Here are some partial solutions to the questions on Review for Midterm #3. For most of the problems, I’ve just provided the answer, so that you can check to make sure that you are doing the problem correctly. In some cases, I’ve provided some of the intermediate steps to help guide you.
1. Along the way you will get the following. x-intercept: , y-intercept: , vertical asymptote: None, horizontal asymptote: , f is increasing on , f is decreasing on , f is concave down on and , f is concave up on , and . Use this information to sketch graph.
2. Remember that you can only use L’Hopital’s Rule on the indeterminate forms or . You don’t need L’Hopital’s Rule on some of these.
(a) Since as , .
(b) Comparing degrees, we get .
(c) Comparing degrees, we get .
(d) Comparing degrees, we get .
(e) This one is in the form . Apply L’Hopital’s Rule once to get .
(f) This one is not in the right form, but it is easy to put it in the right form. We get .
(g) On this one, get common denominators and apply L’Hopitals’s Rule twice. The answer is 0.
(h) Here, you need to take the natural log of , bring exponent down, and then apply L’Hoptital’s Rule. You get . Then .
3. The output of an indefinite integral is a family of functions, whereas the output of a definite integral is a single number (which can be thought of as the area of the graph of a function over an interval where the function is nonnegative).
4. For the function , . If we use in the formula, we would end up dividing by 0. What is ? Why?
5. First, we get , , and . Using this information, we get . Plugging in , we get .
6. Similar to the above problem, but use the limit as n goes to infinity.
7. On the indefinite integrals, don’t forget the C!!!
(e) Replace with . Answer: .
(f) Let . Answer: .
(g) Let . Answer: 4.
(h) Let . Answer:
(i) Let . Answer: .
(j) Turn into 2 fractions and integrate as usual. Answer: .
(k) Let . Answer: .
(l) Let . Answer: .
(m) Let and also substitute in , then separate into 2 fractions. Answer: .
(n) Split into 2 integrals. Evaluate the first integral as usual. Use for the second integral. Answer: 4.
11. . Use .
12. Use the Fundamental Theorem of Calculus.