Deck 1: Functions,graphs,and Limits

Question 1

A closed box with a square base is to have a volume of 10 cubic meters.The material for the top and bottom of the box costs $5 per square meter,and the material for the sides costs $1 per square meter.Express the construction cost of the box as a function of the length of its base,x.

A) $C ( x ) = 10 x ^ { 2 } + \frac { 40 } { x }$ dollars
B) $C ( x ) = x ^ { 2 } + 40 x + 2$ dollars
C) $C ( x ) = 2 x ^ { 2 } + \frac { 2 } { x } + 10$ dollars
D) $C ( x ) = 2 x ^ { 2 } + \frac { 10 } { x }$ dollars

Answer 1

$C ( x ) = 10 x ^ { 2 } + \frac { 40 } { x }$ dollars

Question 2

Find the indicated limit if it exists. $\lim _ { x \rightarrow 3 } ( x + 4 )$

A)3
B)Does not exist
C)-1
D)7

Answer 2

7

Question 3

Find the slope (if possible) of the line that passes through the given pair of points. (28,7)and (8,3).

A) $\frac { 1 } { 5 }$
B)The slope is undefined.
C)5
D)0

Answer 3

$\frac { 1 } { 5 }$

Question 4

Find $\lim _ { x \rightarrow + \infty } f ( x )$ and $\lim _ { x \rightarrow - \infty } f ( x )$ .If the limiting value is infinite indicate whether it is + $\infty$ or - $\infty$ . f (x)= (5 - 9x)(x - 2)

A)- $\infty$ ، - $\infty$
B)- $\infty$ ، 0
C)+ $\infty$ ، + $\infty$
D)+ $\infty$ ، 0

Question 5

Find the indicated limit if it exists. $\lim _ { x \rightarrow 1 } \frac { x ^ { 2 } + 3 x + 2 } { x ^ { 2 } - 1 }$

A)Does not exist
B)6
C) $\frac { 3 } { 2 }$
D)3

Question 6

Write an equation for the line through (5,4)and parallel to the x-axis.

A)y = 4
B)y = -4
C)x = -5
D)x = 5

Question 7

Find the indicated composite function. f (x + 6)where $f ( x ) = \frac { 1 } { x }$

A)f (x + 6)= $x + 6 + \frac { 1 } { x }$
B)f (x + 6)= $\frac { 1 } { x + 6 }$
C)f (x + 6)= $1 + \frac { 3 } { x }$
D)f (x + 6)= $\frac { 1 } { x } + 6$

Question 8

Find the distance between the given points. (-2,6)and (3,7)

A)D = 6
B)D = 26
C) $D = \sqrt { 26 }$
D) $D = \sqrt { 6 }$

Question 9

A closed cylindrical can has a surface area of 120 $\pi$ square inches.Express the volume of the can as a function of its radius,r.

A)V(r)= 60 $\pi$ r cubic inches
B) $V ( r ) = 120 \pi r ^ { 2 }$ cubic inches
C) $V ( r ) = \pi \left( 60 - r ^ { 3 } \right)$ cubic inches
D) $V ( r ) = \pi r \left( 60 - r ^ { 2 } \right)$ cubic inches

Question 10

Each unit of a certain commodity costs p = 47x + 12 cents when x units of the commodity are produced.If all units are sold at this price,express the revenue derived from the sales as a function of x.

A)46x + 12 cents
B) $47 x ^ { 2 } + 12$ cents
C)48x + 12 cents
D)x(47x + 12)cents

Question 11

Plot the given point in a rectangular coordinate system. (1,-5)

A) (Each gridline represents one unit.)
B) (Each gridline represents one unit.)
C) (Each gridline represents one unit.)
D) (Each gridline represents one unit.)

Question 12

Find $\lim _ { x \rightarrow + \infty } f ( x )$ and $\lim _ { x \rightarrow - \infty } f ( x )$ .If the limiting value is infinite indicate whether it is + $\infty$ or - $\infty$ . $f ( x ) = \frac { 4 - 2 x ^ { 3 } } { 6 x ^ { 3 } - 6 x + 1 }$

A)+ $\infty$ ،- $\infty$
B)- $\infty$ ، + $\infty$
C) $- \frac { 1 } { 3 }$ , $- \frac { 1 } { 3 }$
D) $- \frac { 1 } { 3 }$ , $\frac { 1 } { 3 }$

Question 13

Find the points of intersection (if any)of the given pair of curves. $y = x ^ { 2 }$ and y = 3x - 2.

