MthSc 103 Test 2 Spring 2009 Version A JIT 7.2 – 10.6; UC

MthSc 103 Test 2 Spring 2009 Version A JIT 7.2 – 10.6; UC 2.1, 2.2

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Student’s Printed Name: _______________________ CUID:___________________ Instructor: ______________________ Section # :_________ Read each question very carefully. You are NOT permitted to use a calculator on any portion of this test. You are not allowed to use any textbook, notes, cellphone, or laptop on either portion of the test. No part of this test may be removed from the testing room. In order to receive full credit for the free response portion of the test, you must:

1. Show legible and logical (relevant) justification which supports your final answer. 2. Use complete and correct mathematical notation. 3. Include proper units, if necessary. 4. Give exact numerical values whenever possible.

You have 90 minutes to complete the entire test.

On my honor, I have neither given nor received inappropriate or unauthorized information during this test. Student’s Signature: ________________________________________________

Do not write below this line.

Free Response Problem

Possible Points

Points Earned

Free Response Problem

Possible Points

Points Earned

1 5 6a 5

2 5 6b 4

3 4 7 3

4a 5 8 5

4b 1 9 1

4c 2 Free

Response 45

5 5 Multiple Choice 55

Test Total 100


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Multiple Choice. There are 22 multiple choice questions. Each question is worth 2 – 3 points and has one correct answer. The multiple choice problems will count 55% of the total grade. Use a number 2 pencil and bubble in the letter of your response on the scantron sheet for problems 1 – 22. For your own record, also circle your choice on your test since the scantron will not be returned to you. Only the responses recorded on your scantron sheet will be graded. You are NOT permitted to use a calculator on any portion of this test.

1. (3 pts.)

Simplify completely:

a)

b)

c)

d)

2. (2 pts.)

Solve for :

a)

b)

c)

d)

3. (2 pts.)

Simplify completely:

a)

b)

c)

d)

4. (3 pts.)

If , find .

a)

b)

c)

d)

5. (3 pts.)

Let . Find .

a)

b)

c)

d)


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6. (3 pts.)

Factor:

a)

b)

c)

d)

7.

(3 pts.) Given the following table with input values and output values for and , find .

a)

b)

c)

d)

8.

(2 pts.) Simplify completely:

a)

b)

c)

d)

9. (2 pts.)

Write as a single simplified logarithm.

a)

b)

c)

d)

10.

(2 pts.) Factor:

a)

b)

c)

d)

11.

(3 pts.) Evaluate:

a)

b)

c)

d)


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12. (3 pts.)

Below is the graph of . Which of the following is the graph of its inverse?

a)

b)

c)

d)

13. (2 pts.)

Evaluate:

a)

b)

c)

d)


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14. (2 pts.)

Let . What is the range for ?

a)

b)

c)

d)

Use the function to answer the next two questions.

15. (2 pts.)

Which of the following is the graph of ?

a)

b)

c)

d)

16. (2 pts.)

has an asymptote where?

a) vertical at

b) vertical at

c) vertical at

d) doesn’t have an asymptote


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17. (3 pts.)

Evaluate:

a)

b)

c)

d)

18.

(2 pts.) Find .

a)

b)

c)

d)

19. (2 pts.)

Evaluate:

a)

b)

c)

d)

20. (3 pts.)

Solve for : .

a)

b)

c)

d)

21. (3 pts.)

Evaluate:

a)

b)

c)

d)

22. (3 pts.)

Simplify and indicate where it is valid.

a)

b)

c)

d)


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Free Response. The Free Response questions will count 45% of the total grade. Read each question carefully. In order to receive full credit you must show legible and logical (relevant) justification which supports your final answer. Give answers as exact answers. You are NOT permitted to use a calculator on any portion of this test. 1. (5 pts) Solve for :

log3

x ! 4( ) + log3

x + 4( ) = 2

log3

x2!16( ) = 2

32= x

2!16

0 = x2! 25

0 = x ! 5( ) x + 5( )x = 5, x = !5

log3!9( ) is undefined

so the only valid solution is x = 5

2. (5 pts) Given is a factor of , completely factor .

x + 3 x3+ 6x

2! x ! 30

x3+ 3x

2

3x2! x

3x2+ 9x

!10x ! 30

!10x ! 30

0

x2+ 3x !10

so y = x + 3( ) x

2+ 3x !10( )

finally P(x) = x + 3( ) x + 5( ) x ! 2( )

Work on Problem Points Awarded Correctly use product law of logs 1 Correctly eliminate logarithm 1 Correctly solve for both x’s 2 Eliminate the x value that is not in the domain 1 Notes: -0.5 to -1 for notation errors depending on severity

Work on Problem Points Awarded

Correctly perform long division of polynomial 2.5 Correctly factor remaining quadratic 1.5 Write completely factored form 1 Notes: • Full credit for correct synthetic division • -0.5 to -1.5 for errors in polynomial long division depending on severity and cascading effect • -0.5 to -1.5 for incomplete work in polynomial long division • -1 for incorrectly factoring the quadratic


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3. (4 pts) Let Find .

