Question 1

If the total cost function for a product is $C ( x ) = 200 + 4 x + 0.09 x ^ { 2 }$ dollars. Find the minimum average cost.

Accepted Answer

A)  $\$ 15.00$ 
B)  $\$ 23.00$ 
C)  $\$ 13.03$ 
D)  $\$ 12.49$ 
E)  $\$ 11.00$ 
A)  $\$ 15.00$ 
B)  $\$ 23.00$ 
C)  $\$ 13.03$ 
D)  $\$ 12.49$ 
E)  $\$ 11.00$

Question 2

An employee of a delivery company earns $\$$ 25.00 per hour driving a delivery van in an area where gasoline costs $\$$ 2.90 per gallon. When the van is driven at a constant speed s (in miles per hour, with $45 \leq s \leq 60$ ), the van gets $\frac { 290 } { s }$ miles per gallon. Determine the most economical speed s for a 100-mile trip on an interstate highway.

Accepted Answer

A) The most economical speed is 52.0 mph. 
B) The most economical speed is 48.0 mph. 
C) The most economical speed is 25.0 mph. 
D) The most economical speed is 50.0 mph. 
E) The most economical speed is 53.0 mph. 
A) The most economical speed is 52.0 mph. 
B) The most economical speed is 48.0 mph. 
C) The most economical speed is 25.0 mph. 
D) The most economical speed is 50.0 mph. 
E) The most economical speed is 53.0 mph.

Question 3

Find the point on the graph of $f ( x ) = x ^ { 2 }$ that is closest to the point (6, 0.5). Round your answer to two decimal places.

Accepted Answer

A) (1.44, 2.07) 
B) (1.82, 3.31) 
C) (2.29, 5.24) 
D) (1.26, 1.59) 
E) (1.14, 1.30) 
A) (1.44, 2.07) 
B) (1.82, 3.31) 
C) (2.29, 5.24) 
D) (1.26, 1.59) 
E) (1.14, 1.30)

Question 4

Find the limit: $\lim _ { x \rightarrow 2 ^ { + } } \frac { x - 9 } { x - 2 }$

Accepted Answer

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

Question 5

You are in a boat 2 miles from the nearest point on the coast. You are to go to point Q located 3 miles down the coast and 4 miles inland (see figure). You can row at a rate of 4 miles per hour and you can walk at a rate of 4 miles per hour. Toward what point on the coast should you row in order to reach point Q in the least time?

Accepted Answer

A) 3 miles 
B) 8 miles 
C) 4 miles 
D) 1 mile 
E) 5 miles 
A) 3 miles 
B) 8 miles 
C) 4 miles 
D) 1 mile 
E) 5 miles

Question 6

Suppose the sales S (in billions of dollars per year) for Proctor & Gamble for the years 1999 through 2004 can be modeled by $S = 2.5931 t ^ { 2 } - 1.5682 t + 39.831,1999 \leq t \leq 2004$ where t represents the year. During which year were the sales increasing at the lowest rate?

Accepted Answer

A) Sales are increasing at the lowest rate in the year 2004. 
B) Sales are increasing at the lowest rate in the year 1999. 
C) Sales are increasing at the lowest rate in the year 2000. 
D) Sales are increasing at the lowest rate in the year 2002. 
E) Sales are increasing at the lowest rate in the year 2001. 
A) Sales are increasing at the lowest rate in the year 2004. 
B) Sales are increasing at the lowest rate in the year 1999. 
C) Sales are increasing at the lowest rate in the year 2000. 
D) Sales are increasing at the lowest rate in the year 2002. 
E) Sales are increasing at the lowest rate in the year 2001.

Question 7

A function and its graph are given. Use the graph to find the vertical asymptotes, if they exist, where $A = 51$ Confirm your results analytically. $$f ( x ) = \frac { 17 x ^ { 2 } } { ( x - 2 ) ^ { 2 } }$$

Accepted Answer

A)  $x = 1$ 
B)  $x = 2$ 
C)  $x = 0$ 
D)  $x = 5$ 
E) no vertical asymptotes 
A)  $x = 1$ 
B)  $x = 2$ 
C)  $x = 0$ 
D)  $x = 5$ 
E) no vertical asymptotes

Question 8

Use a table utility with x-values larger than 10,000 to investigate $\lim _ { x \rightarrow + \infty } f ( x )$ .What does the table indicate about $\lim _ { x \rightarrow + \infty } f ( x ) ?$ $f ( x ) = \frac { 16 x ^ { 3 } - 5 x } { 5 - 2 x ^ { 3 } }$

