General recommendations:

- Look for patterns (especially in book problems, WeBWorK)
- What has not been assessed yet?
- Think about what the class found difficult. This might be assessed again to measure improvement.
- Remember that definitions and theorems are important.

Limits

- Average rate of change (secant lines) vs.

Instantaneous rate of change (tangent lines) - Intuitive understanding:
- one-sided
- numerical tables
- graphing

- L→∞ and vertical asymptotes
- Limit computation
- Limit laws
- Scalar (constant multiplier)
- Sum & difference
- Multiplication & division (no division by zero)
- Power

- Substitution property (*needs continuous)
- Factoring
- Multiply by conjugate
- Squeeze Theorem

- Limit laws
- \(x \to \infty\) and horizontal asymptotes
- Definition (epsilon-delta)

Continuity

- Continuous at a number
- Removable discontinuity
- Examples: polynomial, rational (no division by zero), sine, cosine, nth-roots, exponential, logarithmic
- Operations: addition, subtraction, multiplication, division, composition, scalar multiplication
- Intermediate Value Theorem

Derivatives ("fancy limit", "slopes of tangent lines")

- Difference quotient
- Definition
- Tangent lines

Derivatives

- Rules:
- Constant
- Sum/difference
- Power (& coefficient)
- Product
- Quotient
- Chain

- Higher derivatives
- Relationship between continuity and differentiability
- Tangent lines
- Derivatives of \(e^x\) and \(\ln(x)\)
- Trigonometric functions

Implicit differentiation

- Derivatives of inverse trigonometric functions
- Logarithmic differentiation
- Related rates

Applications

- Rates of change (position, velocity, acceleration; speed)
- Distance vs displacement
- Linearizations

Formula list on exam 2:

- \(x^2 + y^2 = z^2 \)
- \(P = 4s\)
- \(P = 2x + 2y\)
- \(C = 2 \pi r\)
- \(A = s^2\)
- \(A = xy\)
- \(A = \frac{1}{2} xy\)
- \(A = \frac{b_1 + b_2}{2} h\)
- \(A = \pi r^2\)
- \(SA = 6 s^2\)
- \(SA = 2xy + 2yz + 2xz \)
- \(SA = 2 \pi r h + 2 \pi r^2\)
- \(SA = \pi r^2 + \pi r l\)
- \(SA = 4 \pi r^2\)
- \(V = (\text{base area}) \cdot h\)
- \(V = s^3\)
- \(V = \pi r^2 h\)
- \(V = \frac{1}{3} \cdot (\text{base area}) \cdot h\)
- \(V = \frac{1}{3} \pi r^2 h\)
- \(V = \frac{4}{3} \pi r^3\)

Applications of derivatives

- Extrema
- Critical numbers
- Global/absolute test
- Local/relative test
- 1st derivative test (increasing/decreasing)
- 2nd derivative test (concavity)

- Optimization
- Area formulas
- Surface area formulas
- Volume formulas

More applications of derivatives

- Rolle's Theorem
- Mean Value Theorem
- L'Hopital's Rule
- Curve sketching: zeros, asymptotes, increasing/decreasing, concave up/down, inflection points

Areas

- Sigma notation (aka. summation notation)
- Proof by induction (randomly chosen from handout or book problems)
- Riemann sum approximations using left/right endpoints

Integration

- Definite integral
- integral notation
- Riemann sums as \(n \to \infty\)
- Indefinite integrals
- antiderivatives
- initial value problems

- Properties of integrals
- constant multiplier
- flip lower/upper bounds
- addition/subtraction of integrals

- Fundamental Theorem of Calculus
- Part I: \(\displaystyle\int_{a}^{b} f(x)\,dx = F(b) - F(a)\)
- Part II: \(\frac{d}{dx} \displaystyle\int_{0}^{x} f(t)\,dt = f(x) \)

- integration with substitution
- distance (total) vs. displacement (net)
- Average value of function
- Numeric approximations