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1 Page 1 of 10 Last Name (BLOCK LETTERS): First Name (BLOCK LETTERS): APSC 171 Term Test 1 October 8, 2014 A. Ableson U. Iqbal P. Li Instructions. ˆ This is a 2 hour test. ˆ No textbooks, notes, or aids other than your calculator are allowed. ˆ There are 8 questions (some with multiple parts). ˆ Show all your work. Partial credit will be awarded according to the completeness of your work. ˆ Do not turn the first page until instructed by your proctor. ˆ While waiting to start, fill in your name and student number above. ˆ If you need more room, there are blank pages at the end of the test. If you use these pages, you must provide clear directions to the marker, e.g. Continued on page 10. Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Total

2 Page 2 of 10 Question 1. Differentiate the following functions. (You may assume that the input variables will always be within the domain of their respective functions.) You do not need to simplify after you have taken the derivative. (a) g(x) = sin(ln(5x)) [/2] (b) r(t) = arccos(3t), 10 3 t e 2t [/3] (c) f(u) = (u 2 + 3) tan(u). Your answer should only be in terms of u. [/4]

3 Page 3 of 10 Question 2. Let r(t) = ln(t), 1t 1 (a) Eliminate the parameter to find a Cartesian equation for the graph of r(t). Put your equation in the form y =.... [/3] (b) What is the domain of r(t)? [/1] (c) Sketch a graph of r(t), and indicate the direction of increasing time with an arrow. Clearly indicate the scale on your axes. [/2] y x

4 Page 4 of 10 Question 3. You borrowed your older sister s compressed air launcher (which shoots foam projectiles), and are planning a sneak attack on your friend next door. Your friend is 10 m away from your launch point, behind a tall fence. y + θ Fence Friend + 10 m x The launcher shoots its projectiles at an initial speed of 20 m/s, but you can choose the launch s angle from horizontal, θ. In flight, the ball is subjected only to gravity. (Assume that the acceleration due to gravity is 9.8 m/s 2 in magnitude, and that wind resistance is negligible.) (a) Find an expression for the velocity vector of the ball over time, factoring in the initial velocity. (Your answer will contain θ.) [/3] (b) Find a launch angle θ for which the ball will land on the target. [/5] Note 1: the identify sin(2x) = 2 cos(x) sin(x) may be helpful. Note 2: your launch angle must be greater than π/4 radians (45 o ) to get over the fence.

5 Page 5 of 10 Question 4. Match the following graphs to their corresponding functions. No explanation is necessary. A: B: C: D: E: F: [/5] (a) r(t) = t cos(t), t sin(t), t is graph (b) r(t) = t, sin(50t), t 3 t is graph (c) r(t) = cos(t), sin(t), t is graph (d) r(t) = t 2, t, cos(t) is graph (e) r(t) = sin(4t), e t, cos(4t) is graph (f) r(t) = 1 t 2 + 1, t2, t 3 is graph

6 Page 6 of 10 Question 5. (a) Find the exact value of arctan(tan( 8π/3)). [/1] (b) Give an alternate form of the function f(x) = tan(arcsin(x)) which does not involve trigonometric functions. (You may assume 1 x 1). [/2] (c) Find the slope of the tangent line to the curve defined by xy = sin(x) arcsin(y) at the point ( π 2, 0.5 ). You can give the answer in decimal form. [/2]

7 Page 7 of 10 Question 6. Two particles are scheduled to move along the trajectories given by the vector-valued functions below: r(t) = 2 t, 2t 4 and u(t) = t + 2, t 2 4t 4 (a) Do the planned trajectories cross? If they do, find the location of the crossing(s). [/3] (b) Would the particles collide? If they do, at what time and at what (x, y) location would the collision(s) occur? [/2]

8 Page 8 of 10 Question 7. Evaluate the following limits, or indicate that the limit does not exist. Support your answer, using L Hopital s rule if appropriate. (a) lim x ln(x 4 ) x [/2] (b) lim x 0 e 5x 1 x 2 [/2] (c) lim t 0 cos(at) cos(bt) t 2, given that a and b are positive constants. [/2]

9 Page 9 of 10 Question 8. Tenderheart Bear, one of the mystical Care Bears, is driving in his Cloud Car, with his position given by r(t) = t + 1, t 2, < t < His friend Funshine Bear is standing at coordinates ( 1, 0). When Tenderheart presses a button in his Cloud Car, the car will send a magical hug straight ahead in the direction the car is moving at that moment. If Tenderheart wants to send a hug to Funshine without having to stop, at what time t should Tenderheart press the button? [/6]

10 Page 10 of 10 Space for additional work. Indicate clearly which Question you are continuing if you use this space.