MAT 21B Midterm: MATH 21B Midterm Exam1 Winter 2010 Solutions 1

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Using the properties of -notation sums and the formula for the sum of the rst n integers, we get: (2k 2) = If f (x) is continuous and f (x) dx = 8, what is. Using the substitution rule for de nite integrals, this gives that f (x2)x dx = Find the area of the region bounded between the curves y = 3x and y = x2. First, it is a good idea to sketch a diagram showing the graphs of the two curves and the region bounded between them. Now we nd the x-ordinates of the points of intersection of the two curves. This leads to the equation x2 = 3x, which has the two solutions x = 0 and x = 3. Finally, the area is computed by integrating the di erence of the two curves between the two points. 4 (30 points) evaluate the following de nite integrals: (a) (b) (c)

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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

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MAT 21B Midterm: MATH 21B Midterm Exam1 Winter 2010 Solutions 1

MAT 21B Full Course Notes

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