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Answered: - Pre cal, answers and steps are required. Steps are mandatory.


Pre cal, answers and steps are required. Steps are mandatory.


EXAM2?100 minutes. Support all answers with your work in the space provided. Name______________________

 

All problems worth 3 pts., unless stated otherwise. Show ALL work in the space provided, regardless if the question is

 

multiple choice or free response. [Correct answer alone will NOT receive credit.]

 


 

PART A?(#1-18) No calculator is to be used on this portion of the exam.

 

1. What is the derivative of the function

 


 

f ( x )=5 x 3?2 x+7 ?

 


 

2. What is the value of the derivative of the function

 


 

f ( x )=5 x 3?2 x+7 at the point where x= 1?

 


 

3. Find the equation of the tangent line to the graph of the function

 


 

4. Graph the following function, whose graph has a HOLE:

 

2

 

( x )= x ?2 x?8 .

 

f

 

x?4

 


 

The location of the HOLE

 

is at: (______, _____)

 


 

3

 


 

f ( x )=5 x ?2 x+7 at the point where x=1.

 


 

5. If f ( x )=

 


 

?x 3?4 x +5

 

, describe the end-behavior of the graph of f (x) . [Note: there are no

 

x +2

 


 

asymptotes..]

 

a. As

 


 

x ? ? , f ( x ) ?+? , and as

 


 

x ??? , f ( x ) ???

 


 

b. As

 


 

x ? ? , f ( x ) ??? , and as x ???, f ( x ) ?+ ?

 


 

c. As

 


 

x ? ? , f ( x ) ?+? , and as x ???, f ( x ) ?+ ?

 


 

d. As

 


 

x ? ? , f ( x ) ? 0 , and as x ???, f ( x ) ?0

 


 

e. As

 


 

x ? ? , f ( x ) ??? , and as x ???, f ( x ) ???

 


 

6. [4 pts] Graph the following function, showing all 4 pre-calculus characteristics (intercepts, asymptotes..etc.)

 


 

f ( x )=

 


 

3 x +6

 

x?2 .

 


 

3 x +6

 

f ( x )=

 

7. What is the derivative of the function

 

x?2 ?

 


 

8.

 


 

Find the equation of the tangent line to

 


 

f ( x )=

 


 

3 x +6

 

x?2

 


 

at the point x= 0. Sketch this tangent line on your

 


 

graph of this function above at the point x=0. [p.s. Does the tangent line seem to have the slope you

 

computed?]

 


 

9.

 


 

Find the derivative for the function

 


 

f ( x )=x 2?2 x using the limit definition of the derivative.

 


 

10. Find the equations of ALL asymptotes (both horizontal or vertical) of the function

 

a) x=0 only

 


 

b) y=0 only

 


 

2

 


 

c)y=0 and x=0

 


 

d) y=0, x=1, x=-1

 


 

f ( x )=

 


 

x

 

x + x +1 .

 

2

 


 

e) None.

 


 

x ?4

 

11. If f ( x )= x+ 2 , when x ??2 , then what value must be defined for f (?2) so that the function is

 

made continuous for all values of x?

 

f. -4

 

g. 4

 

h. -2

 

i. 2

 

j. 0

 


 

( )

 

12. If f x =

 


 

13. Let

 

a)

 

b)

 

c)

 

d)

 

e)

 


 

7 x2 + 4 x+ 16

 

, then f ' (4)=?

 

?x

 


 

f ( x )=(x2 ?2 x +2)( x3 +1) , find f ' ( 2 ) .

 

42

 

0

 

16

 

17/2

 

9

 


 

Refer to the diagram below for questions 14-15:

 


 

14. What of the following is TRUE about the function above?

 

??

 


 

a.

 


 

x ? 4 f ( x )=1

 

.

 

lim ?

 

?

 


 

[Hint: you do not need quotient rule?]

 


 

b.

 


 

x ? 0+? f ( x )=5

 

lim ?

 


 

c.

 


 

x ? 2 f ( x )=2

 

lim ?

 


 

d.

 


 

x ? 2+? f ( x )=2

 

lim ?

 


 

e.

 


 

x ? 4 f ( x )=2

 

lim ?

 


 

?

 


 

??

 


 

?

 


 

?

 


 

??

 


 

15.

 

a.

 

b.

 

c.

 

d.

 


 

?

