Math 22 - Fall 203, Lecture,2,3 & 4 Dec 8, 203 Math 22-Final Exam Name: Your discussion session time: Your TA: (circle one) Suzan Afacan Marc Conrad Daniel Hast In Gun Kim Yu Li Zhennan Zhou Zheng Lu Jeff Poskin Keith Rush Paul Tveite Jenny Yeon Adrian Tovar-López Christine Lien Ivan Ongay-Valverde Jason Steinberg Jing Hao Kyriakos Sergio Laura Cladek Mona Jala Nathan Clement Yihe Dong Directions Please Read Carefully! There are 8 problems and pages. No calculators/electronic devices. Cellphones OFF. No outside resources. No scratch paper, you have enough space to show your work. Show all your work. Justify your answers and write clearly. Unsubstantiated/illegible answers will receive little or no credit. If more space is needed, use the back of the previous page to continue your work. Be sure to make a note of this on the problem page so that the grader knows where to find your answers. Raise your hand if more scratch paper is needed or if you have any questions. Good Luck!!! Problem Points Score 5 2 9 3 4 4 5 Problem Points Score 5 9 6 4 7 5 8 9 Total 00
. (5 points) Calculate the following limits (a) lim x 0 cos(2x) sin 2 x (do not use l Hopital in this problem. Little or no credit if you do); cos(2x) cos(2x) x 2 lim x 0 sin 2 = lim x x 0 4x 2 sin 2 x 4x 2 x 2 = 4 2 = 2. (b) lim x x x 2 x 2 ln x = lim x x x 2 ln x =. (This problem continues in the next page) 2
(c) One of the following statements is true and the other is false. Indicate which one is which and explain in detail why. Most credit goes to the explanation. i. If lim x 0 f(x) x = 0, then lim x 0 f(x) = 0 also. ii. If lim x 0 f (x) = and lim x 0 g (x) = 2, then lim x 0 f(x) g(x) = /2. f(x) i. is true, if lim x 0 f(x) 0, then lim x 0 x will be either plus or minus infinity, or it will not exist. ii. is false, choose f(x) = x + and g(x) = 2x + for a counterexample. 3
2. (9 points) Find the values of the parameter a and b that ensure that the function below is differentiable. f(x) = { ae x2 + 2 if x 0 ln(bx + ) if x > 0. For continuity you need both left and right limits at 0 to coincide. That is or a = 2. a + 2 = ln = 0 To be differentiable you need the slope on the right and on the left of x = 0 to coincide also. That is, you need the derivative ae x2 2x at zero, to be equal to b bx + at zero. Thus, you need b = 0. 4
3. (4 points) Differentiate the following functions (a) f(x) = tan(x 2 + ) + cos(ln x) x (No need to simplify); f(x) = 2x sec 2 (x 2 + ) x 2 sin(ln x) cos(ln x) x2 (b) f(x) = x x + u 2 du (No need to simplify); f (x) = + x 2 + (/x) 2 ( x 2 ). (c) Find the derivative of the function y = f(x) at the point (, ) where y is defined by the equation e xy = x 2 + y 2. e xy (y + x dy dy ) = 2x + 2y dx dx which at x = y = results in + dy dx = 2 + 2 dy dx and so dy dx =. 5
4. (5 points) Consider the function f(x) = ex x. (a) Determine on which intervals the function is positive or negative. This problem was in the practice list, check the practice. (b) Determine on which intervals the function increases or decreases. (c) Find all local maxima and minima. (it continues in the next page) 6
(d) Find all asymptotes. (e) Make a sketch of the graph of the function. Are max and min global? Explain. 7
5. (9 points) Find the dimensions of the rectangle with the largest area if one of its sides is on the x-axis and the vertices opposite to this side are on the curve y = 9 x 2, as in the picture. If A is the area of the rectangle, then Also A = 2x(9 x 2 ) = 8x 2x 3 A = 8 6x 2 = 0 = x = ± 3. A = 2x so A ( 3) < 0 and so we have a maximum. Thus the dimensions are base = 2 3 and height 6 units. 8
e 6. (4 points) Compute the following integrals. (a) 2 + ln x dx = {u = 2 + ln x; du = x } x dx; x = = u = 2, x = e = u = 3 = 3 2 udu = 2 u2 3 2 = 9 2 4 2 = 5 2 (b) x sin(x 2 ) cos(x 2 )dx = { u = sin(x 2 ), du = 2x cos(x 2 )dx } = 2 udu = 4 u2 +C = 4 (sin(x2 )) 2 +C. 9
7. (5 points) Consider the region in the first quadrant bounded above by the function y = x, below by the x axis, and to the right by the line y = x 2. (a) Make a drawing and write an integral that describes the area of the region. Do not integrate! 2 0 (y + 2 y 2 )dy = 2 0 4 xdx + ( x x 2)dx 2 (b) Write an integral describing the volume of the solid generated by rotating the area around the y-axis. No need to integrate. 2 0 (π(y + 2) 2 π(y 2 ) 2 )dy (c) Write an integral describing the volume of the solid generated by rotating the area around the x-axis. No need to integrate. 2 0 π( x) 2 dx + 4 2 (π( x) 2 π(x 2) 2 )dx 0
8. (9 points) A particle of mass m is moving along the x axis with an acceleration given by a(t) =, t > 0. The particle started to move 0 meters from the origin, at a velocity of 2 meters (t + ) 2 per hour. (a) Find the velocity. How does the velocity change at t increases? (that is, find the limit of the velocity as t ). t v(t) = v(0) + 0 (s + ) 2 ds = 2 (t + ) + = 3 (t + ). As t increases the velocity increases from 2 to 3. (b) Calculate the position of the particle for any time t. x(t) = x(0) + t 0 (3 )ds = 0 + 3t ln(t + ). (s + ) (c) Calculate the work used to move the particle from t = 0 to t = 2. You do not need to simplify the answer. K(2) K(0) = 2 mv(2)2 2 mv(0)2 = 2 m(3 3 )2 2 m22.