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MATH 106 Exam 3 Review: Taylor Series, Differential Equations, and Euler's Method, Exams of Calculus

National Institute of Industrial EngineeringCalculus

Key points of this past exam of Calculus are: Estimate, Taylor Polynomial, Second Degree, Estimate, Non Zero, Taylor Series, Summation Notation, Complete Series, Evaluate, Second Degree

Typology: Exams

2012/2013

Uploaded on 03/16/2013

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gannak 🇮🇳

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MATH 106 Exam 3 Review November 17, 2004
1. Find the second degree Taylor polynomial for f(x) = 3
√xabout x= 1000 and use it to estimate 3
√999.
2. Write out the first three non-zero terms of the Taylor series about x= 0 for each of the following.
Then write out the complete series for each in summation notation.
(a) cos 3x
(b) xe−x2
(c) 1
1 + x2
(d) arctan x
pf3

Partial preview of the text

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MATH 106 Exam 3 Review November 17, 2004

  1. Find the second degree Taylor polynomial for f (x) = 3

x about x = 1000 and use it to estimate 3

  1. Write out the first three non-zero terms of the Taylor series about x = 0 for each of the following. Then write out the complete series for each in summation notation.

(a) cos 3x

(b) xe−x

2

(c)

1 + x^2

(d) arctan x

  1. Use Taylor series to evaluate the following limits.

(a) lim x→ 0

1 − 4. 5 x^2 − cos 3x 7 x^4

(b) lim x→ 0

xe−x 2 − x + x^3 x^5

  1. Use appropriate second-degree Taylor approximations to estimate a solution near x = 0 to the equation e^5 x^ − cos 2x = sin 4x.
  2. Compute the following derivatives for f (x) =

1 + x^2

(a) f (2004)(0)

(b) f (2005)(0)

  1. Is y = 5e−^4 x^ a solution to the differential equation y′′^ + 3y′^ − 4 y = 0?