t 0. Show the necessary work and make sure that your conclusion is clear. (10 points) MTH 254 Test 1 No Calculator Portion Given: October 14, 2015

Transcription

1 MTH 254 Test 1 No Calculator Portion Given: October 14, 2015 Name 1. Figures A F on page 2 of the supplement show portions of six different vector valued functions along with one surface upon which the graphed curve lies. The formulas for three of the functions are given below. For each formula, identify the figure number for the graph of the curve and state an equation for the surface that is graphed. No work need be shown for this problem. (12 points total) 2 The function r1 t t,, t t is shown in Figure In this figure an equation for the surface is. 2 2 The function r2 t t,, t t is shown in Figure In this figure an equation for the surface is. 3 sin,,cos The function r t t t 2 t is shown in Figure. In this figure an equation for the surface is. 2. Find the unit tangent vector for the function rt 5, sin t cos t, t cos t t 0. Show the necessary work and make sure that your conclusion is clear. (10 points) at the point where Test 1 No Calculator Portion 1

2 3. Suppose that r is a three dimensional vector valued function which neither itself, nor any of its derivatives, ever evaluates to 0. Answer each of the following questions about the motion described by r. Please note that each question is separate from all the other questions. e.g., when answering question d do not assume that anything is true other than the statement made in question d. (6 points total) a. What must be true about the motion if rt r' t for all values of t? b. What must be true about the motion if rt rt for all values of t? c. What must be true about the motion if rt r ' t for all values of t? d. What must be true about the motion if the smallest angle formed when r' t drawn tail to tail is greater than 90 O for all values of t? and r t are 2 T est 1 No Calculator Portion

3 4. Suppose that for a certain function r you know that r 5 3,12, 4. Answer each of the following questions about the motion described by r at the instant that t 5. You may simply write the values in the provided blanks no explanation is required. (5 points total) What is the maximum possible value for a T? What is the minimum possible value for a T? What is the maximum possible value for a N? What is the minimum possible value for a N? 5. Suppose that for a certain function, r, you know that r 2 0,7,5, 1 5 2, and r 2 8,1,6, ˆ 3 4 N 2,0,. Find the center of the osculating circle to r at the point 5 5 0,7,5. Show the necessary work and make sure that your conclusion is clear. (8 points) Test 1 No Calculator Portion 3

4 6. For a certain function, r 9, 4 r t dt 2,3, 5. What is the shortest possible arclength traversed along r between the points where t 4 and t 9? Make sure that your reasoning and conclusion are both clear. (6 points) State a vector valued function for the curve that lies both on the cylinder y z 4 and the plane y 3 x. To receive full credit, each of your components must include a sine expression or a cosine expression. You do not need to show any work, just state the formula for your function. (6 points) 8. A certain vector valued function, r, describes motion confined to this sheet of paper. A portion of the curve described by the function is shown below. The velocity vector is shown for the moment when t 4 (and it s drawn at the point r 4 ). At that moment the speed of the motion is decreasing. Draw a possible vector representation of r 4. That s it no explanation required. Please note that there are many correct answers just draw one of them. (3 points) 4 T est 1 No Calculator Portion

5 MTH 254 Test 1 Calculator Portion Given: October 14, 2015 Name General Directions Unless otherwise stated, for each problem you need to show all relevant work in an organized manner and you need to make sure that both your reasoning and your conclusion are clear. Where relevant, this includes clear annotations on graphs that accompany the problem Suppose that r t t, t, t. Find each of the following. Please read the general directions before proceeding with the test. (7 points each) a. Find an equation for the normal plane to r when t 1. Write your final answer in the form axbycz d where all of the constants are integers. b. Find the rate at which the speed is changing when t 1. Test 1 Calculator Portion 1

6 2. Consider the function r t 3sin t,7cost. Determine the maximum curvature that occurs anywhere along this curve. Please note that there are many correct ways to approach this problem. Having said that, some approaches are easier to execute than others. Whatever approach you choose, make sure that your reasoning is clear, that you show all of the relevant work, and that your conclusion is clear. (9 points) 2 T est 1 Calculator Portion

7 Suppose that r t 2t 3t 36t 1,2t 9t 12t 2,2t 12t 18t 5. There is exactly one point on r where the tangent line to r is parallel to the z axis. Determine that point. Make sure that your reasoning is clear, that you show all of the relevant work, and that your conclusion is clear. (9 points) Test 1 Calculator Portion 3

8 Let r t t t, t t. a. Find each of the following on your calculator. Write the results in the provided blanks. All answers should be exact no decimals. (2 points each) T ˆ 1 N ˆ 1 a T 1 a N 1 b. Calculate, by hand, the vector a 1 Tˆ 1 a 1 Nˆ 1 T N. Show all of the work! What is the significance of this vector i.e. what is a much simpler way we could have come up with the same vector? (4 points) 4 T est 1 Calculator Portion

9 The binormal vector is perpendicular to the osculating plane. The unit tangent vector is perpendicular to the normal plane. The unit normal vector is perpendicular to the rectifying plane. Curvature: t r t r t r t 3 Components of acceleration: a T t x r t r t f x 2 1 f x 3/2 and an t r t r t r t r t Double angle identities: sin 2t 2sin tcost 2 2 cos 2tcos t sin t 2 cos2t 2cos t1 2 cos2t1 2sin t Test 1 Supplement 1

10 Figure A Figure B Figure C Figure D Figure E Figure F 2 T est 1 Supplement