# 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

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**preview**shows half of the first page. to view the full**2 pages of the document.**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 x≥√3 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 ,Z−2

−3

2+3x2

xdx ,Z2

−1|x−1|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 x≤1,

4x−1 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 x≥0 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

6−x4

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

x→0

1−cos(5x)

1−cos(7x), lim

x→0Rx

0sin(t2)dt

x3.

(11) Find the intervals where e−x4is concave up and the intervals where it is concave

down.

(12) Find the absolute maximum of x5(1 −x)4over the interval [0,1].

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