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Math 106B Exam 2 - Integration, Taylor Polynomials, Diff. Equations (Mar 9, 2007), Exams of Calculus

Birla Institute of Technology and ScienceCalculus

The march 9, 2007 exam for math 106b, which covers topics such as integration, taylor polynomials, and differential equations. The exam includes six problems, some of which involve integrating trigonometric expressions using reduction formulas and finding the maximum approximation error of a degree-2 taylor polynomial. Other problems require determining function values, finding tangent lines, and evaluating improper integrals.

Typology: Exams

2012/2013

Uploaded on 03/16/2013

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NAME:
Math 106B - Exam 2 - March 9, 2007
INSTRUCTIONS: Show all of your work and circle your solutions. Cross out any unnecessary work.
Calculators are allowed, but notes and books are not.
Reduction formulas for integrals involving trigonometric expressions are given on the last
page of this exam.
1. (10 points) Integrate the following. (Note: Your final answer should not contain any function compo-
sitions of a trigonometric function with an inverse trigonometric function - for example, tan(arcsin x)
- as you can simplify these.)
Zx2
√9−x2dx.
pf3
pf4
pf5

Partial preview of the text

Download Math 106B Exam 2 - Integration, Taylor Polynomials, Diff. Equations (Mar 9, 2007) and more Exams Calculus in PDF only on Docsity!

NAME:

Math 106B - Exam 2 - March 9, 2007

INSTRUCTIONS: Show all of your work and circle your solutions. Cross out any unnecessary work. Calculators are allowed, but notes and books are not.

Reduction formulas for integrals involving trigonometric expressions are given on the last page of this exam.

  1. (10 points) Integrate the following. (Note: Your final answer should not contain any function compo- sitions of a trigonometric function with an inverse trigonometric function - for example, tan (arcsin x)
    • as you can simplify these.) (^) ∫ x^2 √ 9 − x^2

dx.

  1. (10 points) Let f (x) = x ln x, and let P 2 (x) be the degree-2 Taylor polynomial for f (x) centered at x 0 = 1. (Note: You do not need to find P 2 (x).) According to Taylor’s theorem, what is the maximum approximation error committed by P 2 (x) on the interval [0. 5 , 1 .75]? (Note: Explain where your value of Kn+1 comes from.)
  1. (10 points) Use a comparison to determine whether the following improper integral converges or di- verges. (^) ∫ ∞

1

x + 5x^3

dx.

  1. (10 points) Evaluate the following integral, or explain why it diverges.

∫ (^3)

2

x √ 3 − x

dx.

  1. (10 points) Find all solutions to the differential equation

y′^ =

x ln x y

y

1 2 3 4 5 6 TOTAL (out of 60)