Question 1

Find the value of the limit $\lim _ { k \rightarrow \infty } \left( 1 + \frac { 1 } { 10 ^ { k } } \right) ^ { 10 ^ { 1 } }$ .

Accepted Answer

A) 0 
B) 1 
C) 2 
D)  $e$ 
E) 3
F) $\pi$ 
G)4
H) $\infty$ 
A) 0 
B) 1 
C) 2 
D)  $e$ 
E) 3
F) $\pi$ 
G)4
H) $\infty$

Question 2

At what value of x does the function $f ( x ) = \frac { 1 } { 1 + x ^ { 2 } }$ change from increasing to decreasing?

Accepted Answer

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

Question 3

Find the most general antiderivative of the function $f ( x ) = \frac { 2 + x ^ { 3 } } { x ^ { 3 } }$ .

Accepted Answer

A)  $\frac { 2 + 3 x ^ { 2 } } { 3 x ^ { 2 } } + C$ 
B)  $\frac { 2 x + \frac { 1 } { 4 } x ^ { 4 } } { x ^ { 4 } } + C$ 
C)  $- \frac { 1 } { x ^ { 2 } } + 1 + C$ 
D)  $- \frac { 1 } { x ^ { 2 } } + C$ 
E)  $\frac { 1 } { x ^ { 2 } } + x + C$
F)
$\frac { 1 } { x ^ { 2 } } + 1 + C$

G) $- \frac { 1 } { x ^ { 2 } } + x + C$

H) $\frac { 2 } { x ^ { 2 } } + C$ 
A)  $\frac { 2 + 3 x ^ { 2 } } { 3 x ^ { 2 } } + C$ 
B)  $\frac { 2 x + \frac { 1 } { 4 } x ^ { 4 } } { x ^ { 4 } } + C$ 
C)  $- \frac { 1 } { x ^ { 2 } } + 1 + C$ 
D)  $- \frac { 1 } { x ^ { 2 } } + C$ 
E)  $\frac { 1 } { x ^ { 2 } } + x + C$
F)
$\frac { 1 } { x ^ { 2 } } + 1 + C$

G) $- \frac { 1 } { x ^ { 2 } } + x + C$

H) $\frac { 2 } { x ^ { 2 } } + C$

Question 4

Find the largest interval on which the function $f ( x ) = x e ^ { - x }$ is concave upward.

Accepted Answer

The answer of Find the largest interval on which the...

Question 5

On the surface of the moon, a ball is thrown directly upward at a speed of 64 feet per second from a cliff 80 feet above the ground. The acceleration due to lunar gravity is 5:2 ft/s $2$ .(a) Find expressions for the velocity and height of the ball t seconds after it was released.(b) At what time does the ball reach its highest point? How high above the ground at the base of the cliff does it reach?
(c) When does the ball strike the ground at the base of the cliff? What is its velocity at that instant?

Accepted Answer

(a) @#IMG-DLM& ft/s; @#IMG-DLM& ft
(b) 1

Question 6

There are 50 apple trees in an orchard. Each tree produces 100 lbs of apples. For each additional tree planted in the orchard, the output per tree drops by 1 lb. How many trees should be added to the existing orchard in order to maximize the total output of trees?

Accepted Answer

A) 15 
B) 10 
C) 25 
D) 0 
E) 5
F)20
G)40
H)None of these 
A) 15 
B) 10 
C) 25 
D) 0 
E) 5
F)20
G)40
H)None of these

Question 7

A company has a cost function $C ( x ) = 200 - 50 x + x ^ { 2 }$ and demand function $p ( x ) = 50 - x$ . How many units should it make to maximize its profit?

Accepted Answer

A) 15 
B) 25 
C) 8 
D) 5 
E) 30
F)10
G)20
H)35 
A) 15 
B) 25 
C) 8 
D) 5 
E) 30
F)10
G)20
H)35

Question 8

Find the largest interval on which the function $f ( x ) = \frac { x } { x ^ { 2 } + 1 }$ is concave upward.

Accepted Answer

The answer of Find the largest interval on which the...

Question 9

Find the point on the line $y = 2 x - 3$ that is nearest to the origin.

