Exam topics

Exam 1

Limits

• Average rate of change (slope of secant line) vs.
  Instantaneous rate of change (slope of tangent line)
• Intuitive understanding:
  • one-sided
  • numerical tables
  • graphing
• Vertical asymptotes when \(\text{limit} \to \infty\)
• Limit computation
  • Limit laws
    • Scalar (constant multiplier)
    • Sum & difference
    • Multiplication & division (no division by zero)
    • Power
  • Substitution property (*needs continuity)
  • Factoring
  • Multiply by conjugate
  • Squeeze Theorem
• Horizontal asymptotes when \(x \to \infty\)
• Definition using \(\epsilon \text{ and } \delta\)

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", "instantaneous rates of changes")

• Difference quotient
• Definition
• Tangent lines

Exam 2 review

Derivatives

• Rules:
  • Constant
  • Sum/difference
  • Power (& coefficient)
  • Product
  • Quotient
  • Chain
• Higher derivatives
• Derivatives of \(e^x\) and \(\ln(x)\)
• Trigonometric functions

Implicit differentiation

• Derivatives of inverse trigonometric functions
• Logarithmic differentiation (both sides have base of "e" or natural log)
• Related rates

Applications

• Rates of change (position, velocity, acceleration; speed)
• Related rates
• Linear approximations

Geometric formulas

• Lengths/perimeters
  • Triangle: \(x^2 + y^2 = z^2 \)
  • Square: \(4s\)
  • Rectangle: \(2x + 2y\)
  • Circle: \(2 \pi r\)


• Areas
  • Square: \(s^2\)
  • Rectangle: \(xy\)
  • Parallelogram: \(\text{base} \cdot \text{altitude} \)
  • Triangle: \(\frac{1}{2} \text{base} \cdot \text{altitude} \)
  • Trapezoid: \(\frac{b_1 + b_2}{2} h\)
  • Circle: \(\pi r^2\)


• Surface areas
  • Cube: \(6 s^2\)
  • Rectangular prism: \(2xy + 2yz + 2xz \)
  • Cylinder: \(2 \pi r h + 2 \pi r^2\)
  • Cone: \(\pi r^2 + \pi r l\)
  • Sphere: \(4 \pi r^2\)
• Volumes
  • Prism: \((\text{base area}) \cdot h\)
  • Cube: \(s^3\)
  • Cylinder: \(\pi r^2 h\)
  • Pyramid: \(\frac{1}{3} \cdot (\text{base area}) \cdot h\)
  • Cone: \(\frac{1}{3} \pi r^2 h\)
  • Sphere: \(\frac{4}{3} \pi r^3\)

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

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)
• Riemann sum approximations using left/right endpoints
• Riemann integral, \( \textstyle\int_{a}^{b} f(x)\,dx = \lim_{x \to \infty} \Sigma_{i=1}^{n} f(a + \frac{b-a}{n} \cdot i) \cdot \frac{b-a}{n}\)

Integration

• Definite integral
• integral notation
• Riemann sums as \(n \to \infty\)
• Indefinite integrals
  • antiderivatives
  • initial value problems
• Properties of integrals
  • constant multiplier, \( \textstyle\int k \cdot f(x)\,dx = k \cdot \textstyle\int f(x)\,dx\)
  • flip bounds changes sign, \(\textstyle\int_{a}^{b} f(x)\,dx = -\int_{b}^{a} f(x)\,dx\)
  • addition/subtraction of integrals, \( \textstyle\int_{a}^{b} f(x)\,dx + \int_{b}^{c} f(x)\,dx = \int_{a}^{c} f(x)\,dx \)
• Fundamental Theorem of Calculus
  • Part I: \(\displaystyle\int_{a}^{b} f(x)\,dx = F(b) - F(a)\), where \(F'(x) = f(x)\)
  • Part II: \(\frac{d}{dx} \displaystyle\int_{0}^{x} f(t)\,dt = f(x) \)
• integration with substitution
• distance (total) vs. displacement (net)