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?
Lecture notes (manual):