MATH 1004 Study Guide - Final Guide: Inverse Trigonometric Functions, Scantron Corporation, Partial Fraction Decomposition

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$MATH 1004 Full Course Notes$

MATH 1004 Full Course Notes

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Please choose only one answer and insert in pencil in your scantron sheet. Multiple-choice questions: [3 marks] evaluate lim x 1 x3 1 x2 1 (a) 1 (b) 1/2 (c) 3/2 (d) 0, [4 marks] let f(x) = (ln x)x. Evaluate f (e), where e = 2. 718 . is euler"s number. Nd the derivative of f at x = e. (a) f (e) = 0 (b) f (e) = 1 (c) f (e) = 1 (d) f (e) = 2: [3 marks] let f(x) = 3|x 1|. December 2005: [3 marks] let f(x) = 3x x3. F (0). (a) 2 (b) 1/2 (c) 0 (d) 1/2: [3 marks] find the derivative of the function f de ned by f(x) = arcsin ( 2 x (a) f (x) = ( (b) f (x) =!4 x2 (c) f (x) = x x2 4 (d) f (x) = x x2 4.

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4. (16 pts) MULTIPLE CHOICE! CIRCLE THE CORRECT ANSWER IN EACH PART. (I) ( 4 pts) Complete the statement of the Intermediate Value Theorem: Suppose f is continuous on the interval [a, b] and L is a number between f (a) and f (b). Then there is at least one number c in (a, b) such that (a) C = L; (b) f(c) = 0; (c) f(c) = L; (a) f' (c) = L; (e) C = 0; (f) f (a) < c < f (b); (g) NONE OF THE PREVIOUS ANSWERS. (II) (4 pts) The figure shows the graph of a function f. At what point (points) c does the conclusion of the Intermediate Value Theorem hold for f (x) on the interval [1, 5] and L = 2. --- - - - -1 - - - 2 - - I- - - (a) c = 1; (b) C = 0; (c) C = 2; (a) c = 3; (e) C = 4; (f) C = 5; (g) c= -1. (III) (4 pts) Given that cos x s f (x) s et, for all x > 0 evaluate lim f (x) x+0+ (a) lim f (x) = 0; (b) lim f (x) = 1; (c) lim f (x) = e; x0+ X0+ (d) lim f(x) DOES NOT EXIST. WE DON'T HAVE ENOUGH INFORMATION TO ANSWER THE QUESTION *40+ (IV) (4 pts) Given the function 1 f (x) = , find its inverse, f+(x) x -2 (a) f (x) = x -2; (b) f«+ (x) = x+2 ; (c) fº+ (x) = -2; (f) f-+ (x) -2; (e) f (x) DOES NOT EXIST.

1 2. (a) Find the tangent line to the graph of s(2)=x+ when r = 1. (b) Find the derivative of p(x) = (ln(5.c) + )(2-3V ). (c) Find the derivative of j(x) = 60(2*+52) (d) Find the derivative of k(2) = (e) The following table gives some values for f(x), f'(x), g(x), and g'(x). | f(x) | 3|1|2|2 f'(x) | 1 46 | 4 g(1) 0-183 () -2 0 4-1 • Let h(x) = f(t)g(2). Find h' (6). • Let r(x) = f(g(x)). Find /'6).

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 1004 Study Guide - Final Guide: Inverse Trigonometric Functions, Scantron Corporation, Partial Fraction Decomposition

MATH 1004 Full Course Notes

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