Dr. Zs Calc2 Handout for Lecture 10: Taylor Polynomials
By Doron Zeilberger
Problem Type P10.1: Find the n-th Maclaurin polynomial for f (x) using the denition.
Example Problem P10.1: Find the 5-th Maclaurin polynomial for f (x) = sin x using the denition
Answers to Dr. Z.s Practice Final For Fall 2012 (Sections 1-3, 7-9)
Disclaimer: Not responsible for any errors. One dollar prize for the rst discoverer of any error.
Version of Dec. 10, 2012, 11:30am (thanks to Eunhee Kim who won a dollar [correcting a
ty
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MATH 152 (01-03, 07-09), Fall 2012, Dr. Z.s , Practice Final
WRITE YOUR FINAL ANSWER TO EACH PROBLEM IN THE INDICATED PLACE (right under the question) (when applicable)
Version of Dec. 10, 2012, 11:00am, correcting
Solutions to Attendance Quiz # 0 for Dr. Z.s Calc2 for Sept. 6, 2012
1. Find f (x) if f (x) = x2 (cos x)ex
Sol. to 1:
First Way: By the product rule for three functions
(f gh) = f gh + f g h + f gh
,
we have
f (x) = (x2 (cos x)ex ) = (x2 ) (cos x)ex +x2 (
MATH 152 (01-03), Dr. Z. , Full Solutions to First Midterm, Thurs. Oct. 18,
2012.
1. (10 points [5 each]) Find the following indenite integrals
(a)
x2 ex dx
;
Ans. to (a): (x2 2x + 2)ex + C
We use integration by parts with u = x2 , v = ex . So u = 2x, v =
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MATH 152 (01-03), Dr. Z. , First Midterm, Thurs. Oct. 18, 2012.
WRITE YOUR FINAL ANSWER TO EACH PROBLEM IN THE INDICATED PLACE (right under the question) (when applicable)
Explain your work! Do not write below this
Dr. Zs Calc2 Handout for Lecture 20 [Solving Dierential Equations]
By Doron Zeilberger
Problem Type P21.1: Show that y = f (x) is a solution of the initial value problem
y + A(x)y = B(x) y(a) = b
Example Problem P21.1: Show that y = sin x cos x cos x is a
Dr. Zs Calc2 Handout for Lecture 14 [The Ratio and Root Tests]
By Doron Zeilberger
n=1
Problem Type P14.1: Determine whether the series
tionally convergent, or divergent.
an is absolutely convergent, condi-
Example Problem P14.1: Determine whether the fol
Dr. Zs Calc2 Handout for Lecture 18 [Parametric Equations and arclength and speed]
By Doron Zeilberger
Problem Type P18.1: Find an equaion of the tangent line to the curve at the point corresponding
to the given vale of the parameter
x = f (t) ,
y = g(t)
Dr. Zs Calc2 Handout for Lecture 13 [Alternating Series, Absolute and Conditional Convergence]
By Doron Zeilberger
Problem Type P13.1: Determine whether the series converges or diverges
(1)n bn
,
n=1
where bn is always positive (or always negative).
Examp
Dr. Zs Calc2 Handout for Lecture 12 [Convergence of series with positive terms]
By Doron Zeilberger
Problem Type P12.1: Use the integral test to determine whether the series is convergent or
divergent.
f (n)
n=1
Example Problem P12.1: Use the integral tes
Dr. Zs Calc 2 Handout for Lecture 17 [Arclength and surface area]
By Doron Zeilberger
Problem Type P17.1: Find the length of the curve
axb
y = f (x) ,
.
Example Problem P17.1: Find the length of the curve
0 x /3
y = ln(cos x) ,
.
Steps
Example
1. Find the
Dr. Zs Calc2 Handout for Lecture 16 [Taylor Series]
By Doron Zeilberger
Problem Type P16.1: Find the Maclaurin series for f (x) using the denition of a Maclaurin
series.
Example Problem P16.1: Find the Maclaurin series for f (x) = sin x using the denition
Dr. Zs Calc2 Handout for Lecture 15 [Power Series]
By Doron Zeilberger
Problem Type P15.1: Find the radius of convergence and interval of convergence of the series
bn (x + c)n
,
n=1
for some sequence bn and specic number c. (x is a generl symbol).
Example
Dr. Zs Math 152 Review Problems for Final Exam, Fall 2005
Due: Fri. Dec. 16, 4:00pm, 2005 (Bring to Final at Scott 123 along with notebook) 1. Let C be the curve y = x5 /5, with 0 x 3/4. It is rotated about the x-axis. (a) Set up an integral for the surfa
Answers to Dr. Z.s Second Practice Final For Fall 2012 (Sections 1-3, 7-9)
Disclaimer: Not responsible for any errors. One dollar prize for the rst discoverer of any error.
