1. Three cars ran a race and their velocities are given below. The race was 200 feet long and the velocities are given in feet per minute. The first car, car 1, had a velocity of 1000sin(0.65t²) feet per minute, t minutes into the race. The second car had the velocities given in the table below.

The third car's velocities are plotted below.

1. For car 1, would TRAP(n) over-estimate or under-estimate the distance traveled by car 1 from t = 0 to t = 1 minutes? Explain.
2. For car 1, would MID(n) over-estimate or under-estimate the distance traveled by car 1 from t = 0 to t = 1 minutes? Explain.
3. Verify that car 1 finishes the race between t = .98 minutes and t = .99 minutes using TRAP(100) and MID(100). Explain your reasoning.
4. For car 2, clearly explain, using LEFT(10) or RIGHT(10), why car 2 did not win the race.
5. What does MID(5) give for an estimate of the distance traveled by car 2 from t = 0 to 1 minutes? Is this estimate too big, or too small? How do you know?
6. Calculate the exact time that car 3 finishes the race.
7. Who won the race?
   
   
   
   
   
2. Evaluate each of the integrals below, using the Fundamental Theorem (i.e. do not use fnInt to do any of the definite integrals) and show your work.
1. 
2. 
3. 
4. 
5. 
6. 
7. 
   
   
8. 
9. 
10. 
11. 

3. Consider , where a graph of f(x) is shown below. Use the graph to answer the following questions.

1. Estimate values for LEFT(4), RIGHT(4), MID(4), TRAP(4) and SIMP(4) for this integral.
2. Draw in the rectangles used to calculate MID(4).
3. Which of your estimates are guaranteed to be over-estimates? Which are guaranteed to be under-estimates? For which estimates can you not tell for sure whether they are under or over-estimates?
4. It turns out that, for the integral above, MID(8) = 11.514 and MID(16) = 11.4944. Assume that the error in MID(16) is exactly one-fourth the error in MID(8). Use this assumption to give a better estimate of .
   
   
   
4. Consider .
1. Verify that this integral is improper. Explain why LEFT(n), RIGHT(n), and TRAP(n) cannot be used directly to estimate this integral.
2. Even fnInt will have trouble with this integral (try it, if your calculator simply stays busy for a long time, press the [ON] key to quit the calculation). To estimate this integral, we will cut it into two pieces, namely I₁ =  and I₂ = . Which of LEFT(n) or RIGHT(n) could be used to estimate I₁? Which of LEFT(n) or RIGHT(n) could be used to estimate I₂? Explain.
3. Verify that  =  (that is, show that I₁ = I₂). HINT: make the substitution u = 1 - x, and completely transform I₁ into terms of u.
4. Part (c) says that  = 2, so we only need to estimate , and double it's value. The trouble spot is x = 0, since f(0) is undefined. This means we cannot use LEFT(n) (so our program won't run), nor can we use TRAP(n) (since it uses LEFT(n)). To remove this trouble spot, we need to estimate , and then we could use any of our estimates from x = 1 / (2n) to x = 1 / 2. I chose the upper bound of integration to be 1 / (2n), since that corresponds to using n rectangles from x = 0 to x = 1 / 2. Show (by hand) that using one midpoint rectangle to estimate  gives the estimate .
5. Evaluate the estimate from part(d) for n = 50. We can now estimate  using any of our methods by adding the part(d) estimate for n = 50, to what we get (for example) by using LEFT(49) from x = 0.01 to x = 0.5. Use this idea to give left, right, mid, trap and simp like estimates for  for "n = 50".
6. Between which two of your estimates from teh last part does the true value of the integral lie? Double your best estimate to give an estimate for the original integral from x = 0 to x = 1.
7. Now we'll calculate the exact value of I = . Draw a right triangle with hypotenuse 1, altitude square root of x, and adjacent side length square root of 1 - x. Name the lower left angle q. Show that this triangle gives the substitution x = sin²(q). Use the triangle to completely transform the integral I into terms of q. Evaluate the resulting integral (it is easy) to calculate the exact value of the integral I.