MAT 2020 Midterm: MAT2020 Wayne State 2020Test1

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turquoisegnu128

15 Feb 2019

School

Wayne State University

Department

Mathematics

Course

MAT 2020

Professor

All

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0 2f (x) + g(x)dx = 9, and 5. [20 pts] find the following derivative (a) (b) (c) d dx x. 1 t2 dt (d) d dx 1 + exdx. [20 pts] evaluating the following integration by substitution method: (a) . 1 (b) 1 + ln(x) x dx (c) t 2t + 1dt. [15 pts] find the area of the region bounded by 4x + y2 = 12 and x = y. [15 pts] find the volume of the solid obtained by rotating the region bounded by the curve y = x, y = x about y = 1. [10 pts] a force 30n is required to maintain a spring stretched form its natural length of. 10 m to a length 15 m. how much work is done in stretching the spring from 15 m to 20 m. [10 pts] find the average value of the function g(x) = 2x (1 + x2)2 on the interval [0, 2].

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1. (45 marks) Short-Answer Questions. Put your answer in the box provided but show your work also. Each question is worth 3 marks, but not all questions are of equal difficulty. Full marks will be given for correct answers placed in the box, but at most 1 mark will be given for incorrect answer. Unless otherwise stated, it is not necessary to simplify your answers in this question. a) Find the equation of the tangent plane to the graph of the function z = f(x,y) = sell at (0,1). Answer b) Evaluate (T, 0) if f(x,y) = sin(ry). Answer c) Graph of a function z = f(x,y) is given by: Which of the following level curves correspondes to this function. Answer d) If f(x,y) = e(3x – 2y), find lim - Math)-f(x,y) e) Find % if f(x, y) = x2 - y3 and 2 = r sin(t), y = r cos(t). Answer f) Evaluate the directional derivative of f(x, y) = !at (1, -1) in the direction of u =(2, 1). Answer 8) Does Simono converge? h) Sketch the direction field of the differential equation = 1-2y. i) Consider the differential equation ay = 15 - 31, y(0) = 0, and use Euler's method with step size 0.1 to estimate y(0.1). Answer j) Knowing that (-3,-3) is a critical point of f(x,y) = 2:3 + y2 – 4.cy -3.0, use the second derivative test to find out whether it is a local minimum, local maximum, or a saddle point. Answer k) Solve the initial value problem, dy dt 642 2y + cos(u) y(1) = . Answer 1) Is ū=(2,2, -1) perpendicular to v = (5,-4, 2)? Answer m) Find the area between the graph of y = 1 + sin(x) and y=1-sin(x) from 0 to 7. Answer n) Determine the lentgh of the curve y = 8/2,0 SIS 3. Evaluate and simplify your answer. Answer o) Express the volume of the solid obtained by rotating the bounded region that lies between curves y = (x-2)2 and y=1,150 $4 about the vertical line 2 = 5. Do not evaluate this integral. Answer

MAT 2020 Midterm: MAT2020 Wayne State 2020Test1

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