APPM 1350 Forgiveness exam III Spring 2007
Books, notes and electronic devices are not permitted. Write your (1) name, (2)
instructor’s name and (3) recitation number on the front of your bluebook. There
are 5 problems of 20 points each, plus a 20-point extra credit problem. Show your
work clearly and box your answers. A correct answer with incorrect or no supporting
work may receive no credit, while an incorrect answer with relevant work may receive
partial credit.
(1) (a) Estimate the integral Z3
0
√x+ 1dx using n= 6 rectangles and left-
hand end-points (you do not need to simplify your answer).
(b) Estimate the same integral using the trapezoidal sum for n= 6 (you do
not need to simplify your answer).
(c)How large do you have to make nto be sure that the corresponding
trapezoidal estimate is within 10−4of the real value of the integral?
Hint: The trapezoidal sum and error bound for the trapezoidal rule are
Tn=h
2(y0+ 2y1+... + 2yn−1+yn)
|ET| ≤ b−a
12 h2M
(2)Find the total area of the two regions enclosed between the graph of the
function y=x2−1 on the interval [0,2] and the x-axis.
(3) Calculate the integrals:
(a)Z3
√t+ 1
√tdt
(b) Z(1 + √x)3
√xdx
(c)Z√7
0
t(t2+ 1)1/3dr
(4) (a) State the Fundamental Theorem of Calculus (both parts).
(b) Calculate the derivative of the function h(x) = Z√x
0
cos θdθ.
(c) Find the critical points of the function hgiven in part (b).
(5) (a) Find the average value of the function f(x) = 3x2−3 on the interval
[0,1].
(b) The Mean Value Theorem states that there is a value x=csuch that
f(c) equals the average value of fin over some interval. Find this value c
for the function and interval in part (a).
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