Spring 99, MTH 131  Practice Problems for Exam 3 updated 4/19
Complete all 10 questions. Be careful to show your work and label
all graphs.
1. Consider a swimming pool that has one end deeper
than the other. Water is being pumped into the swimming pool. Let r = f(t)
represent the rate in inches per hour at which the depth of the water in
the deep end is increasing. Write a definite integral expressing the depth
of the water in the deep end after the first 3 hours.
2. Let f(x) = 3x^{2} 1 be a parabola and
g(x) = 2x be a line.
a Sketch graphs of f(x) and g(x)
on the same set of axes and shade in the area bounded by the yaxis, f(x)
and g(x).
b Express this area as a definite
integral.
3. A graph of y = f(x) is given below. Estimate
4. Let
Is f(x) monotonic on the interval 2 £
x £ 4? Using the Riemann Sums program
on your graphing calculator, compute the value of the following definite
integral, accurate to 2 decimal places. Say how many subdivisions you use.
5. When a certain car was tested to see how fast it could
go, the following velocity data was recorded. Assume the velocity is increasing.

t(sec) 
0.0 
0.5 
1.0 
1.5 
2.0 
2.5 
3.0 
3.5 
4.0 
4.5 
v(mi/hr) 
0.0 
15.0 
23.7 
31.0 
40.5 
48.1 
54.1 
59.3 
63.3 
67.0 

Calculate an underestimate for the distance taveled by the car between
times t = 1 and t = 3.
b Calculate an overestimate for the distance traveled
by the car between times t = 2.5 and t = 4.5
6. Let f(x) = 2x^{5}  ^{5}/_{3}
x^{3}. Use the second derivative to determine the exact values
of the inflection points of f(x). Be sure to justify your answer.
7. Let f(x) = x^{2}  e^{x} + 2x.
a Find all critical points of
f(x).
b Use the first derivative test
to find all local extreme values.
8. Let f(x) = x^{3}  ^{3}/_{2}
x^{2}  18x + 4.
a Find all critical points of
f(x).
b Use the second derivative test
to determine local minima and local maxima.
9. Evaluate the following derivatives.
a Find f¢(t)
when
f(t) = (t^{3}  2t) 

_____
Ö1  t^{2} 
. 

b Find g¢(s)
when
10. Suppose the revenue earned from a certain product
is R =  p^{2} +2p +599 where p is the unit price and R is in dollars.
What price should be charged to earn the highest revenue? Hint: find the
global maximum.
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