y =3x + 4

x

solve for x: xy = 3x + 4

xy ! 3x = 4

x( y ! 3) = 4

x =4

y ! 3

interchange: f !1 x( ) =4

x ! 3

OR

y =3x + 4

x

interchange: x =3y + 4

y

solve for y: xy = 3y + 4

xy ! 3y = 4

y(x ! 3) = 4

y =4

x ! 3

so f !1(x) =4

x ! 3


Notation change 0.5 Solve for x 2 Interchange 1 Notation change 0.5 Notes:

can simplify to y = 3 +

4

x first


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4. Consider the function and the given point P on the curve. a. (5 pts) Find and completely simplify

f (2 + h) ! f (2)

h=

3 2 + h( )2

! 4 2 + h( ) +1! 5

h

=3 4 + 4h + h2( ) ! 8 ! 4h ! 4

h

=12 +12h + 3h2

! 4h !12

h

=3h2

+ 8h

h= 3h + 8 h " 0

b. (1 pt) Find

limh!0

f 2 + h( ) " f 2( )h

= limh!0

3h + 8( ) = 8

c. (2 pts) Find an equation of the tangent line at the given point.

y ! 5 = 8 x ! 2( )

Work on Problem Points Awarded Correctly use the given point in the equation 1 Correctly use the slope from part b 1 Notes: -0.5 if change to slope-intercept form incorrectly -1 for solving y = mx + b incorrectly for b and never writing a correct equation


Correctly substitute 2+h 1 Correctly set up difference quotient for this problem

1

Correctly distribute and simplify 2 Correctly factor and cancel 1 Notes: h ! 0 not required -1 arithmetic error -1 poor notation, more than one instance -1 took limit in both part a and part b


Correctly evaluate the limit 1 Notes: • Student does not have to rewrite the limit because it is stated in the problem. It is acceptable to just write the answer. • Graded based on part a unless part a simplified or changed the question severely.


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5. (5 pts) Let . Find the average rate of change in over the interval

f!6

"#$

%&'( f (

!3

"#$

%&'

!6( (

!3

=

sin!6

"#$

%&'

2

( sin(!3

"#$

%&'

2

!6+

2!6

=

1

2

"#$

%&'

2

( (3

2

"

#$

%

&'

2

3!6

=

1

4(

3

4

!2

= (2

4÷!2= (

1

2)

2

!= (

1

!


Average rate of change formula applied to this problem 1 Correctly trig evaluation 1 each Simplify numerator further 0.5 Reduce denominator 0.5 Final answer 1 Notes: -1 arithmetic error -1 poor notation, more than one instance


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6. (5, 4 pts) Find the following limits: a.

limx!4

2x " 4 " 2

x " 4#

2x " 4 + 2

2x " 4 + 2

= limx!4

2x " 4 " 4

x " 4( ) 2x " 4 + 2( )

= limx!4

2 x " 4( )x " 4( ) 2x " 4 + 2( )

= limx!4

2

2x " 4 + 2

=2

8 " 4 + 2

=2

4 + 2

=2

4=

1

2


Correctly used the conjugate to write a form of 1 1 Correctly simplifies the transformed function 3 Correct limit 1 Notes: -4 writes 0/0 and does nothing else correctly

-5 writes

0

0

= 0 and does nothing else correctly

-0.5 for 0/0 left in line with a correctly worked limit -1 for missing or poorly used limit notation -0.5 for inappropriate limit notation -0.5 for missing or poorly used equals

b.

limx!"1

3x2" x " 4

x2"1

= limx!"1

3x " 4( ) x +1( )x "1( ) x +1( )

= limx!"1

3x " 4

x "1="7

"2=

7

2

Work on Problem Points Awarded Expression correctly factored 1 each Simplified 1 Correct limit 1 Notes: -3 writes 0/0 and does nothing else correctly

-4 writes

0

0

= 0 and does nothing else correctly

-0.5 for 0/0 left in line with a correctly worked limit -1 for missing or poorly used limit notation -0.5 for inappropriate limit notation -0.5 for missing or poorly used equals -4 for limit technique for y ! "


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7. (3 pts) If for , use the Sandwich Theorem to find

Use proper notation - there are more points for the work than for the "answer".

3x ! f x( ) ! x3+ 2

limx"1

3x ! limx"1

f x( ) ! limx"1

x3+ 2( )

3 ! limx"1

f x( ) ! 3

so limx"1

f x( ) = 3 by the Sandwich Thm

8. (5 pts) Use the graph of below to find the following limits. If one does not exist, state why.

a. limx!"3

f x( ) = "3 b. limx!"2

f x( ) = "1 c. limx!"1

f x( ) dne

d. limx!0

f x( ) = 1 e. limx!2

f x( ) = "1

9. (1 pt) Check to make sure your Scantron form meets the following criteria. If any of the items are NOT satisfied when your Scantron is handed in and/or when your Scantron is processed one point will be subtracted from your test total. . . . rest removed for space

Work on Problem Points Awarded Left limit 1 Right limit 1 Equality of limits for theorem 1 Notes: do not have to say “by the Sandwich Thm”


Correct answer (each part) 1 Notes: -0.5 for 1 instance of missing or inappropriate equals signs -1 for more than 1 instance

Documents

MthSc 103 Test 2 Spring 2009 Version A JIT 7.2 – 10.6; UC