Math 1300: Calculus I, Instructor: Dana Ernst 

Review for Midterm Exam  #3

Here is a review for your upcoming test.  The test covers sections 4.7 through 5.8.  I will not collect this.  Do what you want with it. This review will give you a good indication of what you will be expected to know for the test.

 

Here is a list of things that you must know and understand:

• Limits at infinity
• Sketch graphs
• L’Hopital’s Rule

• Approximate area under a curve using Riemann sums
• Find exact area under a curve using limits of Riemann sums
• Definition of definite integral (limit of Riemann sums)
• Approximate definite integral
• Fundamental Theorems of Calculus
• Basic integration techniques including substitution
• Evaluate integrals using u-substitution
• Average value of a function on a closed interval

• Solve basic differential equations given initial conditions
• Find area of regions bounded by continuous functions

 

Reminders:  During the exam, SHOW ALL YOUR WORK and JUSTIFY YOUR ANSWERS when necessary.  Also, be organized and neat!

 

Answer each of the following questions.

 

1.     Sketch the graph of the following functions by finding x-intercepts, y-intercept, vertical asymptotes, horizontal asymptotes, intervals of increase and decrease, coordinates of turning points (local max/min), intervals of concavity, coordinates of inflection points.

 

 

2.     Determine each of the following limits.

(a)    

(b)   

(c)    

(d)   

(e)    

(f)    

(g)    

(h)   

 

3. Explain the difference between an indefinite integral and a definite integral.

 

4. Explain why the Power Rule for Integration,
   
   , does not apply to the function
   
   .

 

5. Using Riemann sums, approximate the area under the graph of
   
    on the interval
   
   .  Use equal width partitions, right endpoints, and
   
   .

 

6. Evaluate the following definite integrals using the limit of Riemann sums.  Use equal width partitions and
   
   .  That is,
   
    is the right hand endpoint of the ith subinterval.

(a)    

(b)   

 

7. Evaluate the following integrals.

(a)    

(b)   

(c)    

(d)   

(e)    

(f)    

(g)    

(h)   

(i)     

(j)     

(k)   

(l)     

(m)  

(n)   

 

8.     Solve the following initial value problem: , .

 

9.     Find f given that , , and .

 

10.  Find the area of the region bounded by the following graphs.

(a)   , .

(b)  , , , .

(c)   , .

 

11.  Find the average value of  on the interval .

 

12.  Find the derivative of the following functions.

(a)    

(b)