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3. Consider the system of equations represented as the matrix equation, AT = b[12 marks]. (a) Use the method of inverses to solve the system of equations, and clearly show your answer: 5.0 – 2y = 1 -4.x + 2y = 2 (b) If A-1 = [14-11 11 2 O 3 , and 5 = | 2 ), then evaluate . Clearly show your answer. [ 7 2 1] (c) If 4-1 - 2 31 1 2 , and b then what is the ORIGINAL system of equa- tions?

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Deanna Hettinger
Deanna HettingerLv2
5 Jun 2018
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1. Put the following system of equations into an augmented matrix and perform row opera- tions to get the matrix into Row Reduced Echelon Form. Give your answer as r = ..., y = ...,2= .... Marks will be deducted if you leave your solution in matrix form! IF the system is inconsistent, it is sufficient to only Row Reduce, and then clearly state your findings [22 marks]. (a) 2.r + 3y - 2 1 + y +2 y - 22 = = = 4 3 -1 11 +262 +23 – 3.04 = 2.0 +6.02 + 3.03 - 24 = -X1 + 12 + 2.24 = ri +12 +13 + 3.04 = 1 Ti + 2x2 - 3.13 +224 = -3.01 - 672 +2.13 - 14 = 12 + 2.13 +2 = -Ii - 22 - 2.13 + 4.14 ܝܙ ܨܝܝ

1. Consider a linear system of equations whose augmented matrix is 1 1 3 2 1 1 2 43 11 3 alb (a) Reduce the augmented matrix to echelon form. [4 marks) (b) For what values of a and b will the system have infinitely many solutions? [2 marks] (c) For what values of a and b will the system have a unique solution? (2 marks] (c) For what values of a and b will the system have no solution? [2 marks]

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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