MATH 1A Final: MATH 1A Harvard Finalreview99

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turquoisegnu128

15 Feb 2019

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Math 1a final review problems -- fall 1999. These problems have been taken from final exams and reviews from previous semesters. Problems from the spring 97 final exam: find derivatives for the following functions (simplify completely): a) f x c) f x e) f x x. 1 3 sin ( b) f ( ) 1 d) f x x (ln ) tan x. 1 x f) f x: find the following integrals: +1 x ln ) z ln z dz ln( e x: find the equation of the tangent line to the curve. 4: consider the graph of the function f (x) = tan-1 x . Consider the region bounded below by the x-axis and above by the graph between x = 0 and x = 1. Circle whether each of the statements below is true or. 2 (ln )x x dx d: sketch the graph of f x x. 2 dt: r(50) t(50) true or false.

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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

MATH 1A Final: MATH 1A Harvard Finalreview99

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