Math 124 Final Examination Winter 2014 !!! READ...INSTRUCTIONS...READ!!!

Size: px

Start display at page:

Download "Math 124 Final Examination Winter 2014 !!! READ...INSTRUCTIONS...READ!!!"

Alexander Robinson
5 years ago
Views:

1 1 Math 124 Final Examination Winter 2014 Print Your Name Signature Student ID Number Quiz Section Professor s Name TA s Name!!! READ...INSTRUCTIONS...READ!!! 1. Your exam contains 8 questions and 10 pages; PLEASE MAKE SURE YOU HAVE A COMPLETE EXAM. 2. The entire exam is worth 100 points. Point values for problems vary and these are clearly indicated. You have 2 hours and 50 minutes for this final exam. 3. Make sure to ALWAYS SHOW YOUR WORK; you will not receive any partial credit unless all work is clearly shown. If in doubt, ask for clarification. 4. There is plenty of space on the exam to do your work. If you need extra space, use the back pages of the exam and clearly indicate this. 5. You are allowed one sheet of handwritten notes (both sides). Graphing calculators are NOT allowed; scientific calculators are allowed. Make sure your calculator is in radian mode. Problem Total Points Score Total 100

2 2 1. (12 Points) Find the derivative of the following functions. You do not have to simplify. Your final answers must give the derivative in terms of x. (a) y = sin(πx)+3x 2 ( π ) (b) Assume a and b are constants. y = e ax2 cos b x (c) y = x sin 1 (2x)

3 2. (12 Points) An object is moving in the plane with parametric equations: y axis x(t) = cos(t) y(t) = tsin(t) for0 < t < π. At a typical point (x(t),y(t)) on the curve, the normal line (dashed line in the picture) will intersect the y-axis at a point P. (Note: The normal line is perpendicular to the tangent line.) P x t,y t x axis (a) Find the equation of the normal line to the curve at the point (x(t),y(t)). The coefficients will depend on t (b) The y-intercept of the normal line in (a) has coordinates P = (0,b). Find a formula for b as a function of t. (c) lim t 0 +b= 3

4 4 3. (12 Points) Danica leaves home and drives her car on a straight road, running some errands. Her car s velocity, v(t), in miles per hour, at t hours after leaving home, for 0 t 1.1 is shown in the graph below. Use this graph to answer the following questions. No need to justify. Let x(t) represent the car s position at t hours, relative to home. Let a(t) denote the acceleration of the car at t hours. (a) During what time interval is the position x(t) maximal? (b) During what time interval is the velocity v(t) maximal? (c) What is the maximum value of the acceleration a(t)? Include correct units. (d) During what time interval(s) is x(t) decreasing?

5 5 4. (8 Points) The graph of f(x) is shown in the picture below. f(x) x Compute each limit, or state that the limit does not exist. Justify your answers. (a) lim h 0 f(1+h)+2 h (b) lim h 0 f(f(6+h)) f(f(6)) h

6 6 5. (10 Points) Island Alpha is ten miles north and some distance east of Island Beta. A ship is sailing west towards Island Beta at a speed of 2 miles per hour. The angle θ is measured between the two lines of sight from the ship to each island. (a) When the ship is 2 miles east (and 10 miles south) of Island Alpha, at what rate is θ changing? Alpha North 10 miles West South East Beta 2 miles/hour θ Ship (b) At that time, is the angle θ increasing or decreasing?

7 7 6. (18 Points) Consider the function f(x) = x2 3 x 3 (a) Find all vertical asymptotes or state that there are none. Justify your answer with limit computations. (b) Find all horizontal asymptotes or state that there are none. Justify your answer with limit computations. (c) Find all critical points (numbers) of f and determine whether each corresponds to a local minimum, local maximum, or neither.

8 8 (d) Find all inflection points of f, and list the intervals on which f is concave up. (e) Carefully and clearly sketch the graph of y = f(x). Include BOTH coordinates of all points on the graph that correspond to critical points and inflection points.

9 7. (12 Points) A cone-shaped drinking cup is made from a circular piece of paper of radius 1 by cutting a sector and joining the edges along radius OA and OA. Find the maximum capacity of such a cup.the volume of a right circular cone is V = 1 3 πr2 h. 9

10 10 8. (16 Points) The eccentric anomaly of a comet orbiting the sun is the function E(t) which satisfies Kepler s implicit equation: E 0.9sin(E) = 0.05t where t is time measured in years. The distance r from the sun to the comet is given (in astronomical units) by r = 10(1 0.9cos(E)). a) When is the eccentric anomaly equal to π? How far is the comet from the sun at 6 this time? b) Find dr dt at the time you calculated in part a) (i.e. when E = π 6 ). c) Find the linear approximation of r(t) at the time from part a). d) Use the linear approximation from part c) to estimate the distance of the comet from the sun one month after the time found in a). Recall that t has units of years.

Math 124 Final Examination Winter 2017

Math 124 Final Examination Winter 2017 Your Name Your Signature Student ID # Quiz Section Professor s Name TA s Name Turn off all cell phones, pagers, radios, mp3 players, and other similar devices. This