New Answer Key Test 2 Version A Spring 2015 - Weebly · 2019. 11. 26. · Use a #2 pencil and completely fill each bubble on your scantron to answer each multiple choice question

MATH 1020 ANSWER KEY TEST 2 VERSION A SPRING 2015

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Printed Name: ______________________________ Section #: _______ Instructor:_________________ Please do not ask questions during this exam. If you consider a question to be ambiguous, state your assumptions in the margin and do the best you can to provide the correct answer. See page two for formulas, general directions, and calculator troubleshooting tips. • Any communication with any person (other than the instructor or the designated proctor) during this

exam in any form, including written, signed, verbal or digital, is understood to be a violation of academic integrity.

• All devices, such as computers, cell phones, cameras and PDAs must be turned off while the student is in the testing room.

• The only calculators to be used are TI-83, TI-83+, TI-84 or TI-84+. You may NOT borrow or share a calculator with another person taking this test.

• Statement of Academic Integrity: I have not and will not give or receive improper aid on this test.

In signing below, I acknowledge that I have read, understand, and agree to these testing conditions. Student’s Signature: _________________________________________________________________ (This test will not be accepted for grading unless it bears the signature of the student.)

FR#1 FR #2 FR #3 FR #4 FR #5 scantron Free Response

Total

Multiple Choice Total

Total

Possible points

6 6 5 5 9 1 32 68 100

Points Earned

Useful Formulas: ( )

0

( )( ) lim

h

f x h f xx

hf

→

+ −=′ ;

( ) ( )( ) lim

x a

f x f af a

x a→

−′ =−


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General Directions: • Show work where possible. Answers without supporting work (where work is appropriate) may

receive little credit.

• Do not round intermediate calculations. • Answers in context ALWAYS require units.

• Round your answers to 3 decimal places UNLESS the answer needs to be rounded differently to

make sense in the context of the problem OR the directions specify another type rounding OR the complete answer has less than 3 decimal places.

• When you are asked to write a model, include all components of a model: an equation, a description

of the input including units, a description of the output including units, and the interval when known • When asked to write a sentence of practical interpretation, answer the questions: when?, what?,

and how much? using ordinary, conversational language. DO NOT use math words, terms, or unnecessary phrases.

• Always use a ruler when estimating values off of a graph.

HINTS FOR TROUBLESHOOTING YOUR CALCULATOR:

• If you lose your L1, L2, etc., you may reinsert them using STAT 5 (set-up editor) enter.

• The SCATTER PLOT will not show unless Plot 1 has been turned on and there is data in L1 and L2.

• ZOOM 0 may not work for graphing if Plot 1 is turned on.

• IF you set the entire x and y window range, use the GRAPH button to see the graph.

• DIM MISMATCH error usually means that the lists in L1 and L2 are not of equal length.

• DATA TYPE error usually means that you already have something in Y1 and you need to clear it before you can paste a new equation.

• INVALID DIM error usually means that your plot(s) are on, but that you have no data in the lists. Refer to the second hint above.

• If your batteries die, raise your hand and hold up your calculator. If your instructor has an extra calculator available, he/she will loan it to you for a few minutes.

• SYNTAX ERROR: Try GO TO. This will happen if you use a subtraction minus sign when you should use a negative sign.


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MULTIPLE CHOICE: 68 points (3 points each unless otherwise noted) B A A D C| B C D D C| C B D B B| A C D C A| D B A C C| A B Use a #2 pencil and completely fill each bubble on your scantron to answer each multiple choice question. (For future reference, circle your answers on this test paper.) There is no penalty for guessing on multiple choice. If you indicate more than one answer, or you leave a blank, the question will be marked as incorrect.

3. Given ( ) 2 xf x e= , then '( )f x = _______.

4. Given ( ) 8xf x = , then '( )f x = _______.

a. 2 xe b. 12 xxe − c. 2ln( )x d. 2e

a. ln( )8xx b. 1(8 )xx − c. ln(8 )x d. (ln8)8x

2. Using the graph to the right, put the points in order from least steep to most steep slope: [2 points]

a. B, C, A b. A, B, C

c. B, A, C d. C, B, A

1. Identify the point(s), on the graph to the right, at which there is a correctly drawn tangent line. [2 points]

a. B only b. A and B only

c. A, B, and C d. D only


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5. Given ( ) 4 ln( )f x x= , then '( )f x = ___________.

