Unformatted text preview: Math 102 Old exam questions dealing with (roughly) Chapters 1-5 of Keshet’s Course Notes For Math 102 Section 102 Fall 2009, this is good practice related to Homework #1. [3pts] (a) During this time interval, when is the particle farthest from its initial position? dec2008q5 December During the time Name: Page direction) [4pts] (b) 2008 Math 102 interval 0 ≤ t ≤ 4, what is the greatest speed (in either 7 out of 13 December 2008 Math 102 Name: Page 7 out of 13 of the particle? Problem 5: Problem 5:motion is described by y (t) = t3 − 6t2 + 9t A particle’s 3 A particle’sis the displacement (in metres), t tis − 6t2 (in t where y (t) motion is described by y (t) = time + 9 seconds), and 0 ≤ t ≤ 4 seconds. where y (t) is the displacement (in metres), t is time (in seconds), and 0 ≤ t ≤ 4 seconds. [3pts] (a) During this time interval, when is the particle farthest from its initial position? [3pts] (a) During this time interval, when is the particle time(s) = At farthest from its initial position? [4pts] (b) During the time interval 0 ≤ t ≤ 4, what is the greatest speed (in either direction) Greatest speed = of the particle? At time(s) backward directions) that [3pts] (c) What is the total distance (including both forward and = the particle has travelled during this time interval? At time(s) = [4pts] (b) During the time interval 0 ≤ t ≤ 4, what is the greatest speed (in either direction) [4pts] particle? the time interval 0 ≤ t ≤ 4, what is the greatest speed (in either direction) of the (b) During of the particle?
Greatest speed = mt1-2008(jun)q3
3. Consider the function p(t) = t2 − 6t. 3.Velocity (a) What is the average rate of change of p(t) between t = 2 and t = 3? (b) Is there any time t0 , between t = 2 and t = 3, where the instantaneous rate of change is equal to the average rate of change between t = 2 and t = 3? If so, solve for t0 . If not, explain why in a sketch or carefully-written sentence. Solution: (a) The average rate of change is given by ∆p p(t2 ) − p(t1 ) = ∆t t2 − t1 (32 − 6(3)) − (22 − 6(2)) = 3−2 (−9) − (−8) = = −1 1 (b) Let’s try to solve for t0 . If we can, then the answer is clearly “yes”. dec2004q1 dec2006q6 2006 Decemb er
 i) ii) iii) iv) v) 6. Mathematics 102 Name 5. Derivatives Page 6 of 11 pages x Sketch the graph of f (x) = on the grid provided, showing all of the following if they 1 + x2 are present: x and y intercepts critical p oints intervals where f is increasing or decreasing p oints of inﬂection intervals where f is concave up or down. dec2004q6 5. Derivatives dec2000q2c 5. Derivatives Problem 2: The following problems require little or no computation. Put the answer in the box provided. You may use space here for supporting reasoning or computations, if any, d which part-marks would be awarded. for ec2008q2a (a) Suppose y = f (x) is a function having the following properties: f (1) = 2, f (1) = 3. Suppose g (x) is the inverse function for f (x). Then the slope of the tangent line to g (x) at the point x = 2 is: slope= (b) A diﬀerential equation describing radioactive decay of a substance is dy = −ky, dt where y (t) = Ce−kt is the amount of radioactivity remaining. A plot of y (t) versus t is found to be a curve that goes through the point (0, 10), and has tangent line of slope −5 at that point. What are the values of the constants? C= , k= mt1-2007(israel)q2 5. Derivatives cemb er 2006 dec2006q8 Mathematics 102 Name 5. Derivatives 8 Page Find a and b such that f (x) = ax3 + bx2 + 1 has an inﬂection p oint at (−1, 2). ,b= mt1-2008(jun)q2 5. Derivatives 2. The graph of the derivative, f (x), of a certain function f (x) is shown below. (a) Where is the original function, f (x), increasing? (b) Where does the original function, f (x), have an inﬂection point? Solution: (a) The sign of the derivative tells us where the function is increasing or decreasing.
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- Derivative, 4pts, 4 seconds, 3pts, ec2008q2a