Accepted Answer

A)  $( 0.5 , - 2 )$ 
B)  $( 7.5 , - 1.5 )$ 
C)  $( 0.875 , - 1.25 )$ 
D)  $( 1 , - 0.5 )$ 
E)  $( 1.1 , - 0.8 )$
F)$( 1.2 , - 0.6 )$
G) $( 1.25 , - 0.5 )$

H) $( 1.5,0 )$ 
A)  $( 0.5 , - 2 )$ 
B)  $( 7.5 , - 1.5 )$ 
C)  $( 0.875 , - 1.25 )$ 
D)  $( 1 , - 0.5 )$ 
E)  $( 1.1 , - 0.8 )$
F)$( 1.2 , - 0.6 )$
G) $( 1.25 , - 0.5 )$

H) $( 1.5,0 )$

Question 10

Explain why L'Hospital's Rule is not applicable to evaluate $\lim _ { x \rightarrow 1 ^ { + } } \frac { x + 1 } { x - 1 }$ , then find the limit.

Accepted Answer

Because @#IMG-DLM& , it is not

Question 11

A frugal young man has decided to extract one of his teeth by tying a stout rubber band from his tooth to the chain on a garage door opener which runs on a horizontal track 3 feet above his mouth. If the garage door opener moves the chain at $\frac { 1 } { 4 } \mathrm { ft } / \mathrm { s }$ , how fast is the rubber band expanding when it is stretched to a length of 5 feet?

Accepted Answer

The answer of A frugal young man has decided to...

Question 12

Find the most general antiderivative of the function $f ( x ) = \frac { 2 e ^ { x } e ^ { 2 x } } { e ^ { x } }$ .

Accepted Answer

A)  $2 x + e ^ { x } + C$ 
B)  $2 x - e ^ { x } + C$ 
C)  $2 - e ^ { x } + C$ 
D)  $\frac { 2 e ^ { x } - e ^ { 2 x } } { e ^ { x } } + C$ 
E)  $2 x - 2 e ^ { x } + C$
F) $\frac { 2 e ^ { x } - \frac { 1 } { 2 } e ^ { 2 x } } { e ^ { x } } + C$

G) $2 + x e ^ { x - 1 } + C$

H) $2 x + \frac { 1 } { 1 + x } e ^ { x + 1 } + C$ 
A)  $2 x + e ^ { x } + C$ 
B)  $2 x - e ^ { x } + C$ 
C)  $2 - e ^ { x } + C$ 
D)  $\frac { 2 e ^ { x } - e ^ { 2 x } } { e ^ { x } } + C$ 
E)  $2 x - 2 e ^ { x } + C$
F) $\frac { 2 e ^ { x } - \frac { 1 } { 2 } e ^ { 2 x } } { e ^ { x } } + C$

G) $2 + x e ^ { x - 1 } + C$

H) $2 x + \frac { 1 } { 1 + x } e ^ { x + 1 } + C$

Question 13

If Newton's method is used to find the cube root of a number a with first approximation $x _ { 1 }$ , find an expression for $x _ { 2 }$ .

Accepted Answer

The answer of If Newton's method is used to find...

Find the value of the limit $\lim _ { k \rightarrow \infty } \left( 1 + \frac { 1 } { 10 ^ { k } } \right) ^ { 10 ^ { 1 } }$ .

At what value of x does the function $f ( x ) = \frac { 1 } { 1 + x ^ { 2 } }$ change from increasing to decreasing?

Find the most general antiderivative of the function $f ( x ) = \frac { 2 + x ^ { 3 } } { x ^ { 3 } }$ .

Find the largest interval on which the function $f ( x ) = x e ^ { - x }$ is concave upward.

There are 50 apple trees in an orchard. Each tree produces 100 lbs of apples. For each additional tree planted in the orchard, the output per tree drops by 1 lb. How many trees should be added to the existing orchard in order to maximize the total output of trees?

A company has a cost function $C ( x ) = 200 - 50 x + x ^ { 2 }$ and demand function $p ( x ) = 50 - x$ . How many units should it make to maximize its profit?

Find the largest interval on which the function $f ( x ) = \frac { x } { x ^ { 2 } + 1 }$ is concave upward.

Find the point on the line $y = 2 x - 3$ that is nearest to the origin.

