Mathematics 1229A/B Study Guide - Final Guide: Augmented Matrix, Matrix Multiplication, Invertible Matrix

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MATH 1229A/B Full Course Notes

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Note: your answers to the problems in part a must be. Use the following information for questions 1, 2 and 3. Let u = (1, 0, 3) and v = (0, 5, 0). Find the length (magnitude) of the vector u + v. A: (x, y) = (5, 7) + t(2, 3) D: (x, y) = (2, 3) + t(7, 5) e: (x, y) = (5, 7) + t(0, 0) B: (x, y) = (5, 7) + t(3, 2) c: (x, y) = (2, 3) + t(5, 7) Consider the following vectors in 3: u = 2i 3j and v = i j. Which one of the following is an equation of the line in 2 which passes through the point. A: (x, y) = (4, 2) + t(1, 1) b: (x, y) = (4, 2) + 12(1, 1) c: (x, y) = (1, 1) + t(2, 4) D: (x, y) = (4, 2) + t(1, 1)

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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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Mathematics 1229A/B Study Guide - Final Guide: Augmented Matrix, Matrix Multiplication, Invertible Matrix

MATH 1229A/B Full Course Notes

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