MATH 105 Midterm: MATH105 Exam 3 Solutions

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turquoisegnu128

24 Oct 2018

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$MATH 105 Full Course Notes$

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1. (a) evaluate or show that it doesn"t exist. We can use the substitution u = x2 + y, so u 12 1 = 0: (x,y) (1, 1) x2 + y lim sin(x2 + y) lim (x,y) (1, 1) sin(x2 + y) x2 + y. = lim u 0 sin(u) sin(0) u 0 d dt sin(t) (cid:12)(cid:12)(cid:12)t=0. Here we used the de nition of derivative, f(cid:48)(a) = lim x a (b) consider the area function a(x) = (cid:82) x (cid:90) 2. By the fundamental theorem of calculus, (cid:90) 2. 3 f (x) f (a) x a. 1 f (t)dt, with a(2) = 6 and a(3) = 5. f (t)dt = a(2) a(3) = 6 5 = 1 . (c) a self-employed software engineer estimates that her annual income over the next. 10 years will steadily increase according to the formula 70, 000e0. 1t, where t is the time in years.

Related Questions

a) Use part I of the Fundamental Theorem of Calculus to find the derivative of
F(x)=â« 4 upper, x down sin(t^5) dt?

b) Use part I of the Fundamental Theorem of Calculus to find the derivative of

y=â« âx upper, â5 down cos (t) / t^7 dt

c) If f(x)=â« 4upper, 1â4x down, sin(t) / (1+t^2) , then fâ²(x)=?

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mauvegnat752

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 105 Midterm: MATH105 Exam 3 Solutions

MATH 105 Full Course Notes

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