Accepted Answer

A)  $- 8$ 
B)  $8$ 
C)  $16$ 
D) 0 
E) does not exist 
A)  $- 8$ 
B)  $8$ 
C)  $16$ 
D) 0 
E) does not exist

Question 9

Use analytic methods to find the limit as $x \rightarrow - \infty$ for the given function. $f ( x ) = \frac { 7000 x } { 3300 - 4 x }$

Accepted Answer

A)  $- 1750$ 
B)  $825$ 
C)  $- 825$ 
D)  $1750$ 
E) does not exist 
A)  $- 1750$ 
B)  $825$ 
C)  $- 825$ 
D)  $1750$ 
E) does not exist

Question 10

Find any horizontal asymptotes for the given function. $y = \frac { 8 x ^ { 3 } } { 8 x ^ { 2 } + 6 }$

Accepted Answer

A)  $y = 1$ 
B)  $y = 8$ 
C)  $y = 6$ 
D)  $y = 0$ 
E) no horizontal asymptotes 
A)  $y = 1$ 
B)  $y = 8$ 
C)  $y = 6$ 
D)  $y = 0$ 
E) no horizontal asymptotes

Question 11

Compare dy and $\Delta y$ for $y = 3 x ^ { 2 } - 2$ at x = 0 with dx = -0.06. Give your answers to four decimal places.

Accepted Answer

A)  $d y = - 0.0100 ; \quad \Delta y = 0.0106$ 
B)  $d y = 0.0000 ; \quad \Delta y = 0.0108$ 
C)  $d y = - 0.0300 , \quad \Delta y = 0.0107$ 
D)  $d y = 0.0200 ; \quad \Delta y = 0.0105$ 
E)  $d y = 0.0200 ; \quad \Delta y = 0.0107$ 
A)  $d y = - 0.0100 ; \quad \Delta y = 0.0106$ 
B)  $d y = 0.0000 ; \quad \Delta y = 0.0108$ 
C)  $d y = - 0.0300 , \quad \Delta y = 0.0107$ 
D)  $d y = 0.0200 ; \quad \Delta y = 0.0105$ 
E)  $d y = 0.0200 ; \quad \Delta y = 0.0107$

Question 12

A function and its graph are given. Use the graph to find the vertical asymptotes, if they exist. Confirm your results analytically. $f ( x ) = \frac { 8 } { x + 2 }$

Accepted Answer

A)  $x = 2$ 
B)  $x = 8$ 
C)  $x = 6$ 
D)  $x = - 2$ 
E) no vertical asymptotes 
A)  $x = 2$ 
B)  $x = 8$ 
C)  $x = 6$ 
D)  $x = - 2$ 
E) no vertical asymptotes

Question 13

Sketch the graph of the relation $x y ^ { 2 } = 4$ using any extrema, intercepts, symmetry, and asymptotes.

Accepted Answer

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

Question 14

A rectangular page is to contain $36$ square inches of print. The margins on each side are 1 inch. Find the dimensions of the page such that the least amount of paper is used.

Accepted Answer

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

Question 15

Find the limit. $\lim _ { x \rightarrow -\infty } \frac { 4 x ^ { 2 } } { x + 3 }$

Accepted Answer

A)  $0$ 
B)  $- \frac { 4 } { 3 }$ 
C)  $\frac { 4 } { 3 }$ 
D)  $\infty$ 
E)  $- \infty$ 
A)  $0$ 
B)  $- \frac { 4 } { 3 }$ 
C)  $\frac { 4 } { 3 }$ 
D)  $\infty$ 
E)  $- \infty$

Question 16

The measurement of the circumference of a circle is found to be 47 centimeters, with a possible error of 0.9 centimeters. Approximate the percent error in computing the area of the circle.

Accepted Answer

A) 4.26 % 
B) 1.91 % 
C) 3.83 % 
D) 7.66 % 
E) 2.13 % 
A) 4.26 % 
B) 1.91 % 
C) 3.83 % 
D) 7.66 % 
E) 2.13 %

Question 17

Sketch the graph of the function given below. Choose a scale that allows all relative extrema and points of inflection to be identified on the graph. $y = \frac { x ^ { 2 } + 1 } { x ^ { 2 } - 2 }$

Accepted Answer

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

Question 18

For the function $f ( x ) = \frac { 4 x + 7 } { 3 x - 7 }$ , use a graphing utility to complete the table and estimate the limit as x approaches infinity. $\begin{array}{lllllll}
x & 10^{0} & 10^{1} & 10^{2} & 10^{3} & 10^{4} & 10^{5}\
f(x)
\end{array}$

