Space Curves
There will be a second ten point quiz given on Thursday Oct 17 which will provide the scores for Quiz 3 and Quiz 4. The quiz will require you to analyze a space curve in the following manner.
• Given r(t), graph it by elliminating the parameter. Remember rules like sine squared plus cosine squared is 1. In 3 dimensions, first graph the xy projection lightly in pencil and then draw the z coordinate above the picture.
• Plot points like r(0) or r(1) and r(pi). These are just dots added to the above graph. You will need to know how to evaluate sine and cosine at simple angles like pi 5pi 3pi/2 etc so learn the unit circle.
• Find r'(t). Be careful to do each component seperately and use product and chain rules as necessary.
• Find the speed. The speed is the magnitude of the velocity r'(t).
• Plot r'(0). First plug the number into the formula for r'(t) and be careful with the parenthesis and trigonometry. Then plot the vector by placing the tail at r(0) and moving up and over according to the value of r'(0). The answer should be an arrow which is tangent to the curve.
• Find the tangent line to r(t) at t=0. The direction is r'(0), the point is r(0), use the parametric equation for a line.
• Find the acceleration. The formula is r''(t) and just be careful to differentiate each component seperately. Note that if the speed is constant r'(t) is perpendicular to r''(t). Note if the path is a circle and the speed is constant then the acceleration vector points inward radially. Remember that when plotting r''(0) you put the tail at r(0). Note that r"(t)=-9.8k when it is acceleration due to gravity in meters per second squared.
• Find the arclength from t=0 to t=5. Use the formula for arclength which is the integral of the speed. You may wish to review integrals involving square roots. Note that a curve is parametrized by arclength if the arclength from 0 to s is s for all values of s. This is the same as having constant speed =1.
The following examples should be practiced:
• r(t)=(2 t -3, t squared)
• r(t)=(t-5, 3t squared)
• r(t)=(5cos(2t+pi/2), 5sin(2t+pi/2))
• r(t)=(3 cos(pi/2 - t), 3 sin(pi/2 -t))
• r(t)=(t squared, t-2)
• r(t)=(2 t -3, t squared, 4t)
• r(t)=(cos(t), sin(t), t)