 


 

Which of the following statements is TRUE about the above function:

 

The limit of f(x) exists at x=2

 

The function is continuous at x=2

 

The function value f(2)=2.

 

The domain of the function is [0,4]

 


 

x ?2+? f ( x)

 

??

 

x ? 2 f ( x )=lim ?

 


 

e.

 


 

lim ?

 


 

?

 


 

?

 


 

16. Which of the following could be the function whose graph is shown here:

 

a)

 


 

f ( x )=2 x 3?15 x 2 +24 x?5

 


 

b)

 


 

f ( x )=?2 x 3?15 x 2 +24 x?5

 


 

c)

 


 

f ( x )=?2 x 3?15 x 2 +24 x+ 5

 


 

d)

 


 

f ( x )=2 x 3?15 x 2 +24 x +5

 


 

e)

 


 

f ( x )=2 x 2?15 x +5

 

JUSTIFY:

 


 

17. At which points does the function

 

a) x=0, -3

 


 

b) x= 23/3, -23/3

 


 

3

 


 

f ( x )=x ?23 x +2 have a slope of 4? (x-values only is ok)

 

c) x=-3, 3

 


 

d) no points

 


 

e) x= 17, -4

 


 

18. Using your knowledge of the derivative, evaluate the expression

 


 

19. Which of the following is TRUE about the function

 

f.

 

g.

 


 

The function has a removable discontinuity at x=1.

 

??

 


 

x ? 1 f ( x )=+?

 

.

 

lim ?

 

?

 


 

h. The limit of the function exists at x=1.

 

i.

 


 

j.

 


 

x ? 1+ ? f (x )=??

 

lim ?

 

.

 

?

 


 

x ?1+? f ( x)

 

x ? 1?? f ( x )=lim ?

 

lim ?

 

?

 


 

?

 


 

f ( x )=

 


 

lim

 


 

4

 

4

 

? x+ h?? x

 


 

h?0

 


 

5?2 x

 

x?1

 


 

?

 


 

h

 


 

.

 


 

20. The Cost function for a new financial instrument is given by

 


 

C ( x )=2 x2 ?16 x+120 , where x is the number

 


 

of shares of preferred stock in the instrument. What is the minimum cost for this function?

 


 

PART B?Calculators allowed. They may be useful on a few of the problems on this section of the exam (but really NOT

 

useful on most of the problems)

 

21. The flight of a projectile fired from a cannon follows the trajectory given by the function below. Find the time to

 

the nearest hundredth of a second that it takes the projectile to reach the ground.

 


 

h (t)=

 


 

?1 2

 

t + ? 3t +5

 

2

 


 

22. What is the domain of the function

 

a)

 


 

(?? ,?3 ] U ?

 


 

b)

 


 

f ( x )=? 9?x2 ?

 


 

(??,?3 ) U (3, ?)

 


 

c) [?3, 3]

 


 

d)

 


 

(?3,3)

 


 

e) All real #s

 


 

23. For the following supply and demand functions, find the point of market equilibrium and the revenue at the

 

point of market equilibrium. (Give your answers to the nearest tenth).

 


 

4 p=?15 ? x+ 110

 

2

 


 

p?x =7

 


 

24. For what value(s) of x is the following function NOT defined?

 


 

f ( x )=

 


 

x ?3

 

x +3 x3 ?15 x +6 .

 

4

 


 

JUSTIFY:

 


 

Round answers to nearest hundredth.

 


 

25. At what point x is the tangent line to

 

a) 1

 


 

b) 6

 


 

f ( x )=x + ? x

 


 

c) 1/36

 


 

f ( x )=

 


 

26. The rational function

 


 

d) 1/6

 


 

x?2

 

2 x ?3 x?2

 

2

 


 

parallel to the line

 


 

? 4 x +15 ?

 


 

e) 36

 


 

has only ONE vertical asymptote and ONE hole. Find the vertical

 


 

asymptote and find the coordinates (x,y) of the hole.

 


 

27. What is the domain of the function

 


 

a)

 


 

(?? ,?10 ] U ?

 


 

b)

 


 

f ( x )=

 


 

1

 

? 100?x 2 ?

 


 

(??,?10 ) U (10, ?)

 


 

c) [?10,10 ]

 


 

d)

 


 

(?10,10)

 


 

e) All real #s

 


 

 


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