Accepted Answer

A) 0.75 
B) 1.333333 
C) 2.333333 
D) 1.75 
E) -0.25 
A) 0.75 
B) 1.333333 
C) 2.333333 
D) 1.75 
E) -0.25

Question 19

This problem contains a function and its graph, where $\mathrm { A } = 10$ Use the graph to determine, as well as you can, the vertical asymptote. Check your conclusion by using the function to determine the vertical asymptote analytically. $$f ( x ) = \frac { 2 ( x - 3 ) } { x - 2 }$$

Accepted Answer

A)  $x = - 5$ 
B)  $x = 0$ 
C)  $x = - 1$ 
D)  $x = 2$ 
E) no vertical asymptote 
A)  $x = - 5$ 
B)  $x = 0$ 
C)  $x = - 1$ 
D)  $x = 2$ 
E) no vertical asymptote

Question 20

Match the function $f ( x ) = \frac { 4 x } { x ^ { 2 } + 2 }$ with one of the following graphs.

Accepted Answer

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

If the total cost function for a product is $C ( x ) = 200 + 4 x + 0.09 x ^ { 2 }$ dollars. Find the minimum average cost.

Find the point on the graph of $f ( x ) = x ^ { 2 }$ that is closest to the point (6, 0.5). Round your answer to two decimal places.

Find the limit: $\lim _ { x \rightarrow 2 ^ { + } } \frac { x - 9 } { x - 2 }$

Suppose the sales S (in billions of dollars per year) for Proctor & Gamble for the years 1999 through 2004 can be modeled by $S = 2.5931 t ^ { 2 } - 1.5682 t + 39.831,1999 \leq t \leq 2004$ where t represents the year. During which year were the sales increasing at the lowest rate?

Use a table utility with x-values larger than 10,000 to investigate $\lim _ { x \rightarrow + \infty } f ( x )$ .What does the table indicate about $\lim _ { x \rightarrow + \infty } f ( x ) ?$ $f ( x ) = \frac { 16 x ^ { 3 } - 5 x } { 5 - 2 x ^ { 3 } }$

Use analytic methods to find the limit as $x \rightarrow - \infty$ for the given function. $f ( x ) = \frac { 7000 x } { 3300 - 4 x }$

Find any horizontal asymptotes for the given function. $y = \frac { 8 x ^ { 3 } } { 8 x ^ { 2 } + 6 }$

Compare dy and $\Delta y$ for $y = 3 x ^ { 2 } - 2$ at x = 0 with dx = -0.06. Give your answers to four decimal places.

Sketch the graph of the relation $x y ^ { 2 } = 4$ using any extrema, intercepts, symmetry, and asymptotes.

A rectangular page is to contain $36$ square inches of print. The margins on each side are 1 inch. Find the dimensions of the page such that the least amount of paper is used.

Find the limit. $\lim _ { x \rightarrow -\infty } \frac { 4 x ^ { 2 } } { x + 3 }$

The measurement of the circumference of a circle is found to be 47 centimeters, with a possible error of 0.9 centimeters. Approximate the percent error in computing the area of the circle.

Sketch the graph of the function given below. Choose a scale that allows all relative extrema and points of inflection to be identified on the graph. $y = \frac { x ^ { 2 } + 1 } { x ^ { 2 } - 2 }$

For the function $f ( x ) = \frac { 4 x + 7 } { 3 x - 7 }$ , use a graphing utility to complete the table and estimate the limit as x approaches infinity. x 1 1 1 1 1 1 f(x)

Match the function $f ( x ) = \frac { 4 x } { x ^ { 2 } + 2 }$ with one of the following graphs.

Fundamental Concepts of Algebra

Equations and Inequalities

Functions and Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Systems of Equations and Inequalities

Matrices and Determinants

Limits and Derivatives

Applications of the Derivative

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Exam 10: Further Applications of the Derivative

If the total cost function for a product is C(x)=200+4x+0.09x2C ( x ) = 200 + 4 x + 0.09 x ^ { 2 }C(x)=200+4x+0.09x2 dollars. Find the minimum average cost.

Find the point on the graph of f(x)=x2f ( x ) = x ^ { 2 }f(x)=x2 that is closest to the point (6, 0.5). Round your answer to two decimal places.