A)There are no points of intersection.
B)(2,4)
C)(1,1),(2,4)
D)(0,0)

Question 14

Find the slope and y-intercept (if they exist)of the line 8y = 7x.

A)Slope is 7 and y-intercept is 8.
B)Slope is $\frac { 8 } { 7 }$ and y-intercept is 0.
C)Slope is $\frac { 7 } { 8 }$ and y-intercept is 0.
D)Slope is 7 and y-intercept is 0.

Question 15

Two car rental agencies are competing.One agency rents cars for 50 dollars per day and 35 cents a mile; the other agency rents cars for 35 dollars per day and 45 cents a mile.For a 10 day trip,how many miles must you travel to have the total cost be the same with each agency? Round to the nearest whole mile ,if necessary.

A)43 miles
B)1,500 miles
C)33 miles
D)150 miles

Question 16

Find the difference quotient, $\frac { f ( x + h ) - f ( x ) } { h }$ . $f ( x ) = \frac { 9 } { x }$

A) $\frac { f ( x + h ) - f ( x ) } { h } = - \frac { 9 } { x ( x + h ) }$
B) $\frac { f ( x + h ) - f ( x ) } { h } = \frac { 9 } { x + h }$
C) $\frac { f ( x + h ) - f ( x ) } { h }$ = 9
D) $\frac { f ( x + h ) - f ( x ) } { h }$ = 1

Question 17

The cost of renting a backhoe at one distributor is $325,plus $35 per day.Write a linear function C(x)that describes the cost of renting the backhoe for x days,then use your function to find how much it would cost to rent it for 12 days.

A) $C ( x ) = 35 x + 313 ; \quad \$ 733$
B) $C ( x ) = 325 x + 35 ; \quad \$ 3,935$
C) $C ( x ) = 12 ( 325 + 35 x ) ; \quad \$ 8,940$
D) $C ( x ) = 325 + 35 x ; \quad \$ 745$

Question 18

Sketch the graph of the given function. $f ( x ) = x ^ { 2 } + 2$

A)
B)
C)
D)
(Tick marks are spaced one unit apart.)

Question 19

Determine the domain of the given function. $f ( x ) = \frac { x ^ { 2 } - 1 } { x + 3 }$

A)all real numbers x except x = 1,x = -1,and x = -3
B)all real numbers x
C)all real numbers x except x = -3
D)all real numbers x except x = 1 and x = 4

Question 20

Compute the indicated value of the given function. $f ( t ) = \left\{ \begin{aligned}- 2 t + 8 & \text { if } t < - 1 \\t ^ { 2 } + 9 & \text { if } t \geq - 1\end{aligned} \right.$ ; f (-3)

A)18
B)0
C)2
D)14

Question 21

List all values of x for which f (x)is not continuous. $f ( x ) = \frac { 2 x + 1 } { x ^ { 2 } + x }$ .

A) $- \frac { 1 } { 2 }$
B)-1
C)None
D)0 and -1

Question 22

List all the values of x for which the given function is not continuous. $f ( x ) = \left\{ \begin{array} { c l } x ^ { 3 } & \text { if } x \leq 5 \\125 & \text { if } x > 5\end{array} \right.$

A)x = 5
B)x = ±5
C)The function is continuous for all values of x.
D)x = 0

Question 23

An efficiency consultant determines that when new workers are hired to wait tables at an upscale restaurant,the average number of tables they can wait on in a 6 hour shift is given by $N ( x ) = \frac { \sqrt { 5.3 x ^ { 2 } + 50 x + 14 } } { 0.1 x + 0.8 }$ where x is the number of shifts they've worked since being hired.What happens to an average waiter's productivity in the long run (as $x \rightarrow \infty$ )?

A)It approaches 53 tables per shift.
B)It increases without bound.
C)It approaches 23 tables per shift.
D)It approaches 14 tables per shift.

Question 24

Find the indicated one-sided limit.If the limiting value is infinite,indicate whether it is + $\infty$ or - $\infty$ . $\lim _ { x \rightarrow 4 ^ { - } } \frac { \sqrt { x } - 2 } { x - 4 }$

A) $- \frac { 1 } { 4 }$
B) $\frac { 1 } { 4 }$
C)2
D)-2

Question 25

Find the indicated one-sided limit.If the limiting value is infinite,indicate whether it is + $\infty$ or - $\infty$ . $\lim _ { x \rightarrow 2 ^ { - } } f ( x )$ where $f ( x ) = \left\{ \begin{aligned}x + 5 & \text { if } x < 2 \\x ^ { 2 } & \text { if } x \geq 2\end{aligned} \right.$

A)There is none
B)7
C)0
D)4