6. Given

33( )

x xh x

x

−= , complete the following ( ) _________ .h x′ =

7. Given 3

1( )

7f c

c= , complete the following _________ .

df

dc=

8. Given ( )g z z z= + , complete the following ( ) _____________ .g z′ =

9. 3( ) 10.2 ln( )xS x e x π= + − hundred units gives a company’s cumulative sales for a new product, x years after the product was introduced to the market in 2000, 2 5.x≤ ≤ Fill in the blank to complete the following sentence: In 2003, the company’s cumulative sales were increasing by _________ hundred units per year.

a. 4 ln( )x b. 4 xe c. 4

x d. 4x

a. 23 x− b. 2x− c. 23 3x− d. 6x−

a. 421c−

− b. 2

1

21c c. 43

7c−

− d. 23

7c−

−

a. 1

2z z+ b. 1

2z

z+ c. 1

12

z+ d. 11

2 z+

a. 175.597 b. 174.965 c. 41.606 d. 205.206


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10. Use the information in the tables below to determine the percentage rate of change for the function

( )f x at 3.x = [2 points]

11. Suppose that f is a continuous function with ( 1) 1f ′ − = − , (0) 0f ′ = , and (1) 1f ′ = . Which one of the following graphs satisfies these conditions and could be a graph of f ?

a. b.

c. d.

x ( )f x ( ) (3)

3

f x f

x

−

−

x ( )f x ( ) (3)

3

f x f

x

−

−

2.9 8.410 5.900 3.1 9.610 6.100 2.99 8.940 5.990 3.01 9.060 6.010 2.999 8.994 5.999 3.001 9.006 6.001 2.9999 8.999 6.000 3.0001 9.001 6.000 2.99999 9.000 6.000 3.00001 9.000 6.000

a. 6.000 b. -33.333 c. 66.667 d. 150.000


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12. Which one of the following graphs is the slope graph for the graph of ( )f x shown below?

a. b.

c. d.


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Use the following to answer the next two questions. [2 points each]

The graph below shows the height ( )f x of a jumping frog, x seconds after jumping.

Use the table below to answer the following questions.

13. The value -3.984 in the darkened cell in the table represents the ____________________________. a. instantaneous rate of change in the frog’s height 0.499 seconds after jumping

b. average rate of change in the frog’s height between 0.499 and 0.501 seconds after jumping

c. instantaneous rate of change in the frog’s height 0.5 seconds after jumping

d. average rate of change in the frog’s height between 0.499 and 0.5 seconds after jumping 14. How quickly is the frog’s height above ground changing 0.5 seconds after it jumped?

15. Complete the following sentence: The ___________________ on a continuous and differentiable function can be calculated by finding the _______________________ between a point and increasingly close points on the graph of the function. a. slope of a secant line at a point; slope of the tangent line b. slope of the tangent line at a point; slope of the secant line c. derivative at a point; slope of the tangent line d. average rate of change at a point, slope of the secant line

a. 2.200 feet b. -4.000 ft/sec c. 0.500 ft/sec d. -4.016 ft/sec

0.5x −→ ( ) (0.5)

0.5

f x f

x

−

−

0.5x +→ ( ) (0.5)

0.5

f x f

x

−

−

0.499 -3.984 0.501 -4.016 0.4999 -3.998 0.5001 -4.002

0.49999 -4.000 0.50001 -4.000

0.499999 -4.000 0.500001 -4.000

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.5

1

1.5

2

2.5

x seconds

Height f (x), in feet


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Use the following to answer the next three questions.

( )f x thousand fish gives the population of a certain species of fish in a lake, x months after the species of fish was introduced into the lake, 1 7x≤ ≤ . Given: (2) 1021.175, (2) 1176.435, (5) 8839.899, (5) 1929.250f f f f′ ′= = = =

16. Which one of the following is the best sentence of interpretation for (2) 1176.435f ′ = ? a. Two months after being introduced into a lake, the population of a certain species of fish was

increasing by 1176.435 thousand fish per month. b. Two months after being introduced into a lake, the population of a certain species of fish was

1176.435 thousand fish. c. Two months after being introduced into a lake, the population of a certain species of fish increased

on average by 1176.435 thousand. d. Between two and five months after being introduced into a lake, the population of a certain species

of fish increased on average by 1176.435 thousand fish per month. 17. Find the percentage rate of change in the certain species of fish population, two months after it was

introduced into the lake. [2 points]

18. What are the units for percentage rate of change in the previous question? [2 points]

a. 2606.241 b. 765.660 c. 115.204 d. 88.448

a. thousand fish b. thousand fish per month

c. percent d. percent per month


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Use the following graph to answer the next two questions.