Find the limit: lim⁡x→2+x−9x−2\lim _ { x \rightarrow 2 ^ { + } } \frac { x - 9 } { x - 2 }limx→2+​x−2x−9​

A function and its graph are given. Use the graph to find the vertical asymptotes, if they exist, where A=51A = 51A=51 Confirm your results analytically. f(x)=17x2(x−2)2f ( x ) = \frac { 17 x ^ { 2 } } { ( x - 2 ) ^ { 2 } }f(x)=(x−2)217x2​

Use analytic methods to find the limit as x→−∞x \rightarrow - \inftyx→−∞ for the given function. f(x)=7000x3300−4xf ( x ) = \frac { 7000 x } { 3300 - 4 x }f(x)=3300−4x7000x​

Find any horizontal asymptotes for the given function. y=8x38x2+6y = \frac { 8 x ^ { 3 } } { 8 x ^ { 2 } + 6 }y=8x2+68x3​

Compare dy and Δy\Delta yΔy for y=3x2−2y = 3 x ^ { 2 } - 2y=3x2−2 at x = 0 with dx = -0.06. Give your answers to four decimal places.

A function and its graph are given. Use the graph to find the vertical asymptotes, if they exist. Confirm your results analytically. f(x)=8x+2f ( x ) = \frac { 8 } { x + 2 }f(x)=x+28​

Sketch the graph of the relation xy2=4x y ^ { 2 } = 4xy2=4 using any extrema, intercepts, symmetry, and asymptotes.

A rectangular page is to contain 363636 square inches of print. The margins on each side are 1 inch. Find the dimensions of the page such that the least amount of paper is used.

Find the limit. lim⁡x→−∞4x2x+3\lim _ { x \rightarrow -\infty } \frac { 4 x ^ { 2 } } { x + 3 }limx→−∞​x+34x2​

The measurement of the circumference of a circle is found to be 47 centimeters, with a possible error of 0.9 centimeters. Approximate the percent error in computing the area of the circle.

Sketch the graph of the function given below. Choose a scale that allows all relative extrema and points of inflection to be identified on the graph. y=x2+1x2−2y = \frac { x ^ { 2 } + 1 } { x ^ { 2 } - 2 }y=x2−2x2+1​

For the function f(x)=4x+73x−7f ( x ) = \frac { 4 x + 7 } { 3 x - 7 }f(x)=3x−74x+7​ , use a graphing utility to complete the table and estimate the limit as x approaches infinity. x 1 1 1 1 1 1 f(x)

Match the function f(x)=4xx2+2f ( x ) = \frac { 4 x } { x ^ { 2 } + 2 }f(x)=x2+24x​ with one of the following graphs.

Fundamental Concepts of Algebra

Equations and Inequalities

Functions and Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Systems of Equations and Inequalities

Matrices and Determinants

Limits and Derivatives

Applications of the Derivative

Derivatives of Exponential and Logarithmic Functions

Integration and Its Applications

Techniques of Integration

Functions of Several Variables

Trigonometric Functions Web

Series and Taylor Polynomials Web

Probability Web

Filters

If the total cost function for a product is $C ( x ) = 200 + 4 x + 0.09 x ^ { 2 }$ dollars. Find the minimum average cost.

Find the point on the graph of $f ( x ) = x ^ { 2 }$ that is closest to the point (6, 0.5). Round your answer to two decimal places.

Find the limit: $\lim _ { x \rightarrow 2 ^ { + } } \frac { x - 9 } { x - 2 }$

Use analytic methods to find the limit as $x \rightarrow - \infty$ for the given function. $f ( x ) = \frac { 7000 x } { 3300 - 4 x }$

Find any horizontal asymptotes for the given function. $y = \frac { 8 x ^ { 3 } } { 8 x ^ { 2 } + 6 }$

Compare dy and $\Delta y$ for $y = 3 x ^ { 2 } - 2$ at x = 0 with dx = -0.06. Give your answers to four decimal places.

Sketch the graph of the relation $x y ^ { 2 } = 4$ using any extrema, intercepts, symmetry, and asymptotes.

A rectangular page is to contain $36$ square inches of print. The margins on each side are 1 inch. Find the dimensions of the page such that the least amount of paper is used.

Find the limit. $\lim _ { x \rightarrow -\infty } \frac { 4 x ^ { 2 } } { x + 3 }$

Sketch the graph of the function given below. Choose a scale that allows all relative extrema and points of inflection to be identified on the graph. $y = \frac { x ^ { 2 } + 1 } { x ^ { 2 } - 2 }$

For the function $f ( x ) = \frac { 4 x + 7 } { 3 x - 7 }$ , use a graphing utility to complete the table and estimate the limit as x approaches infinity. x 1 1 1 1 1 1 f(x)

Match the function $f ( x ) = \frac { 4 x } { x ^ { 2 } + 2 }$ with one of the following graphs.