Explain why L'Hospital's Rule is not applicable to evaluate $\lim _ { x \rightarrow 1 ^ { + } } \frac { x + 1 } { x - 1 }$ , then find the limit.

Find the most general antiderivative of the function $f ( x ) = \frac { 2 e ^ { x } e ^ { 2 x } } { e ^ { x } }$ .

If Newton's method is used to find the cube root of a number a with first approximation $x _ { 1 }$ , find an expression for $x _ { 2 }$ .

Functions and Models

Limits and Derivatives

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Exam 4: Applications of Differentiation

Find the value of the limit lim⁡k→∞(1+110k)101\lim _ { k \rightarrow \infty } \left( 1 + \frac { 1 } { 10 ^ { k } } \right) ^ { 10 ^ { 1 } }limk→∞​(1+10k1​)101 .

At what value of x does the function f(x)=11+x2f ( x ) = \frac { 1 } { 1 + x ^ { 2 } }f(x)=1+x21​ change from increasing to decreasing?

Find the most general antiderivative of the function f(x)=2+x3x3f ( x ) = \frac { 2 + x ^ { 3 } } { x ^ { 3 } }f(x)=x32+x3​ .

Find the largest interval on which the function f(x)=xe−xf ( x ) = x e ^ { - x }f(x)=xe−x is concave upward.

There are 50 apple trees in an orchard. Each tree produces 100 lbs of apples. For each additional tree planted in the orchard, the output per tree drops by 1 lb. How many trees should be added to the existing orchard in order to maximize the total output of trees?

A company has a cost function C(x)=200−50x+x2C ( x ) = 200 - 50 x + x ^ { 2 }C(x)=200−50x+x2 and demand function p(x)=50−xp ( x ) = 50 - xp(x)=50−x . How many units should it make to maximize its profit?

Find the largest interval on which the function f(x)=xx2+1f ( x ) = \frac { x } { x ^ { 2 } + 1 }f(x)=x2+1x​ is concave upward.

Find the point on the line y=2x−3y = 2 x - 3y=2x−3 that is nearest to the origin.

Explain why L'Hospital's Rule is not applicable to evaluate lim⁡x→1+x+1x−1\lim _ { x \rightarrow 1 ^ { + } } \frac { x + 1 } { x - 1 }limx→1+​x−1x+1​ , then find the limit.

Find the most general antiderivative of the function f(x)=2exe2xexf ( x ) = \frac { 2 e ^ { x } e ^ { 2 x } } { e ^ { x } }f(x)=ex2exe2x​ .

If Newton's method is used to find the cube root of a number a with first approximation x1x _ { 1 }x1​ , find an expression for x2x _ { 2 }x2​ .

Functions and Models

Limits and Derivatives

Differentiation Rules

Integrals

Applications of Integration

Differential Equations

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Filters

Find the value of the limit $\lim _ { k \rightarrow \infty } \left( 1 + \frac { 1 } { 10 ^ { k } } \right) ^ { 10 ^ { 1 } }$ .

At what value of x does the function $f ( x ) = \frac { 1 } { 1 + x ^ { 2 } }$ change from increasing to decreasing?

Find the most general antiderivative of the function $f ( x ) = \frac { 2 + x ^ { 3 } } { x ^ { 3 } }$ .

Find the largest interval on which the function $f ( x ) = x e ^ { - x }$ is concave upward.

A company has a cost function $C ( x ) = 200 - 50 x + x ^ { 2 }$ and demand function $p ( x ) = 50 - x$ . How many units should it make to maximize its profit?

Find the largest interval on which the function $f ( x ) = \frac { x } { x ^ { 2 } + 1 }$ is concave upward.

Find the point on the line $y = 2 x - 3$ that is nearest to the origin.

Explain why L'Hospital's Rule is not applicable to evaluate $\lim _ { x \rightarrow 1 ^ { + } } \frac { x + 1 } { x - 1 }$ , then find the limit.

Find the most general antiderivative of the function $f ( x ) = \frac { 2 e ^ { x } e ^ { 2 x } } { e ^ { x } }$ .

If Newton's method is used to find the cube root of a number a with first approximation $x _ { 1 }$ , find an expression for $x _ { 2 }$ .