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MATH 106 Final Exam Review Part II: Taylor Series, Differential Equations, and Limits, Exams of Calculus

Guru Jambheshwar University of Science and TechnologyCalculus

A review of various mathematical concepts covered in a final exam, including computing radii and intervals of convergence, estimating roots using taylor polynomials, finding taylor series, evaluating infinite series, working with differential equations, and sketching slope fields. It also includes problems related to population dynamics.

Typology: Exams

2012/2013

Uploaded on 03/16/2013

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parni 🇮🇳

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MATH 106 Final Exam Review, Part II
1. Compute the radius and interval (including endpoints) of convergence for
∞
X
n=1
(x+ 3)n
(5n)(2n).
2. Use a second-degree Taylor polynomial to estimate √101.
3. Find the complete Taylor series (in summation notation) for f(x) = ln (1 −x) about x= 0.
4. Evaluate the following exactly.
(a) 1 −1+1/2−1/6+1/24 −1/120 + ...
(b) 8/3−8/9+8/27 −8/81 + ...
(c) π−π3/6 + π5/120 −...
pf3
pf4

Partial preview of the text

Download MATH 106 Final Exam Review Part II: Taylor Series, Differential Equations, and Limits and more Exams Calculus in PDF only on Docsity!

MATH 106 Final Exam Review, Part II

  1. Compute the radius and interval (including endpoints) of convergence for

∑^ ∞

n=

(x + 3)n (5n)(2n)

  1. Use a second-degree Taylor polynomial to estimate
  1. Find the complete Taylor series (in summation notation) for f (x) = ln (1 − x) about x = 0.
  2. Evaluate the following exactly.

(a) 1 − 1 + 1/ 2 − 1 /6 + 1/ 24 − 1 /120 + ... (b) 8/ 3 − 8 /9 + 8/ 27 − 8 /81 + ... (c) π − π^3 /6 + π^5 / 120 − ...

  1. Let f (x) = x^3 sin (− 5 x^2 ).

(a) Write out the first 3 non-zero terms in the Taylor series about x = 0 for f (x).

(b) Write out the complete series for f (x) in summation notation.

(c) Compute f (13)(0).

(d) Compute lim x→ 0

5 x^5 + f (x) 5 x^9

  1. Use the appropriate second-degree Taylor polynomials to estimate a solution near x = 0 to 1 + sin 8x = e^10 x.
  1. A colony of endangered sea otters has an annual birth rate of 10% and an annual death rate of 15%. In an attempt to sustain the colony, activists bring in otters from another region where the animals are plentiful. They do this at a rate of 50 otters per year.

(a) Write a DE whose solution is P (t), the otter population t years from now.

(b) Find any and all equilibrium solutions.

(c) Find the general solution of your DE.

(d) Find and sketch the particular solution if the current population is 400 otters.

  1. A population obeys the differential equation

dP dt

=. 001 P (3000 − P ). Sketch solutions for P (t) for the following initial populations: P (0) = 0, P (0) = 100, P (0) = 2000, P (0) = 4000.

See old exams and quizzes at http://abacus.bates.edu/˜etowne/mathresources.html