Math 132, Calculus II

Practice Problems for Exam 2

Tom Linton, Fall 2006

The following questions cover most of the important topics from sections 6.4 through 7.7 of the Hughes-Hallett (4th Ed) text. If you can answer most of these questions in a timely fashion, you are likely well prepared for our second midterm exam.

  1. Batman and Robin are out directing traffic at a busy intersection in Gotham City when the Riddler zooms by at a constant speed of 1.34 miles per minute (well over the speed limit, this is just above 80 mph). It takes Batman and Robin exactly 2 minutes to fire up the Batmobile, buckle their seat belts, and start chasing the Riddler (who continues to drive at the constant velocity of 1.34 miles per minute). Assume that the Batmobile has a velocity given by MATH miles per minute, at time $t$ (measured in minutes, with $t=0$ corresponding to the time when Batman and Robin begin their pursuit of the Riddler). Our main goal is to figure out whether or not Batman and Robin catch up to the Riddler, and if they do, we'd like to estimate how long it takes.

    1. Explain why the distance traveled b the Riddler at time $t$ is given by the area function MATH, and then evaluate this definite integral to find a "standard formula" for $R(t)$.

    2. A plot of Batman and Robin's velocity is shown below.


      132ex2pf06__7.png

      The distance taveled by Batman and Robin by time $t$ is given by the area function MATH (where the integrand is plotted above). Use TRAP(40) to estimate the value $B(20)$. After chasing the Riddler for 20 minutes, have Batman and Robin actually traveled farther than your estimate, or have they actually traveled a shorter distance than your estimate? Explain.

    3. Calculate exactly how for the Riddler has traveled by the time $t=20$. Can you be absolutely certain that by the time $t=20$, Batman and Robin have "caught up to" (i.e. traveled farther than) the Riddler?

    4. Calculate a "standard formula" for the distance traveled by Batman and Robin by time $t$, as follows. In the equation MATH, make the substitution $w=\frac{x}{2}+1$, and transform the integral (bounds and all) over to terms of $w$. Now use integration by parts (with $u=\ln (w)$ and $dv=dw$) to evaluate this integral.

    5. You should now have standard formulas (non-area functions) for both $R(t)$ and $B(t)$. Use these formulas to verify that sometime before $t=17$, Batman and Robin "catch up to" the Riddler. Use some form of technology to estimate the time when Batman and Robin first catch the Riddler, accurate to 2 decimal places.

  2. Consider the definite integral MATH.

    1. Is this integral improper? Explain.

    2. A plot of the integrand is shown below. The plot suggests that perhaps if we split this integral into two parts, namely
      MATH
      where MATH and MATH, then $I_{1}=I_{2}$. Verify rigorously that this is indeed the case as follows. Start with the integral MATH and make the substitution $w=1-x$ (transforming bounds and all over to terms to $w$). This substitution may seem like it doesn't quite work out, but if you are persistent, you should be able to eliminate all occurences of $x$ in $I_{1}$ and the resulting definite integral should "become" equivalent to $I_{2}$, thus establishing that $I_{1}=I_{2}$.


      132ex2pf06__34.png

    3. We now know that
      MATH
      so, in order to calculate the value of MATH, we can simply calculate MATH and double its value. Explain why we cannot use LEFT (n), TRAP(n), nor SIMP(n) to estimate the value of $I_{1}$, but that we can use MID(n) to estimate this definite integral.

    4. Since our calculator program actually calculates all five estimates at once, we're stuck, and we'll have to "get things started" by hand. The only trouble spot for estimating MATH is the left endpoint $x=0$. If we were to use MID(n) to estimate MATH, the first sub-interval would run from $x=0$ to $x=\frac{1}{2n}$, with a midpoint of $x=\frac{1}{4n}$. This corresponds to using MID(1) to estimate MATH. By hand, show that MID(1) gives MATH.

    5. Part (d) says that if we use $n=50$ midpoint rectangles to approximate MATH, then the first rectangle yields the approximation MATH. By looking at the plot of the integrand above, does this correspond to an over-estimate or an under-estimate of the true area? Explain.

    6. We can now use any of our estimation techniques to estimate MATH, because we can simply double an estimate of MATH, and we know that
      MATH
      and the last definite integral is NOT improper. Use this idea, along with Simpson's Rule, where $n=50$ to estimate MATH.

    7. Finally, we can also evaluate MATH exactly using the following strategy. Draw a right triangle and label one of the acute angles as $\theta $. Make the hypotenuse have a length of 1, the length of the leg adjacent to the angle $\theta $ should be $\sqrt{1-x}$, and the length of the leg opposite the angle $\theta $ should be $\sqrt{x}$. The triangle suggests the substitution MATH. Use this idea (or the triangle) to show that MATH.

  3. Evaluate each fo the integrals below using the Fundamental Theorem (i,e, do NOT use Mathematica, fnInt, your calculator, or the table of integrals). and show your work.

    1. MATH

    2. MATH

    3. MATH

    4. MATH

    5. MATH

    6. MATH

    7. MATH

    8. MATH

    9. MATH

    10. MATH

    11. MATH

  4. A car starts with initial velocity $5\ \frac{m}{\sec }$ and accelerates at MATH for $t=0$ to 12 seconds.

    1. Find a formula for $v(t)$, the car's velocity after $t$ seconds, and $x(t)$ the position of the car after $t$ seconds.

    2. How "far" does the car travel in 12 seconds?

    3. How long does it take the car to travel 100 meters? How fast is the car traveling at this time?

  5. Consider MATH

    1. What is $F^{\prime }[x]$? How about $F^{\prime }[3]$?

    2. Make a plot of the integrand, from $t=2$ to $5$.

    3. Use MID[8] to estimate $F[3]$. Is this an over-estimate, or an under-estimate? How do you know?

    4. Use MID[16] to give another estimate of $F[3]$.

    5. Assume that the error in MID[16] is exactly one fourth the error in MID[8] to give an even better estimate of $F[3]$.

  6. Three graphs of $f(x)$ are shown below, with vertical lines drawn at $x=1$ and $x=4$ (and no vertical scale shown). Approximating MATH, using LEFT(3), MID(3), and TRAP(3) gave the values (listed from smallest to largest, NOT necessarily in the order stated above), I got 11.8198, 12.0904, and 13.8701.

    132ex2pf06__93.png 132ex2pf06__94.png


    132ex2pf06__95.png

    1. On the first graph above, draw in the 3 rectangles for LEFT(3). On the 2nd graph above, draw in the 3 rectangles for MID(3). On the 3rd graph above, draw in the 3 trapezoids for TRAP(3).

    2. Match each of the rules (LEFT, MID, and TRAP) with the approximate value it produced. State how you decided which rule produced which value, and then decide where the true value of the integral lies (between which two approximations).

    3. What is RIGHT(3)? Hint: you know LEFT(3) and TRAP(3) and hopefully you know a formula that relates TRAP(3) to LEFT(3) and RIGHT(3), which you can solve for RIGHT(3).

    4. What is SIMP(3)?

    5. The y tickmark on the graphs is at an integer value (the same one for all 3 plots). It is also at the average value of f(x) from x = 1 to 4. What is the y-value of that tickmark?

  7. Use the substitution $w=1+x^{3}$ to completely transform (bounds and all) MATH, over to terms of $w$. Then evaluate the integral.

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