MATH 220 CALCULUS I SPRING 2018, MIDTERM I FEB 16, 2018

Size: px

Start display at page:

Download "MATH 220 CALCULUS I SPRING 2018, MIDTERM I FEB 16, 2018"

Dustin Patterson
5 years ago
Views:

1 MATH 220 CALCULUS I SPRING 2018, MIDTERM I FEB 16, 2018 DEPARTMENT OF MATHEMATICS UNIVERSITY OF PITTSBURGH NAME: ID NUMBER: (1) Do not open this exam until you are told to begin. (2) This exam has 12 pages including this cover and one intentionally blank page for your use. There are 10 problems. You have 50 minutes. (3) No notes, books or calculators are permitted. (4) Please turn off all cell phones. (5) Place your ID card on your desk for inspection. (6) Show your work and justify the steps. Good luck! PROBLEM POINTS SCORE TOTAL 100 1

2 1. Find the values of the limit, if they exist. Justify your answers. x2 + 3x 2x (i) lim x 1 x 1 ( 1 ) (ii) lim (x 2) sin. x 2 x 2 2

3 2. Find the vertical and horizontal asymptotes of the function f(x) = 2x2 3x + 5 x 2 4x

4 3. In the picture below, the curve represents the graph of the function f. (i) Determine the values of a for which f is NOT continuous at x = a. (ii) Determine the values of b for which the derivative f (b) does NOT exist. (iii) Determine the values of c for which the derivative f (c) = 0. (iv) One of the two numbers f (0) and f (4) is negative. Which one is it? 4

5 4. Find numbers a, b such that the function ax 2 2x + 2 if x 3, f(x) = sin(π(x 2)) π(x 2) if 3 < x < 7 2 b cos(2πx) if x 7 2, is continuous everywhere. Justify your answer. 5

6 5. The equation of motion of a particle on a straight line is given by x(t) = 3t 2. (i) Find the average velocity of the particle over the time interval [2, 2.1]. (ii) Calculate the velocity at time t = 2 of the particle, using ONLY the limit of appropriate average velocities. Show your work. (iii) Write the definition of the derivative x (t) of an arbitrary function x(t). (You have to write the mathematical expression. Do NOT explain in words what the derivative is.) x (t) = 6

7 6. Applying the rules of differentiation, find the derivative f (z) of the function Do NOT simplify your final answer. f(z) = (z3 + 2z) 1/3. cos z 7

8 7. By implicit differentiation, find the slope of the tangent line to the curve at the point ( π, π) on the curve. y sin(x 2 ) x 2 sin(y) = 0 8

9 8. A man runs along a straight path. A searchlight is located on the ground 40m from the path and is kept focused on the man. When the bell tolls midnight, the man is located at a 20m distance from the point on the path closest to the searchlight, and exactly at the same moment, the searchlight s speedometer shows that it is rotating with a rate of 5Rad/min. What is the velocity of the man at midnight? 9

10 9. Let g(x) = sin 2 (πx). (i) Find g (x) and g (x). (ii) Use linear approximation to estimate g(0.26). (Remember that sin(π/4) = 2/2.) 10

11 10. Circle the correct answer among the 4 possible choices A, B, C, and D. (i) What is the domain of the function f(x) = x 5 x 6? A) All real numbers except 6 B) The interval (, 5) C) The interval (6, ) D) The intervals [5, 6) and (6, ) (ii) Let v = sin x and assume that du dv = v 2. What is du dx? A) cos x B) cos x sin 2 x C) 2 cos x 1 sin 3 D) x sin x (iii) Which could be a correct completion of the sentence f (a) exists,? A) and the tangent to the graph at (a, f(a)) is vertical. B) and f is not continuous at x = a. C) and the tangent to the graph at (a, f(a)) is horizontal. D) and f(x) = x a. (iv) A function f, defined for all real values x is given. Alice truly claims that for all ε > 0, she can find M > 0 such that if x > M, then f(x) 5 < ε. Which of the following choices is the correct expression of that fact? A) lim x 5 f(x) =. C) lim x 5 f(x) = 5. B) lim x f(x) = 5. D) lim x f(x) =. 11

12 (This page is intentionally left blank.) 12

Multiple Choice Answers. MA 113 Calculus I Spring 2018 Exam 2 Tuesday, 6 March Question

Multiple Choice Answers. MA 113 Calculus I Spring 2018 Exam 2 Tuesday, 6 March Question MA 113 Calculus I Spring 2018 Exam 2 Tuesday, 6 March 2018 Name: Section: Last 4 digits of student ID #: This exam has 12 multiple choice questions (five points each) and 4 free response questions (ten