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01:640:151 Study Guide - Final Guide: B Movie, Marsupial, List Of Association Football Teams To Have Won Four Or More Trophies In One SeasonExam


Department
Mathematics
Course Code
01:640:151
Professor
All
Study Guide
Final

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Math 151, Fall 2009, Review Problems for the Final Exam
Your final exam is likely to have problems that do not resemble these review
problems. You should also look at the review problems for the first two exams.
(1) Find the largest interval [a, b] such that sin x3 cos xfor all xin [a, b]. Find the
area of the region bounded by y= sin x,y=3 cos xbetween x=aand x=b.
(2) A continuous function f(x) on the interval [1,10] has the properties R8
1f(x)dx = 14,
R10
4f(x)dx = 7, R10
1f(x)dx = 2. Find R8
4f(x)dx.
(3) Evaluate Z(1 + x)(2 + 3x)dx ,Z2
3
2+3x2
xdx ,Z2
1|x1|dx.
(4) Find Zx2ex3+4 dx,Zsin xcos x dx,Ztan xsec2x dx.
(5) A bacterial population quadruples in size every 7 days. How many days does it take
for this population to triple in size?
(6) Explain why the function f(x) = n2x+ 1 if x1,
4x1 if x > 1is continuous but not differen-
tiable.
(7) A continuous function f(x) is defined by f(x) = |x|ln |x|if x6= 0,
aif x= 0, where ais a
constant. Find a. Is f(x) a differentiable function? Find the intervals where f(x) is
increasing and the intervals where f(x) is decreasing. Hint: Look at the case x0 and
use symmetry.
(8) Let f(x) be defined for x > 0 by f(x) = x2ln x. Find the intervals where f(x) is
concave up and the intervals where f(x) is concave down.
(9) Consider f(x) = x6
6x4
4. Find the local maxima, local minima and inflection points.
Find the intervals where f(x) is increasing, the intervals where it is decreasing, the intervals
where it is concave up and the intervals where it is concave down.
(10) Find lim
x→∞
ln x
x, lim
x→∞
x3
ex/10 , lim
x0
1cos(5x)
1cos(7x), lim
x0Rx
0sin(t2)dt
x3.
(11) Find the intervals where ex4is concave up and the intervals where it is concave
down.
(12) Find the absolute maximum of x5(1 x)4over the interval [0,1].
1
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