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

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

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

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

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.

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.

 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?