Math 1300: Calculus I, Instructor: Dana Ernst

Partial Solutions to Review #3 

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.

(a)   1

(b)  2

 

7.     On the indefinite integrals, don’t forget the C!!!

(a)    

(b)   

(c)    

(d)   

(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.

 

8. If you integrate
   
   , you get
   
    (this is the general solution to the differential equation).  Now, we find the particular solution by finding C.  Since
   
   ,
   
   , and so
   
   .  Therefore, the answer is
   
   .

 

9. Integrating and using
   
   , we get
   
   .  Then integrating again and using
   
   , we get the final answer of
   
   .

 

10. For these 3 problems, draw a picture, determine interval of integration, and use
    
    .  In the first 2 problems, the bottom is
    
    .

(a)   .

(b)  .

(c)   .

 

11.  .  Use .

 

12.  Use the Fundamental Theorem of Calculus.

(a)    

(b)