Graph: f (x) =
x3 ln(1 + x2 )
One way to do this is to use the approximation: ln(1 + x2 ) x2 This is the first term of the Taylor series expansion of ln(1 + x2 ) centered at x = 0. Taylor series is super important. You'll see these show up again e
MATH 31B SECTION 2 FIRST PRACTICE MIDTERM
Please note: Show your work. Correct answers not accompanied by sufcent explanations will receive little or no credit. Please call one of the proctors if you have any questions about a problem. No calculators, com
MATH 31B SECTION 2
FINAL EXAM SOLUTIONS.
Please note: Show your work. Correct answers not accompanied by sufﬁcent explanations will receive
little or no credit. Please call one of the proctors if you have any questions about a problem. No calculators,
com
Unocial Math 31B Practice Final by TA Ning Khamsemanan Disclaimer: this practice exam is my attempt to help you study for the nal. Things in here might not be in the real exam and vice versa. Dont use this as your main study. There might be some typos and
Unocial Math 31B Practice Final by TA Ning Khamsemanan Disclaimer: this practice exam is my attempt to help you study for the nal. Things in here might not be in the real exam and vice versa. Dont use this as your main study. There might be some typos and
MATH 31B/2 PRACTICE FINAL EXAM SOLUTIONS. Please note: the aim of this practice nal is to give you several problems on the material not covered by the rst two midterms and practice midterms. You should therefore treat the rst two midterms and practice mid
wer QIfGP egsgi psxev iewF
the (rst two midterms nd prtie midterms s prt of the prtie (nlF
lese noteX the im of this prtie (nl is to give you severl prolems on the mteril not overed y the (rst two midterms nd prtie midtermsF ou should therefore tret
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MATH 31B SECTION 2 SECOND PRACTICE MIDTERM Problem 1. Evaluate the denite integral
0
1
2
sin2 x cos 2x dx
MATH 31B SECTION 2
SECOND PRACTICE MIDTERM
2
Problem 2. Find the limit
x
0
lim xx
MATH 31B SECTION 2
SECOND PRACTICE MIDTERM
3
Problem 3. Use a trigo
MATH 31B SECTION 2 SECOND PRACTICE MIDTERM Problem 1. Evaluate the denite integral
2 0
Hence
2 0
0
1
8
sin2 x cos 2x dx
2
1 cos 2x 2
1 4
sin2 x cos 2x
Using the identity cos2 2x
1 2
1
1 cos 2x 2
1 1 4
cos 4x
1 cos 4x dx 4
sin2 x cos 2x
1 1 cos 2x cos
MATH 31B SECTION 2 SECOND MIDTERM
Please note: Show your work. Correct answers not accompanied by sufcent explanations will receive little or no credit. Please call one of the proctors if you have any questions about a problem. No calculators, computers,
p595, number 29. Find the surface area of the ellipse y2 x2 + 2 =1 2 a b for a > b when rotated about x-axis. - Notice that this ellipse has the points (-a, 0) and (+a, 0) as its x-intercepts and (-b, 0), (+b, 0) as y-intercepts. Check this. So solve
Problem 2 on review sheet.
Graph f (x) =
x3 ln(1+x2 )
on (-, ).
First, we note that f (x) = 0 at x = 0 and that limx f (x) = , limx- f (x) = -. This is done by L'hopital's rule. This info will help draw the graph as x goes off to . Now, we need t
MATH 31B SECTION 2 FIRST MIDTERM
Please note: Show your work. Correct answers not accompanied by sufcent explanations will receive little or no credit. Please call one of the proctors if you have any questions about a problem. No calculators, computers, P