1. MATH 110 Differential Calculus
Functions are everywhere. They are fundamental objects in calculus. The objective of this course is to discover the properties of a given function by using its derivative functions.
There are three major components in this course:
Computing limits – Chapter 2
Computing derivatives – Chapter 3
Applications of derivatives – Chapter 4

2. Topics to be covered Appendix A, B, and D: High school math review
Chapter 1: Functions
Fundamental objects in calculus
Like numbers, there are different ways to represent a function
Chapter 2: Limits and Derivatives
Limits lay a foundation for calculus
Derivative is defined as the limit of a special expression

3. Topics to be covered (Continued)
Chapter 3: Differentiation Rules and formulas
Instead of using the definition to computer derivatives, we establish some rules, called differentiation rules, and some formulas, and then apply these rules and formulas to compute derivatives
Each of you must know how to compute derivatives

4. Topics to be covered (Continued)
Chapter 4: Applications of Differentiation
(For your convenience, all topics of application are collected here)
Equation of tangent line
Linear approximations and differentials
Related rates
Maximum and minimum value problems
The mean value theorem
Curve sketching
Optimization problems
Antiderivatives

5. Notations used throughout this course
“A  B” means that “the statement A implies the statement B”
“A  B” means that “statement A implies the statement B and the statement B implies the statement A”. So “A  B” means that two statements A and B are equivalent.

8. Example for fun: In the following, we want to
show that the weight of an elephant equals the
weight of a mosquito. Of course, it’s a absurdity.
Can you find the mistake in the process of “proof”?
Proof:
Let E = the weight of an elephant
m = the weight of a mosquito
w = E – m
Then
w = E – m  E = m + w
 E(E – m) = (m + w) (E – m)
--- both sides multiplied by E – m.

9.  E2 – mE = mE + wE – m2 – mw
--- multiply out both sides
 E2 – mE – wE = mE – m2 – mw
--- subtract wE from both sides
 E (E – m – w) = m(E – m – w)
--- take out the common factor E on left side
and the common factor m on right side
 E = m
--- dividing both sides by E – m – w
Thus, the weight of the elephant is the same as the
weight of the mosquito. What’s going wrong? Can
you find out the error?