Exam topics

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

• 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

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)