{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

final-02 - f using the information from this question 5...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Faculty of Arts and Science University of Toronto April/May 2002 Examinations MAT 137Y, Calculus! Time Alloted: 3 hours 1. Evaluate the following expressions. (6%) (i) lim t ∞ t 2 t 3 sin 1 t . (6%) (ii) 1 0 xe x dx . (6%) (iii) 3 x 4 x 2 3 x 2 dx . (6%) (iv) dx 1 x 1 3 . (6%) 2. Find the volume of the solid generated by revolving the region bounded by the curves y e x 2 x 0 x 2 y 0 about the y -axis. (10%) 3. A solid consists of a hemisphere of radius r that is glued onto the top face of a right circular cylinder of height h and base radius r . If the surface area (including the bottom face of the cylinder) of the solid is a constant, show that the volume is maximized when h r . (It may help to remember that the surface area of a sphere with radius r is 4 π r 2 . 4. Consider the function f x x 1 3 x 4 3 . (2%) (a) Find f x and f x . (4%) (b) Find all vertical and horizontal asymptotes and find all x and y -intercepts. (4%) (c) By locating all critical points of f , find the intervals for which f is increasing and decreasing, and classify all critical points. (4%) (d) Find the intervals of concavity of f and locate all inflection points. (4%) (e) Sketch the graph of
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Background image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: f , using the information from this question. 5. Consider the function f x sin x . (3%) (a) Give the Taylor polynomial P 2 n 1 x of sin x of degree 2 n 1 (in powers of x ). (5%) (b) Use an estimate on the error R 2 n 1 x sin x P 2 n 1 x to ±nd a value of n such that P 2 n 1 1 5 approximates sin 1 5 with error less than 10 4 . 6. Determine, with justi±cation, whether the following series converges or diverges. (5%) (a) ∞ ∑ k 1 ln k k . 1 (5%) (b) ∞ ∑ k 1 2 k k ! k k . 7. Consider the function f x e x . (3%) (a) Write down the Taylor series (or the power series expansion in powers of x ) for f x . Include the k-th (non-zero) term in your answer. (3%) (b) Express 1 e x 2 dx as a series; include the k-th non-zero term in your answer. (3%) (c) Find the radius of convergence of the power series found in part (a). Justify your answer. (10%) 8. Use the ε-δ de±nition of limit to prove that lim x 3 x 2 9. 2...
View Full Document

Report

{[ snackBarMessage ]}