MATH 141 Midterm: MATH141 South Carolina 141 96 2 nospace

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turquoisegnu128

15 Feb 2019

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Mathematics

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MATH 141

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In problem 2 you must use the de nition of the derivative; in the other problems you may use any legitimate derivative rule. Section: (the penalty for each mistake is ve points. ) Let f (x) = (a) graph y = f (x) . (b) fill in the blanks: How high does the object get: let y = x2 cos2(4x5 + 19x) . Find dy : let y = q4x3 + 9x + sin3(cos(5x4 + 3x)) . Find dy dx : let 3x2y3 = sin(xy2) + 3x5 . Find dy dx : let y = 3 x + 15 4 x . Find d2y dx2 : the area of a square is growing at the rate of 4 square feet per second. How fast is the length of each side growing when each side has length 6 feet: a 30 foot ladder is leaning against a wall.

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[ -2 e* if x< 0 3. (24 pts) Let g(x) = 1 X+6 if x20. [x - 3 (I) Evaluate the limit. (Note: Possible answers include + or - .) [You may not use a table of values, a graph, or L'Hospitals's Rule to justify your answer. ] SHOW YOUR WORK. (a) (3 pts) lim g(x) = ; (b) (3 pts) lim g(x) = x 10 x-10+ (c) (3 pts) lim g(x) = ; (d) (3 pts) lim g(x) = 14 = * 3+ (e) (3 pts) lim g(x) = ; (f) (3 pts) lim g(x) = X + (II) (6 pts) Using the definition of continuity, show that gis continuous at 0. SHOW YOUR WORK.

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 141 Midterm: MATH141 South Carolina 141 96 2 nospace

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MATH 141 Midterm: MATH141 South Carolina 141 96 f nospace

MATH 141 Midterm: MATH141 South Carolina 141 95 2 nospace

MATH 141 Midterm: MATH141 South Carolina 141 95 f nospace

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