MA104 Lecture Notes - Trigonometric Substitution, Hyperbolic Function, Riemann Sum

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25 Feb 2013

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Page 1 of 2: evaluate each of the following integrals. Show all of your work. cos3(x) sin2(x)dx tan2(t)dt dx x x2 4. 0 (a) r /2 (b) r /4 (c) r (d) r /4 (e) r ln 2. 0: give the riemann sum argument, using regular partitions, to derive the formula for volumes of revolution. Assume the area under y = f (x) (where f is a continuous function), above the x-axis, between x = a and x = b (b > a) is rotated about the x-axis. [5 marks: suppose that the function y = f (x), 4 x 6, is twice continuously di erentiable on. [4, 6] and |f (x)| 3 on [4, 6]. Determine the number n of subdivisions necessary using the trapezoid rule to ensure the error in approximating r 6. 4 f (x)dx is less than 0. 01: compute each of the following integrals, using the method of improper integration.

Related Questions

1.Evaluate the following indefinite integrals. Give your answers in most general form.

a. cos(x) + e^(5 xâ10) dx

b. e^t +sin(5t-11) dt

c. 7/x^2 +(x)^1/3 dx

d. 1/x+ 1/x^6 dx

e. 9x^4+1/x dx

f. (1+(2x+1)^-1) dx

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

MA104 Lecture Notes - Trigonometric Substitution, Hyperbolic Function, Riemann Sum

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