# Exam topics

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.

## Exam 1

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
• $$x \to \infty$$ 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

## Exam 2 review

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)
• Linearizations

Formula list on exam 2:

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

## Exam 3 review

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

## Final exam review

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)