1. (a) 1 ln 2 . (b)
2
2
12
+ (
3
2
1)
2. divergent (by Limit Comparison Test and
Some Answers to Dr. Zs Math 152 Review Problems for Final Exam, Fall 2005
1. Cant give the answer without giving away the solution.
2. y = 3 x + 2 3 , y = 3 x + 2 + 3 ,
2
2
2
2
3. y = 8ex3 (x 2)/(x 1), 1 < x < .
4. (a) 1/2 (b) /4 (c) /2
1
5. (a) 2 ex (sin
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MATH 152 (01-03, 07-09), Fall 2012, Dr. Z.s , Second Practice Final
WRITE YOUR FINAL ANSWER TO EACH PROBLEM IN THE INDICATED PLACE (right under the question) (when applicable)
Explain your work! Do not write below t
Dr. Zs Calc2 Handout for Lecture 8: Improper Integrals
By Doron Zeilberger
Problem Type P8.1: Determine whether the following integral is convergent or divergent. Evaluate it if it is convegent.
f (x) dx
,
a
where f (x) is easy to integrate.
Example Probl
Dr. Zs Calc2 Handout For Lecture 11 (Part B) [Series]
By Doron Zeilberger
Problem Type 11B.1: Determine whether the series is convergent or divergent if it is convergent,
nd its sum
an
,
n=1
where lim an does not exist or exists but is not zero.
Example P
Dr. Zs Calc2 Handout for Lecture 5: Trigonometric Integrals
By Doron Zeilberger
Problem Type P5.1: Integrate an odd power of a sine or cosine.
sin2n+1 x dx
cos2n+1 x dx (n = 1, 2, . . .)
OR
Example Problem P5.1: Evaluate the integral
sin3 x dx
Steps
Examp
Dr. Zs Calc2 Handout For Lecture 2 [Volumes and Average of a Function]
By Doron Zeilbegrer
Problem Type P2.1: Find the volume of the solid obtained by rotating the region bounded by
the given curves about the x-axis
y = f (x), x = a, x = b, y = 0
Example
Dr. Zs Calc2 Handout for Lecture 6: Trigonometric Substitution
By Doron Zeilberger
Problem Type P6.1: Integrate expressions involving
a is some number.
a2 x2 , or
a2 + x2 , or
x2 a2 , where
Example Problem P6.1: Evaluate the integral
1 x2 dx
Steps
Example
Dr. Zs Calc2 Handout for Lecture 7
[Integration of Rational Functions by Partial Fractions]
By Doron Zeilberger
Problem Type P7.1: Evaluate the integral
C1 x + C2
dx
(x a)(x b)
,
where a, b, C1 , C2 are numbers.
Example Problem P7.1: Evaluate the integral
Solutions to MATH 152 (07-09), Dr. Z. , First Midterm, Thurs. Oct. 18, 2012.
1. (10 points [5 each]) Evaluate the following denite integrals:
(a)
1
xe2x dx
;
0
Ans. to (a): 3 e2 +
4
1
4
.
We rst evaluate the indenite integral, using integration by parts.
Dr. Zs Calc2 Handout for Lecture 9: Numerical Integration
By Doron Zeilberger
Problem Type P9.1: Use (a) The Trapezoid Rule, (b) The Midpoint Rule, and (c) Simpsons
Rule to approximate the given integral with the specied value of n
b
f (x) dx
,
n = EvenIn
Dr. Zs Calc2 Handout For Lecture 11 (Part A) [Sequences]
By Doron Zeilberger
Problem Type P11A.1: Determine whether the sequence
an = f (n)
(where f (x) is a nice function) converges or diverges. If it converges nd its limit.
Example Problem P11A.1: Deter
Dr. Zs Calc2 Handout for Lecture 3: Volumes by Cylindrical Shells
By Doron Zeilberger
Problem Type P3.1: Use the method of cylindrical shells to nd the volume generated by rotating
the region bounded by the given curves about the y-axis.
y = f (x) ,
y=0 ,
Dr. Zs Calc2 Handout for Lecture 4: Integration by Parts
By Doron Zeilberger
Problem Type P4.1: Evaluate the intgeral
Something(x) SomethingElse(x) dx
Example Problem P4.1: Evaluate the intgeral
x cos x dx
Steps
Example
1. Start with a blank table
1. Alwa