A portion of the graph of function ( )f x is shown below.

19. Carefully draw a tangent line to the graph at ( )2, 1.5− − and calculate its slope. Which one of the following is closest to the slope of the tangent line?

20. Complete the graph using the following information:

• The function is decreasing most quickly at the point 0x = . •

1

0x

df

dx=

=

• 2

0x

df

dx=

>

a. b.

c. d.

a. 0.33 b. 1.25 c. 3.0 d. 4.5


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Use the following to answer the next four questions.

( )f x gives the month of April’s average atmospheric temperature at the equator, in kelvin (K), where x is the number of kilometers (km) above sea level, 0 92x≤ ≤ .

21. Locate the point on the above graph at which atmospheric temperature at the equator during the month of April is increasing the most rapidly. Which one of the following calculations should be used to determine how quickly the atmospheric temperature at the equator is changing at that point?

a. The slope of the graph at 45 km. b. The function value at 45 km. c. The slope of the line connecting the points between 30 and 70 km above sea level. d. The slope of the tangent line at 30 km above sea level. 22. On which interval(s) will the slope graph of ( )f x have only positive values? [2 points]

23. On which interval(s) will the slope graph of ( )f x be increasing? [2 points]

24. On the interval (0, 92), how many relative minimum and maximum points in all will there be on the slope graph of ( )f x ? [2 points]

a. (0,30), (70,92) b. (15,45), (85,92) c. (30,70) d. (0, 92)

a. (0,30), (70,92) b. (15,45), (85,92) c. (30,70) d. (0, 92)

a. 0 b. 1 c. 2 d. 3


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Use the following to answer the next three questions. [2 pts each]

( ) 17.22(0.754 )tP t = billion cod gives the population of Northern Canadian cod, where t is the number of decades since 1963, 0 3.t≤ ≤ These next three questions involve writing a best rate-of-change model.

25. Begin writing a rate of change model. The sentence will continue in subsequent questions.

26. Continue writing a rate of change model. The sentence will continue in the next question.

27. Complete the rate of change model. a. … gives the population of Northern Canadian cod, where t is the number of decades since 1963,

0 3.t≤ ≤ b. … gives the rate of change in the population of Northern Canadian cod, where t is the number of

decades since 1963, 0 3.t≤ ≤ c. …gives the population of Northern Canadian cod decreasing per decade since 1963, where t is the

number of decades since 1963, 0 3.t≤ ≤ d. … gives the average rate of change in the population of Northern Canadian cod, where t is the

number of decades since 1963, 0 3.t≤ ≤

End of Multiple Choice

a. 1( ) 17.22( )(0.754 )tP t t −′ = b. ( ) 17.22(0.754 )tP t′ = c. ( ) 17.22(ln 0.754)(0.754 )tP t′ = d. ( ) 17.22(ln )(0.754 )tP t t′ =

a. …billion cod per decade… b. … billion cod since 1963…

c. …decades per billion cod… d. …billion cod per year…


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FREE RESPONSE: 32 points

Show work where possible. Read the directions on rounding, inclusion of units, and writing models and sentences at the front of the test. 1. Find the derivative for each of the given functions. Simplify exponents and coefficients, and use exact

numbers. For full credit, include proper use of derivative notation and equal signs. ( )_______ / 6pts

a. 2 3( ) 1.5 8f x x x e= − +

( ) 3 8f x x′ = −

b. 1123( ) 6 8f x x x−

= +

2 33 2( ) 2 4f x x x

− −

′ = −

2. The function ( )f t relates the optimal angle tilt toward the South, in degrees, of a solar panel in Logan, Utah on June 21st, t hours after midnight, 6 21t≤ ≤ .

a. Graphically estimate the rate of change in the optimal angle tilt of a solar panel at 2:30 pm, on June 21st

in Logan, Utah. Show work and round the answer to three decimal places.

