MATH 112- Final Exam Guide - Comprehensive Notes for the exam ( 205 pages long!)

Show all your work for full credit. The textbook, lecture notes, and elec- tronic devices with symbolic computing capacity are NOT allowed 1. Evaluate cos(x + 2y)dA, whe 2. Evaluate the following double integral Jo Jz 3. Evaluate the double integrals by converting them to polar coordinates. 2心ー昏 4. Evaluate the double integrals by converting them to polar coordinates. 5. Evaluate the triple integrals dxdydz, 2 drdzdy, 3. rdzdrd9. 6. Find the volume between the graph of the function f(r, y) - zy and the rect- angle R = {(z, y)1-2 3and-1 7. Evaluate the following double integrals y 5) 0 z+2 3. 2ded 8. Let S be the triangular region with vertices (0,0), (-1, 1), and (1, 1). (Sketch the region first!) . Find the centroid ofS . If the density at each point in S is proportional to the point's distance from the y-axis, find the mass of S.

Please read the directions for each problem carefully. You may work together and receive help on this exam; however, each student must turn in his or her own exam. This exam is due by 11:30am on Monday November 27th. Late exams will not be accepted. If you are unable to attend class on Monday November 27th, you must scan and email the exam to me by 11:30am (11 points) Given the function f(x) =- (a) Find the domain of the function in interval notation. (x-1)2 x2+1 I. Answer (in interval notation):0o o (b) Find the intercepts of the function. All work must be shown. Write your answers as ordered pairs. o Answer(s) (written as ordered pairs): L101-10.1) (c) Find the equations of any asymptotes. All work must be shown. L91 C110) Answer(s) (written as equations) (d) Find the critical numbers. All work must be shown

Fall 2017 Math 2150 Final Exam Take 1 Part 2 Section 8 Closed book, closed notes. No caleulators. Show your work. No loitering, no smoking, no walking on the grass, no feeding the animals, no parking, no stopping or standing, no dumping, no running, no fishing, no vacancy. All problems weighted equally. (1) Use the power series method to solve (1-z)y" +zy-y = 0. Here's a standard power series that you will find useful in your answer: 23 2 3 4! n! nao (2) A 64 Ib object stretches a spring 4 ft in equilibrium. It is attached to a dashpot with damping constant c-8 lb-sec/ft. The object is initially displaced 18 inches above equilibrium and given a downward velocity of 4 ft/sec. Find its displacement for t>0. Assume the motion is free. Since the weight is 64, for the mass you can use m =64/32 = 2. The following equations will be useful: The equation of motion for this system is my" + c/ + ky= F, where m is the mass of the object, c is the damping constant, k is the spring constant, and F is the forcing function. mg = kar, where g 32n/s2 is the gravitational constant, and Al is the distance by which a spring is stretched from its natural length when the object is attached to it. Remember to convert from inches to feet, so all units are compatible.