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Name: ______________________________________ CU ID#: ________________________
Section #: ____________ Instructor’s Name:_________________________________________
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. Statement of Academic Integrity: I have not and will not give or receive improper aid on this test.
Signature: _____________________________________________
Free Response Problem
Possible Points
Points Earned 1 4
2 4
3 5
4 4
5 6
6a 6
6b 4
6c 3
7 1
Free Response Total
Multiple Choice Total
TOTAL SCORE 100
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.
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.
Use the graphs of f ( x ) and g t ( ) shown below to answer questions 5 and 6.
- Which ONE of the following statements is true?
a. The tangent line drawn at the point (1,2 )on f ( x )will lie above the graph of f ( x )
and the tangent line drawn at the point ( 1,2 )on g t ( ) will lie above the graph of g t ( ).
b. The tangent line drawn at the point (1,2 )on f ( x )will lie above the graph of f ( x )
and the tangent line drawn at the point ( 1,2 )on g t ( ) will coincide with the graph of g t ( ).
c. The tangent line drawn at the point (1,2 )on f ( x )will lie above the graph of f ( x )
and the tangent line drawn at the point ( 1,2 )on g t ( ) will lie below the graph of g t ( ).
d. The tangent line drawn at the point (1,2 )on f ( x )will lie below the graph of f ( x )
and the tangent line drawn at the point ( 1,2 )on g t ( ) will coincide with the graph of g t ( ).
- Given f (1) = 2 and g (1) = 2.
Suppose that you use f '(1) to estimate f (2) and g '(1) to estimate g (2).
Which ONE of the following statements is supported by the graphs shown above?
a. The estimate of f (2)will be too high and the estimate of g (2)will be accurate.
b. Both the estimate of f (2) and the estimate of g (2)will be too high.
c. The estimate of f (2)will be too high and the estimate of g (2)will be too low.
d. The estimate of f (2)will be too low and the estimate of g (2)will be accurate.
Use x t ( ) = 3 t + 3 e 5 t and f ( x ) = ln x to answer questions 7 and 8.
f ( x ) g t ( )
- Construct an equation for f (^) ( t (^) ) = f (^) ( x t ( ))using x t ( ) = 3 t + 3 e^5 t and f ( x ) = ln x.
a. (^) ( 3 t + 3 e^5 t )( ln t ) b. 3ln (^) ( x (^) ) + 3 e (^5 x ) c. ln 3( t + 3 e^5 t ) d. (^) ( 5 )
3 t 3 e t t
+ ^
- Find
df dt
a.
t 5 et
5 5
t t
e t e
c.
t
15 e x x
- The cost of printing flyers to advertise an upcoming concert is given by D n ( )dollars where
n is the number of flyers that are printed.
Give the notation for the marginal analysis that would estimate the additional cost to produce the 200th unit at this factory.
a. D ' 200( ) b. D ' 199( ) c. D ' 201( ) − D '(200)d. D (201) − D (200)
- Find the derivative of (^) ( ) 4
m x ( ) = 2 π + 4 x
a. ( )
3 16 2 π + 4 x b. ( ) 3 4 2 π + 4 x c. ( ) 3 8 π + 16 x d. ( ) ( ) 3
4 2 π + 4 x 2 π + 4
Use the following information to answer questions 11, 12, 13, and 14.
Given: x (3) = 10 , x '(3) = −2.1 , f (10) = 0.67, and f '(10) =1.
Find
df dt
at t = 3.
a. -2.52 b. 10.593 c. 2.52 d. -0.
Use the following information to answer 16, 17, and 18.
Given h x ( ) = f ( x ) ⋅ g x ( ); f (2) = 7 , f '(2) = 17 , g (2) = 4 , and g '(2) = −0..
- Evaluate h (2).
a. 28 b. 11 c. 24 d. 66.
- Evaluate h '(2).
a. 8 b. 66.25 c. -4.25 d. 24
- Estimate g (1.5)using g (2) = 4 , and g '(2) = −0..
a. 3.75 b. 3.875 c. 4.125 d. 4.
Use the following information to answer 19, 20, and 21.
The demand for a new toy x weeks after its introduction is given by D x ( ) = ln ( x )+ 3 thousand
toys.
The price of the new toy is (^) P x ( ) = −0.2 ln( x ) + 15 dollars per toy, x weeks after its introduction.
Recall that revenue = price ⋅ demand.
