## 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) = 3x2 -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 y-axis, f(x) and g(x).

b    Express this area as a definite integral.

3.    A graph of y = f(x) is given below. Estimate

 ó õ 3 0 f(x)  dx.
4.    Let

f(x) =  1
 _____ Öx2 -1
.
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.
ó
õ
4

2

1
 _____ Öx2 -1
dx
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
1. 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) = 2x5 - 5/3 x3. 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) = x2 - ex + 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) = x3 - 3/2 x2 - 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) = (t3 - 2t) _____ Ö1 - t2 .
b    Find g¢(s) when

 g(s) = sin(s) e5s .
10.    Suppose the revenue earned from a certain product is R = - p2 +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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