These questions range from easy to challenging. 

Don't expect to answer the all of the hardest questions correctly.  Most students will not be able to do so.  Everyone should be able to answer the easiest questions, but almost everyone will bog down at some point.  Don't stay bogged down for too long before you move on to the next question.  However do consider every question, and think about it for at least a couple of minutes. 

Do explain how you get your results.  Your instructor won't be able to tell much of anything from just a wrong answer, and even if the answer is right won't be able to tell from just the answer whether you got it by a process you can later build on.

You are also welcome to insert your own questions into the document.  If you don't understand a problem, tell me as much as you can about what you think you do understand, and what you don't understand.  The more information you give me the more likely I will be to be able to respond in a helpful way.

Insert each response on the line after **** and before #$&*.   Submit using the Submit Work Form.

`q001.  Consider the vector function

r(t) = 4 cos( pi / 12 * t) i + 4 sin(pi / 12 * t) j.

• Sketch the vectors r (0), r(2), r(4) and r(6).  Do the vectors 'move' in the clockwise or counterclockwise direction?  Do they get longer, shorter or stay the same length?

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• Sketch the path defined by r(t) for 0 <= t <= 6.

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• What are r( 3 ) and r( 4)?

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• For the interval between t = 3 and t = 4, what is the average rate of change of r with respect to t?

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• What is the expression for r ' (t)?

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• What are the values of r ' (3), r ' (4) and r ' (3.5)?

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• How are your answers to the preceding question related to the average rate you calculated previously?

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• What is the expression for || r(t) || ?

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• What is the expression for r(t) / || (r(t) ||?

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• What is the magnitude of the vector ( r(t) / || (r(t) || )?

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• What is the dot product r(t) dot r ' (t) ?

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`q002.  For the vector function

r(t) = 4 cos( pi / 12 * t) i + 8 sin(pi / 12 * t) j

• Sketch the vectors r (0), r(2), r(4) and r(6).  Do the vectors 'move' in the clockwise or counterclockwise direction?  Do they get longer, shorter or stay the same length?

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  #$&*

• Sketch the path defined by r(t) for 0 <= t <= 6.

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  #$&*

• What are r( 3 ) and r( 4)?

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  #$&*

• For the interval between t = 3 and t = 4, what is the average rate of change of r with respect to t?

  ****

   

  #$&*

• What is the expression for r ' (t)?

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  #$&*

• What are the values of r ' (3), r ' (4) and r ' (3.5)?

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  #$&*

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  #$&*

  How are your answers to the preceding question related to the average rate you calculated previously?

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  #$&*

• What is the expression for || r(t) || ?

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  #$&*

• What is the expression for r(t) / || (r(t) ||?

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  #$&*

• What is the magnitude of the vector ( r(t) / || (r(t) || )?

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  #$&*

• What is the dot product r(t) dot r ' (t) ?

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`q003.  For the vector function

r(t) = 4 cos( pi / 12 * t^2) i + 8 sin(pi / 12 * t^2) j

• Find the values of t for which pi/12 * t^2 takes the values 0, pi/6, pi/3 and pi/2.

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• Sketch the path defined by r(t) for 0 <= t <= sqrt(6).

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• What are r( sqrt(3) ) and r( 2)?

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• For the interval between t = sqrt(3) and t = 2, what is the average rate of change of r with respect to t?

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• What is the expression for r ' (t)?

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• What are the values of r ' (sqrt(3)), r ' (2) and r ' (sqrt(3.5))?

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  #$&*

• How are your answers to the preceding question related to the average rate you calculated previously?

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  #$&*

• What is the expression for || r(t) || ?

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  #$&*

• What is the expression for r(t) / || (r(t) ||?

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  #$&*

• What is the magnitude of the vector ( r(t) / || (r(t) || )?

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• What is the dot product r(t) dot r ' (t) ?

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`q004.  For the foam piece you were given:

• What are the lengths of the sides of the quadrilateral you traced on you paper, starting with the two sides that meet at your chosen point?

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• What is the altitude of the piece at your chosen point

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• What are the altitudes of the piece at the other three corners of the quadrilateral you trace, listed in the counterclockwise order of the corners?

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`q005.  The sides of my quadrilateral, as drawn on the paper, were 4, 2.7, 4, 2.7.  My quadrilateral didn't really form a perfect rectangle, but we'll assume it did.
 

The estimated altitudes of my four points, listed in the order specified above (going counterclockwise around the quadrilateral), were 5, 3, 2.5 and 4.6.

• The vector forming the edge of my foam piece, above the side of the quadrilateral which has length 4 and intersects my initial point, has i component 4, with its k component running from altitude 5 to altitude 3.  Thus the k component of this vector is 3 - 5 = -2.  The vector is therefore 4 i - 2 k.  What is the vector which forms the corresponding side of your quadrilateral?

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• The vector forming the edge of my foam piece, above the side of the quadrilateral which has length 2.7 and intersects my initial point, has j component 2.7, with its k component running from altitude 5 to altitude 4.6.  Thus the k component of this vector is 4.6 - 5 = -.4.  The vector is therefore 2.7 j - .4 k.  What is the vector which forms the corresponding side of your quadrilateral?

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• Let's call these two vectors v and w. The cross product of my vectors is v X w = (4 i - 2 k) X (2.7 j - .4 k) = 5.4 i + 1.6 j + 10.8 k.  The k component is the greatest, the j component the least, which is consistent with the direction of the vector perpendicular to the upper surface of the foam piece.  What is the cross product of your two corresponding vectors? 

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• That cross product is perpendicular to both of your two vectors v and w, so it should be perpendicular to the upper surface of the foam piece.  Is the order of the magnitudes of your i, j and k components consistent with what you see when you point in the direction perpendicular to the surface?

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`q006.  The paper has dimensions close to 28 cm x 21 cm.  If you measured using a reduced ruler, each level of reduction increases these dimensions by a factor of about 1.6. 

I estimate that a vector from the origin to the first point I marked on the paper is 9 i + 12 j.  The point directly above this, when I placed the foam piece on the paper, was 5 cm above the paper, so the coordinates of that point were 9 i + 12 j + 5 k.

When I laid a flat piece of foam on my original foam piece, one of the points where its corner met the paper was at position 16 i + 24 j relative to the origin.  The point was on the paper so its k coordinate was zero.

Estimate the coordinates of a point where a flat piece of material, laid across the top of your foam piece, would intersect the paper.  If you left your foam piece behind, as many of you did, use your memory to estimate where such a point might lie.

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The vector from my point 9 i + 16 j + 5 k to the point 16 i + 24 j was (16 i + 24 j) - (9 i + 12 j + 5 k) = 7 i + 12 j - 5 k.  What vector do you get when you do the same?

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The flat piece used to obtain the second point was parallel to the top of the foam piece, so it should be perpendicular to the vector v X w.   Two vectors are perpendicular if their dot product is zero.  Calculating the dot product of these two vectors, we obtained a result of about 3.  Both vectors had magnitudes greater than 10, so 3 is much less than the product of the two magnitudes, and is therefore quite close to 0.

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Do the same for your v X w vector and the vector between your two points on the paper.  Is your result much smaller than the product of the magnitudes of these vectors?

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