## 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.
• 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?
• How to locate a root using Intermediate Value Theorem?

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?

Chapter 4: Application of derivatives (4.2-4.6)

• What are possible locations of extreme points?
• Are critical points necessarily local min or local max?
• How to determine increasing and decreasing parts of a function?
• How to determine concave up and concave down parts of a function?
• Understand the meaning of horizontal, vertical, oblique asymptotes. Give an example for each of them.
• Understand the meaning of dominant terms in a rational function. Give an example of a quadratic dominant term.
• For optimization, pay attention to the domain. What are possible locations where the function can attain absolute maximum or minimum?
• If f'(x_0)=0, is it true that f(x_0) must be a local minimum or local maximum? Are there other possibilities?
• Sometimes people get confused between Intermediate Value Theorem and Mean Value Theorem. Make sure that you know the difference. What is the relation between Mean Value Theorem and Rolle's Theorem? Memorize them!

Chapter 5: Integration (5.1-5.6)

• Definition of Riemann Sum.
• Meaning of the limit " || P || -> 0".
• (Impotent) Be able to express a definite integral as a limit of Riemann sum.
• (More impotent) Be able to express the limit of a Riemann sum as a definite integral and evaluate the limit.
• (Most Important) The statement and applications of `Fundamental Theorems of Calculus' Part 1 and 2. Memorize them!
• How to take derivatives with respect to the upper and lower limits of integration? (page 304)
• The meaning of indefinite integrals.
• How to solve Initial Value Problems using definite or indefinite integrals?
• How to use the chain rule in integration?

Lecture notes (manuscript):

Lecture notes (typed):