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Math 106: Calculus
Final - Fall 2010
Duration : 150 minutes
Name
Student ID
Signature
- Put your name, student ID and signature in the space provided above.
- No calculators or any other electronic devices are allowed.
- This is a closed-book and closed-notes exam.
- Show all of your work; full credit will not be given for unsupported answers.
- Write your solutions clearly; no credit will be given for unreadable solutions.
- Mark your section below.
Section 1 (Azadeh Neman, MW 12:30-13:45) Section 2 (Emre Mengı, MW 15:30-16:45) Section 3 (Selda K¨uc¸¨ukc¸ifc¸i, MW 11:00-12:15) Section 4 (Selda K¨uc¸¨ukc¸ifc¸i, MW 14:00-15:15) Section 5 (Azadeh Neman, TuTh 12:30-13:45) Section 6 (Azadeh Neman, TuTh 15:30-16:45)
Question 1. Let f (x) = √^1 x.
(a) Use a linear approximation to estimate √^11. 1.
(b) Find the Taylor series for f (x) centered at 1.
(c) Determine the interval and radius of convergence for the Taylor series that you determined in part (b).
Question 3.
(a) Find the x-coordinates of the critical points of the function
F (x) =
∫ (^) x 2 1
sin t t dt defined for x > 1.
(b) Evaluate the limit
hlim→ 0
−h
sin t dt h^2.
Question 4.
(a) Evaluate the indefinite integral given below. ∫ x cos(7x) dx
(b) Is the integral given below convergent or divergent? ∫ (^) ∞ −∞
f (x) dx, where f (x) =
e^2 x^ if x < 0 e−x^ if x ≥ 0 Evaluate the integral if it is convergent. State your reasoning and show the details of your work.
Question 5. Consider the region R bounded by the curves y = e^2 x^ and y = e−x, and the vertical lines x = −1 and x = 1.
(a) Find the area of the region R.
(b) Find the volume of the solid obtained by revolving R about the x-axis.
Question 6. In each part indicate whether the series is convergent or divergent. Explain your answer. In particular state explicitly which test you are using.
(a)
∑^ ∞
n=
n^2 n^3 · √n + 1
(b)
∑^ ∞
k=
k · (ln k)^6
(c)
∑^ ∞
n=
sin n