28.62 19.572.586

18 14.5

−=

−

b. Write a sentence of interpretation for the rate of change found in part (a). At 2:30 pm, the optimal angle tilt of a solar panel in Logan, Utah on June 21st is increasing by 2.586 degrees per hour ( )_______ / 6pts

a. 3.5 pts b. 2.5 pts

1 pt input; 1 pt output description; 1 pt “increasing by”; 2 pt slope; 1 pt units


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3. Pest-Buster is a company working to remove mice from homes in a town where there has been an infestation. Suppose that

3 2

4 2

100 ln( )( )

( 2 6)

x xf x

x x

⋅ −=

+ − mice gives the average number of mice in a home,

x years after 2000, 2.5 6.x≤ ≤ checkpoint: (3) 3.108f =

a. Use the numeric method to determine how quickly the average number of mice in a home is changing in 2003. Fill in the table below with entries rounded correctly to THREE decimal places. (Answers will be marked wrong if not rounded to three decimal places.)

x → 3− f (x)− f (3)

x − 3 x → 3+ f (x)− f (3)

x − 3 2.9 -2.904 3.1 -2.618

2.99 -2.770 3.01 -2.741 2.999 -2.757 3.001 -2.754

2.9999 -2.756 3.0001 -2.756 2.99999 -2.756 3.00001 -2.756

b. Using the numeric method, how quickly is the average number of mice per home is changing in 2003? Round the answer to three decimal places and include units with the answer.

-2.756 mice per year

( )_______ /5pts

4. Consider the graph of ( )f x below:

a. This graph has a vertical tangent. Draw the vertical tangent directly on the graph, using a straight-edge. b. The instantaneous rate of change does not exist at __ __1x = and __ __6x = .

c. A tangent line drawn at 3x = lies above /below (circle one) the graph of ( )f x . Its slope is positive/negative/zero/does not exist (circle one). ( )_______ /5pts

3 pts entries in table, correct to three decimal places 1 pt slope, correct to three decimal places 1 pt slope units

a. 1 pt correctly drawn vertical tangent at x=1 b. 1 pt for each blank c. 1 pt each circle


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5. Use the limit definition of the derivative to find the derivative of 2( ) 5 3 2f x x x= − + . (Check your answer using simple rate of change formulas.)

For full credit, continue from the general limit definition (provided below), clearly showing all necessary algebraic steps (cancellations, expansions, etc.) and including proper use of notation and equal signs.

0

( ) ( )( ) lim

h

f x h f xf x

h→

+ −′ =

2 2

0

2 2 2

0

2 2 2

0

2

0 0

0

[5( ) 3( ) 2] [5 3 2]lim

[5( 2 ) 3( ) 2] [5 3 2]lim

5 10 5 3 3 2 5 3 2lim

10 5 3 (10 5 3)lim lim

lim(10 5 3)

10 3

, ( ) 10 3

h

h

h

h h

h

x h x h x x

h

x xh h x h x x

h

x xh h x h x x

h

xh h h h x h

h hx h

x

Thus f x x

→

→

→

→ →

→

+ − + + − − +

+ + − + + − − +=

+ + − − + − + −=

+ − + −= =

= + −

= −′ = −

=

( )_______ /9pts 6. 1 point for correctly filling out and bubbling the scantron with a #2 pencil, a correct XID, a correct test

version, AND the front of the test is completed with your signature on the academic integrity statement. END OF TEST

3 pts: Find slope of secant using given function: [f(x+h) –f(x)]/h 1 pt: Square (x+h) correctly 2 pts: Distribute 5, -3, 2 and the -1 (the minus sign) correctly 1 pt: Combine like terms and show the result; factor out “h” and “cancel”; then show the limit of a

completely simplified expression 1 pt: Evaluate limit of simplified expression to find derivative. ½ pt correctly written limit notation throughout proof ½ pt: equal signs in correct places and used throughout proof

Documents

New Answer Key Test 2 Version A Spring 2015 - Weebly · 2019. 11. 26. · Use a #2 pencil and completely fill each bubble on your scantron to answer each multiple choice question