MATH241 Study Guide - Final Guide: Maxima And Minima, Relative Growth Rate
University of Delaware
Department of Mathematical Sciences
MATH 241 FINAL EXAM – version A Spring 2016
Name:
Lecture Section: Instructor:
Instructions:
•Check your examination booklet before you start. There should be 10 items on 10 pages.
•Partial credit will not be given if appropriate work is not shown.
•Answer questions in the space provided. If you need more space for an answer, continue
your answer on the back of the page, or/and use the margins of the test pages. Do not take
pages apart from the booklet.
•Carefully work out each problem and clearly indicate your final answer to every problem.
•The time allowed for completing this exam is 120 minutes = 2 hours.
•You may NOT use calculators, dictionaries, notes, or any other kinds of aids.
•SCHOLASTIC DISHONESTY WILL NOT BE TOLERATED.
Cheating receives a Failing grade.
page1 page2 page3 page4 page5 page6 page7 page8 page9 page10
q1 q2 q3 q4 q5 q6 q7 q8 q9 q10 Total
1. (12 points) Evaluate each of the following limits:
(a) (3 points) lim
x→2
x2−3x+ 2
x2−x−2
(b) (3 points) lim
x→+∞√x2+ 4x−√x2+ 2x
(c) (3 points) lim
x→1+
x2−x−2
x2−3x+ 2
(d) (3 points) lim
h→0
ex2+2hx+h2−ex2
h
Hint: Use the fact that this limit represents the derivative of a certain function.
1
2. (10 points) A function is given by the piecewise definition:
f(x) =
x2for x < 0
1
2for 0 ≤x≤1
x−1
x2−1for x > 1
(a) (1 point) Compute lim
x→1−
f(x)
(b) (2 points) Compute lim
x→1+f(x)
(c) (2 points) Is fcontinuous at x= 1? Justify your answer using the mathematical
definition of continuity.
(d) (1 point) Compute f0(x) for x < 0.
(e) (1 point) Compute f0(x) for 0 <x<1.
(f) (3 points) Is fdifferentiable at x= 0? Justify your answer.
2
Document Summary
Instructions: check your examination booklet before you start. There should be 10 items on 10 pages: partial credit will not be given if appropriate work is not shown, answer questions in the space provided. If you need more space for an answer, continue your answer on the back of the page, or/and use the margins of the test pages. Cheating receives a failing grade. page1 page2 page3 page4 page5 page6 page7 page8 page9 page10 q1 q2 q3 q4 q5 q6 q7 q8 q9 q10. Hint: use the fact that this limit represents the derivative of a certain function. 1: (10 points) a function is given by the piecewise de nition: x2 for x < 0. 2 x 1 x2 1 for. 2: (9 points) di erentiate each of the given functions with respect to x: (a) (3 points) f (x) = e2x x4 + x (b) (3 points) g(x) = e x cos(3x) (c) (3 points) q(x) = ln (arctan(x))
