## Math 1300: Calculus I

### Review for Midterm Exam #1

Here is a review for your upcoming test.  The test covers sections 1.1 through 3.3.  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.  Obviously, this review is longer than the exam will be.  In a few days, I will post partial solutions to the questions in Part II.

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

• Shapes of basic graphs
• Find domain
• Combine functions together (sum, difference, product, quotient, composition) and find domain
• Find slope
• Find equation of line passing through 2 given points
• Find equation of line parallel/perpendicular to a given line and passing through a given point
• Limit Laws
• Evaluate limits (both sides, from the left, and from the right)
• Squeeze Law
• Continuity
• Intermediate Value Theorem
• Limit definition of derivative
• Find slope of tangent line to a given function at a given point
• Find equation of tangent line to a graph at a given point
• Power rule
• Product rule
• Quotient rule
• Chain rule

Reminders:  During the exam, SHOW ALL YOUR WORK and JUSTIFY YOUR ANSWERS when necessary.  Also, be organized and neat!  Don’t forget to write limits where you need them.  Also, don’t forget to ask yourself if you need to rewrite a function before finding the derivative.  And, if you rewrite a function before finding the derivative, do not label the function as the derivative.

Part I: Give a brief answer to each of the following.

1. When you are trying to find the domain of a function, what are the two major things that you need to think about avoiding?

1. Given functions   and   , how to you find the domain of   ?

1. What is the first thing that you should do when evaluating limits?  What are the three possibilities and what do you do in each case?

1. When doing limits involving trig functions, what is the theorem that you want to try to use?

1. When using the Squeeze Law, how do you go about picking the “top” and “bottom” functions?  What work do you need to show?

1. What would a question look like that would require you to use the IVT?  What are the hypotheses that you need to satisfy to use the IVT?

1. What is the limit definition of the derivative?  What the heck is the derivative?

1. Think of some examples where it’s a good idea to rewrite the equation before finding the derivative.

1. When do you need to use the product rule? Quotient rule? Chain rule?

1. How do you find the equation of the tangent line to a graph at a given point?

Part II: Answer each of the following questions.

1. Exercise #19, page 49

1. Exercise #23, page 49

1. Exercise #6, page 32

1. Use limit laws to evaluate:   .

1. Evaluate the following limits if they exist.

(a)   (b)   (c)   (d)   (e)   (f)   (g)   (h)   (i)  for    (j)   6.     Exercise #27, page 85

7.     Is the following function continuous for all real numbers?  Why or why not?   8.     Prove that the following function has at least one zero in the interval  .    9.     Find all vertical asymptotes of the following function.   10.  Explain why the limit as x approaches 0 of the following function does not exist.   11.  Use the limit definition to find the derivative of the function  . 12.  Given  , prove that  does not exist.  13.  Differentiate each of the following functions, but do not simplify.

(a)   (b)   (c)   (d)   (e)   (f)   (g)   (h)   1. Find an equation of the tangent line to the graph of   at   .