University:
University of Michigan
Course:
MATH 116 | Calculus II
Academic year:

2013
Views:

202

Pages:

2
Author:

gevaj0

Math 116 / Exam 1 (February 11, 2013) page 2 1. [10 points] Indicate if each of the following is true or false by circling the correct answer. No justification is required. a. [2 points] Let (x̄,ȳ) be the center of mass of the metal plate bounded by the line y = 1−x, the x-axis and the y-axis for 0 ≤ x ≤ 1. If the plate has uniform mass density, then x̄ = ȳ. True False Solution: Since the plate has uniform density and it is symmetric about the line y = x, then ȳ = x̄. It also follows using the formulas for the coordinates of the center of mass R1 R1 δy(1 − y)dy 0 δx(1 − x)dx = R0 1 = ȳ. x̄ = R 1 δ(1 δ(1 − x)dx − y)dy 0 0 b. [2 points] The function F (x) = Z x2 sin(et )dt is an even function. 1 True False Solution: The function F (x) is even if it satisfies F (−x) = F (x). Since F (−x) = Z (−x)2 t sin(e )dt = 1 Z x2 sin(et )dt = F (x). 1 then F (x) is even. c. [2 points] Let h(x) be an antiderivative of g(x). If g(x) is measured in kg and x in inches, then the units for h(x) are kg per inch. True Solution: The second fundamental theorem of calculus says that h(x) = False Z x g(t)dt for a some contant a. The units of g(x) and x are kg and inches respectively, then the units of h(x) are kg · inches. Z 1−t 3 d. [2 points] The function R(t) = ex dx is decreasing for all values of t. t True False Solution: 3 3 R′ (t) = −e(1−t) − et < 0 for all values of t. Hence R(t) is decreasing. Winter, 2013 Math 116 Exam 1 Problem 1 Solution Math 116 / Exam 1 (February 11, 2013) page 3 e. [2 points] The length of the curve y = x2 from x = 0 to x = 2 is smaller than 4. True False Solution: The length L of the curve y = x2 from x = 0 to x = 2 is given by s 2 Z 2 Z 2p dy 1+ dx = 1 + (2x)2 dx. L= dx 0 0 Using your calculator you can find L ≈ 4.64. Or, you can argue without using your calculator. The length L of the curve y = x2 from x = 0 to x = 2 isplarger than the length 2 2 L √1 of the line connecting the points (0, 0) and (2, 4). Since L1 = (2 − 0) + (4 − 0) > 42 = 4. Hence L > L1 > 4. Winter, 2013 Math 116 Exam 1 Problem 1 Solution

Math 116 Exam

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