Download AB Calculus Summer Paper

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AB Calculus Summer 2009 Paper
Guidelines
1.
Due Date: The first "hour" period of the first cycle of school.
Late work is not acceptable, but the paper may be handed into the switchboard at any
time during the summer.
2.
All work must be neat (some will type), accurate, and complete. In general, show all
your work. Justify steps when appropriate. Show pictures/graphs when appropriate.
Spell correctly, label axes, and include units (always to three decimal place accuracy).
3.
Commence each problem with a statement of the problem. (This can be cut and pasted.)
This also implies that the work is not done on the hand-out, but on separate paper.
4.
This will be the first evaluation of your AB course and will constitute a significant
portion of the first term grade.
5.
Plagiarism is defined as the sharing/ copying of written work. You are encouraged,
however, to form study groups and to discuss problems, without limitation, amongst
yourselves. Papers, however, will be evaluated for identical work and conclusions.
6.
Except where noted, you may use your calculator freely.
7.
Numerous hours are required to successfully complete this paper. Please manage your
time carefully.
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1. The population of fish in a certain lake at time t months is given by the function
20, 000
, where t ≥ 0. There is an upper limit on the fish population due to the oxygen
P(t ) 
t

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1  24e
supply, available food, etc.
a) What is the initial population of fish?
b) When will there be 15,000 fish?
c) What is the maximum number of fish possible in the lake?
2. Find the exact values of sine, cosine, and tangent without using a calculator of each of the
following radian measures.
a)

23
6
b)
9
2
d)

10
3
e)

25
4
c)
3
f)

3
2
3. The population of a southwesten “sunbelt” state is growing and its population equals
S  1,500, 000(20.02t ) where t is in years, whereas a northeastern “smokestack” state is losing
population and its population is given by N  8, 000, 000(20.03t ) . In how many years will the
two states have the same population?
4. For the graph y = f(x) shown at the right, arrange the following numbers in ascending
(i.e., smallest to largest) order.






The slope of the graph, f(x) at A.
The slope of the graph, f(x) at B.
The slope of the graph, f(x) at C.
The slope of the line AB.
The number 0.
The number 1.
f(x)
B
C
A
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5. The Richter Scale that measures the intensity of earthquakes is logarithmic. Thus if quakes of
strength S1 and S2 measure m1 and m2 on the scale, then the difference in the scale readings is the
S
logarithm of the ratio of the quakes’ strengths, i.e., m2 – m1 = log 2 . Suppose a very small
S1
earthquake measured 1.5 and the largest on record measured 8.5. How much more powerful was
the larger quake than the smaller one?
f ( x  h)  f ( x )
.
h
Find the difference quotient for the function f ( x)  2 x  x3 and express it as a single
fraction in lowest terms.
6. Given the Difference Quotient:
a)
b)
Use the difference quotient in part (a) to determine the average rate of change of
f(x) as x changes from 1 to 1.1, from 1 to 1.01, and from 1 to 1.001. Estimate the
instantaneous rate of change of f(x) at x = 1.
7. Water is being poured into the container shown at a constant rate (measured in volume/unit
time). Sketch a graph of the depth h of the water as a function of time t.
1

8. Given f ( x)   100  x 2
 x  10

x  10
 10  x  10
x  10
a) Sketch a graph of f(x)
b) Solve for x: f(x) = 8
c) For what x is f continuous?
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9. Find possible formulae for the following graphs.
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10. Write a function that models the given graph:
11. Determine the domain of the function f ( x) 
x( x  2)
x2  4
 3x 
3 x2  1  x 

x2  1 

12. Consider the function f ( x) 
x2  1
a) Simplify f(x) as much as possible.
b) What is the domain of f?
c) What (if any) are the zeros of f?
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13.
Temperature readings T (in degrees Celsius) were recorded every hour starting at
midnight on a day in April in Whitefish, Montana. The time x is measured in hours from
midnight. The data are given in the following table.
x (h)
0
1
2
3
4
5
6
7
8
9
10
11
12
a)
b)
c)
T(oC)
6.5
6.1
5.6
4.9
4.2
4.0
4.0
4.8
6.1
8.3
10.0
12.1
14.3
x (h)
13
14
15
16
17
18
19
20
21
22
23
24
T(oC)
16.0
17.3
18.2
18.8
17.6
16.0
14.1
11.5
10.2
9.0
7.9
7.0
Sketch a graph of T vs x.
Find the average rate of change of temperature with respect to time
i)
from noon to 3 p.m.
ii)
from noon to 2 p.m.
iii)
from noon to 1 p.m.
Estimate the instantaneous rate of change at noon.
14. A weight hanging from a spring is set in motion moving up and down. Its distance (in cm)
above or below the equilibrium point at time t seconds is given by d  5(sin 6t  4 cos 6t ) . At
what time during the first two seconds is the weight at the equilibrium position (d = 0)?
15. The curve y  ab x  c contains the points (1, 5), (2, 11) and (3, 35). Find the values
a, b, and c algebraically. Show all work for credit.
16.
a) Sketch a graph of f(x) = [x] where [x] is the greatest integer function.
b) Then sketch on a second plane, g ( x) 
[ x]
for 0 < x ≤ 4.
x  [ x]
17. Solve for x: |ln x| > 1. Use exact values and justify your answer.
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18. The number of deer P at any time t (in years) in a federal game reserve is given by
P(t ) 
800  640t
20  0.8t
a) Find the number of deer when t = 15, 70, 100.
b) Find the horizontal asymptote of the graph y = P(t)
c) According to the model, what is the largest possible deer population and why?
19. Use factoring, the quadratic formula, angle formulas or identities to solve the equations
below. Find all solutions on the interval 0, 2  .
a) 3sin 2 x  cos 2 x  0
b) tan 2 x  sec x  1
d) cos 2x  sin x  0
e) 2(sin x  cos x)  sec x
20. Let f (tan  ) 
c) 2sin 2x  tan x
2 tan 
.
1  tan 2 
 1 
a) find f 

 2
b) If f has an inverse, find it. Show all work. Otherwise, explain why the inverse
doesn't exist.
c) Show that f (tan   cos  ) 
1
21. Simplify:
1
1
1
1
1  sec 2 x
2 tan 
cos 