Mr. Simonds’ MTH 254 Class – Fall Term 2016
MTH 254 – Test 1 Given: October 20, 2016 Name You may use the full functionality of your calculator – e.g. to calculate cross products, derivatives, etc. 
1. Determine a vector‐valued function that graphs to the tangent line to r  t   t 2 ,7  3 t at the point  9,16  . Show all relevant work and make sure that both your reasoning and your conclusion are clear. (10 points) 
2. Find an equation for the osculating plane for the function r  t   2 sin  t  , 2sin  t  , 2 3 cos  t  at the point where t 
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. Show all relevant work and make sure that both your reasoning and your conclusion are clear. (12 points) Test 1|1
Mr. Simonds’ MTH 254 Class – Fall Term 2016
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3. Find each of the following for the curve r shown in Figure 1. That’s it … you do not need to show any other work … just state the answer to each question. (2 points each)  6,9,9 
a. State Tˆ  2  . b. State Nˆ  2  ? c. What is the equation of the osculating plane when t  2 ? Figure 1
The moving object is at the
top of the ellipse at t  2 .
4. Figures A‐F on the graph supplement show portions of six different vector‐valued functions along with one surface upon which a given 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. Please note that the cylinders shown in the figures B, D, and F are all indeed circular. (15 points total) 
The function r1  t   cos  t  ,sin  t  , cos  2 t  is shown in Figure The function r2  t   cos  t  , 2 sin 2  t  , cos  2 t  is shown in Figure 2 cos  t  , 2 sin  t  , 2 cos  t  is shown in Figure In this figure an equation for the surface is . . . 
In this figure an equation for the surface is 
The function r3  t  
In this figure an equation for the surface is 2|Test 1
Mr. Simonds’ MTH 254 Class – Fall Term 2016
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5. For a certain function r , r   7   2, 2,  4 and the osculating plane at the point where t  7 has equation x  y  z  8 . Determine the two possible values (vectors) for Nˆ  7  . Make sure that both your reasoning and your conclusion are clear. (7 points) 1
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6. For a certain function r , r  9   1, 2,7 , Tˆ  9   3, 4,0 , and Bˆ  9   0,0,1 . 5
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a. Determine an equation for the osculating plane to r when t  9 . Make sure that both your reasoning and your conclusion are clear. (5 points) b. Determine Nˆ  9  . Make sure that both your reasoning and your conclusion are clear. (5 points) Test 1|3
Mr. Simonds’ MTH 254 Class – Fall Term 2016
7. An object is continuously moving clockwise around a circle. The acceleration function maintains a constant magnitude of 2 ft/s/s and a constant angle with the radius of 30O (as illustrated in the diagram below). Explain why the described situation is actually impossible. Note, you will want to use an appropriate formula in your answer. (5 points) 8. As discussed in class, whether or not the function associated with circular motion has constant magnitude is dependent upon the relative location of the axis‐system’s origin. a. Draw an axis system relative to the circle on the left so that the function that describes the 
circle, r , does have constant magnitude. Draw a different axis system on the right so that the 
function that describes the circle, q , does not have constant magnitude. Keep the two axis systems on their own side of the dividing line. (3 points) 
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b. State something that is true about r  t  and r   t  that is not true about q  t  and q  t  . Also, state the theorem that is being illustrated. (4 points) . 4|Test 1
Mr. Simonds’ MTH 254 Class – Fall Term 2016
9. An object is moving from left to right at constant speed along the curve shown below. Draw possible acceleration vectors at each of the two indicated points. Make sure that you consider the relative lengths of the vectors you draw. (3 points) 
10. Determine Nˆ 1 for the function r  t   t 3  t 2 , t 4  t ,2 t 5 . You may simply write down the result – no other work need be nor should be shown. Exact value only – no decimals. (Note, the correct answer is not attractive – but it’s also not crazy – if you’re getting a crazy answer (lots and lots and lots of digits) try again.) (5 points) 
11. For a certain function r , the velocity vector when t  1 is 2,1,1 and the acceleration vector is 3,  1, 2 . The linear unit is meters and the time unit is seconds. At what rate is the speed changing at that moment? Make sure that both your reasoning and your conclusion are clear. Make sure that you include the correct unit in your stated value. (8 points) Test 1|5
Mr. Simonds’ MTH 254 Class – Fall Term 2016
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12. Find the center of the osculation circle for the function r  t   2 2 sin  t  ,3 2 cos  t  at the point where t 
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. Show all relevant work and make sure that both your reasoning and your 4
conclusion are clear. (14 points)  3
 4
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 points in the direction of 3, 2 ‐ you do not need to use your calculator to 
 3 
determine Nˆ 
 . (12 points)  4 
Note: Nˆ 
6|Test 1