- Twelve weeks after the new toy’s introduction, the revenue was ___________thousand dollars.
a. 5.485 b. 1.168 c. 79.548 d. 14.
- Twelve weeks after the new toy’s introduction, revenue was increasing by _________________.
a. $ 1.029 thousand dollars per week b. 0.083 thousand toys per week
c. $ 1.117 thousand dollars per week d. 1.117 thousand toys per week
- Twelve weeks after the new toy’s introduction, the number of toys sold was increasing by __________ toys per week.
a. 83 b. 77 c. 83,000 d. 1.404%
END OF THE MULTIPLE CHOICE
2 (^5 1 4 1) (or 4 ) 2 2
x x h x x x x x
=^ −^ + = − +^ −
h x = − x −
Points earned Points possible 4
- P x ( ) = 41(1.103 ) x eagles gives the breeding population of female Golden Eagles near the Altamont Pass Wind Resource Area in California between 1997 and 2000, x years after
Write a completely defined rate of change model for P x ( ). Your answer must include the derivative equation, the units for the derivative, the description for the output and units, the description for the input with units, and the interval.
P '( x ) = 41ln(1.103)(1.103 ) x eagles per year
gives the rate of change in the breeding population of female Golden eagles
between 1997 and 2000
where x is the number of years after 1997.
Points earned Points possible 6
Okay to use quotient rule. award ½ pt for 1
4 (or ) 2 2
x x x
− + − if that’s all they
did
deduct ½ pt for missing f '( x )if rest is correct
1 point per term
Equation and underlined words (or similar) should appear in the answer. 2 pts for equation; deduct ½ for no (^) P '( x ) ; 41ln(1.103) is worth 1 pt;^ (1.103 ) x^ is worth 1 pt
eagles per year is worth 1 pt (A/N) rate of change in eagles is worth 1 pt (A/N)
between 1997 and 2000 (or similar) is worth 1 pt x years after 1997 (or similar) is worth 1 pt
-1/2 if parentheses error -1 if rewrite is wrong but then took derivative correctly based on rewrite -1/2 if no f'(x)
1 pt per term if used product rule or quotient rule as in:
( )(^ )
( ) ( ) (^) ( ( ) )
2 1
1 2 2
( ) 5 8 2
'( ) (10 1) 2 5 8 2 2
h x x x x
h x x x x x x
−
− −
= − + →
= − + − + −
-2 pts for the wrong derivative (1pt for 41 ln(1.103) and 1 pt for (1.103^x); -1/2 if no P'(x) = ; -1/2 if they miswrote a number in the derivative; -1 if they take the derivative but don't put it in the model correctly -1 if the mix up the units and the output description inaccurately; -1 if units are not eagles per year (-1/2 if they had population per year) -1 if there is no rate of change (-1 if they only have change or average rate of change) -1 if there is no between 1997 and 2000 (or 0<=x<=3) (-1/2 if only one is incorrect); -1 if 1997 <= x <= 2000 -1 if there is no input description (x years after 1997)
a. Use the information above to find the value for each of the following notations. Give the units with each answer. Where appropriate, you may circle the correct choice from the options given.
C (4) = __________________ ____________________________________________
4
_________________
p
dC dp (^) =
= ____________________________________________
2012
________________
t
dp dt (^) =
= ____________________________________________
p (2012) = _______________ ____________________________________________
Points earned Points possible 6
b. A function to find the level of carbon monoxide (CO) in the air as a function of time can be
constructed using ____________. ( addition, subtraction, multiplication, composition .)
In 2012, the level of CO in the air will be ___________( increasing , decreasing ) by
_______________ ( number value ) _____________________________( roc units ).
Points possible 4
- An environmental study in a suburban community found that the average daily level of carbon monoxide could be modeled by C ( p ) parts per million (ppm), where p thousand people was the population.
- The projected population of the suburban community was modeled by p ( t ) thousand people in year (^) t.
- In 2012, the population was projected to be 4.0 thousand and at that time it will be increasing by 0.323 thousand people per year.
- In 2012, when the population was projected to be 4.0 thousand, the daily level of carbon monoxide in the air was projected to be 4.635 ppm and at that time the level will be increasing by 0.6 ppm per thousand people.
4.635 (1 pt) ppm (0.5 pt)
0.6 (1 pt) ppm/thousand people (0.5 pt) A/N
0.323 (1 pt) thousand people/year (0.5 pt) A/N
4 (1 pt) thousand people (0.5 pt)
0.1938 (0.194) ppm/year
1 pt per box A/N
