MATH 1131Q Lecture 15: MATH 1131Q , Lecture 15 , Section 6.2

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MATH 1131Q Lecture 1: 2.2

MATH 1131Q Lecture Notes - Lecture 2: Horse Length, Classification Of Discontinuities, Intermediate Value Theorem

MATH 1131Q Lecture 15: MATH 1131Q , Lecture 15 , Section 6.2

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Math 1131q , lecture 15 , section 6. 2. Cylinder volume = area of base * height. Rectangular box volume = lwh r^2 = x^2 + y^2. If keep adding disks together , it will form the sphere. (r^2 - x^2) dx = [ r^2x - x^3/3 ] -r r. = [ (r^3 - r^3/3) - (-r^3 + r^3/3) ] 4/3 r^3 = volume of the sphere. : let s be a solid that lies between x = a and x = b. Definition of s in the plane p x , through x and perpendicular to the x-axis , is a(x) , where a is a continuous function, then the volume of s is n. Example : volume of solid with washer cross section. The region r enclosed by the curves y = x and y = x^2 is rotated about the x-axis.

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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 1131Q Lecture 15: MATH 1131Q , Lecture 15 , Section 6.2

MATH 1131Q Lecture 1: 2.2

MATH 1131Q Lecture Notes - Lecture 2: Horse Length, Classification Of Discontinuities, Intermediate Value Theorem

MATH 1131Q Lecture 15: MATH 1131Q , Lecture 15 , Section 6.2

Document Summary

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