Study Guide for Calculus (I)

Chapter 2: Limits and Continuity

• Reading suggestion: Focus on 2.4-2.5.
• Section 2.1-2.3 are high school materials. We will not go into details in class. Pay attention to examples where the limit does not exist.
• How to locate a root using Intermediate Value Theorem?
• Must know the formal definition of limits (Section 2.5) and know how to verify it (like Example 5 on p 104).
• Why is x=c excluded in the definition of limits?
• How to apply Sandwich Theorem?
• Definition of continuity. Examples of functions discontinuous at a point c.
• Compare the definitions of "lim_{x ->c} f(x) = L" and "f(x) is continuous at c". Why is x = c now included in the definition of continuity?

Chapter 3: Derivatives

• We will very briefly go over 3.1-3.3 as a review for what you learned in high school. Focus your reading on 3.4-3.8.
• Able to give examples of non-differentiable functions of various type (p 125 of textbook).
• Does continuity imply differentiability? Does differentiability imply continuity?
• Understand and memorize the product rule and the quotient rule.
• Understand and memorize the derivatives of all six trigonometric functions.
• Understand and memorize the chain rule.
• Practice chain rule by computing the derivatives of g(f(x)) where f, g are elementary functions chosen by yourself such as polynomials, rational functions, square root and trigonometric functions. You should practice as much as possible till you can write the solution directly for very complicated expression, ie. without defining intermediate variables.
• Why implicit differentiation?
• Be able to memorize the formula of linear approximation, ie. the equation for the tangent line.
• Understand and memorize the the linearization of (1+x)^k for any number k.
• How to estimate the error between f(x) and its linear approximation?
• Understand the meaning of 'differential'. It's just a notation, but you might see it again in the future.
• How to estimate the change of a function using differentials?
• Understand how Newton's method works and be able to derive it (IMPORTANT).
• When does Newton's method NOT work?

Lecture notes (manual):

Lecture notes (typed):