MATH 139 Midterm: MATH 139 McGill Examf02
15 Feb 2019
School
Department
Course
Professor
Math 106 Fall 2013
Final Exam (75 points)
Name:
Show all your work to receive full credit for a problem and keep your written answers
brief and clear. Points will be taken off if you do not show how you arrived at your
answer, even if the final answer is correct.
Do not use the calculator integral function. Whenever possible, find the exact values
of integrals by finding antiderivatives or using the table of integrals. When you use a
formula from the table of integrals, mention the formula number and the value(s) of
any constant(s) that you may need.
Give exact answers. If needed, round off your answers to four decimal places.
There are twelve questions on five pages. Questions are printed on both sides of a
page.
You may use any of the following facts:
Arclength=Zb
aq1 + (f′(x))2dx Zu dv =uv −Zv du
|I−Ln| ≤ K1(b−a)2
2n|I−Rn| ≤ K1(b−a)2
2n
|I−Tn| ≤ K2(b−a)3
12n2|I−Mn| ≤ K2(b−a)3
24n2
T(x) = f(x0) + f′(x0)(x−x0) + f′′(x0)
2! (x−x0)2+···+f(n)(x0)
n!(x−x0)n+···
|f(x)−Pn(x)| ≤ Kn+1
(n+ 1)! |x−x0|n+1
Z∞
1
1
xpdx converges for p > 1 and diverges for p≤1.
Z1
0
1
xpdx converges for p < 1 and diverges for p≥1.
∞
X
n=1
1
npconverges for p > 1 and diverges for p≤1.
sin x=x−x3
3! +x5
5! −x7
7! +···for xin (−∞,∞).
cos x= 1 −x2
2! +x4
4! −x6
6! +···for xin (−∞,∞).
ex= 1 + x+x2
2! +x3
3! +x4
4! +···for xin (−∞,∞).
1. (6 points) Solve the following initial value problem:
(ln x)2dy
dx =1
x, y(2) = 0.
2. (6 points) Evaluate the following integral exactly. (You may use formulas 1-18 only from the
table of integrals for this problem.)
Z3x2+x+ 23
(4 −x)(x2+ 9) dx
80
MATH 139 Full Course Notes
Verified Note
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Document Summary
Show all your work to receive full credit for a problem and keep your written answers brief and clear. Points will be taken o if you do not show how you arrived at your answer, even if the nal answer is correct. Whenever possible, nd the exact values of integrals by nding antiderivatives or using the table of integrals. When you use a formula from the table of integrals, mention the formula number and the value(s) of any constant(s) that you may need. If needed, round o your answers to four decimal places. Questions are printed on both sides of a page. You may use any of the following facts: Arclength=z b a q1 + (f (x))2 dx. T (x) = f(x0) + f (x0)(x x0) + f (x0) Z u dv = uv z v du. 0 dx converges for p > 1 and diverges for p 1. dx converges for p < 1 and diverges for p 1.
