the height of the water is rising is

Let $2 x + 3 y = 8$ If $\frac { d y } { d t } = 4,$ then $\frac { d x } { d t }$ is

Let $y = f ( x )$ be a differentiable function for which the graph of its derivative, ƒ ', is given below: At what x-value(s), if any, does the graph of f have a local minimum?

Let $A = \frac { 1 } { 2 } b h .$ If $\frac { d h } { d t } = - 4 , \frac { d b } { d t } = 6,$ when and $h = 6,$ then $\frac { d A } { d t }$ is

Determine which of the following is not true for the graph of $f ( x ) = x ^ { \frac { 2 } { 3 } }$

Determine which of the following is not true for the graph of $f ( x ) = x ^ { 4 } - 2 x ^ { 3 } + x$

Let $f ( x ) = \sqrt { x - 1 }$ on $[ 1,3 ]$ Then the set of all c in (1,3) guaranteed by the Mean Value Theorem is

Determine which of the following is not true for the graph of $f ( x ) = - x ^ { 4 } + 6 x ^ { 2 } - 4$

Let $y = f ( x )$ be a differentiable function for which the graph of its derivative, ƒ ', is given below: At what x-value(s), if any, does the graph of f have a horizontal tangent?