Calculus Exam Notes

Limits and Derivatives

• $f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$ (Definition of the Derivative/Difference Quotient)
• $f'(c) = \lim_{x \to c} \frac{f(x)-f(c)}{x-c}$ (Alternate form of the Derivative Definition)
• Continuity at a Point (c, f(c)) requires $f(c)$ to be defined, $\lim<em>{x \to c} f(x)$ to exist, and $\lim</em>{x \to c} f(x) = f(c)$.
• Special Trig Limits: $\lim<em>{x \to 0} \frac{sin x}{x} = 1$ and $\lim</em>{x \to 0} \frac{1 - cos x}{x} = 0$
• One-sided Limits: $\lim<em>{x \to c^-} f(x) = L = \lim</em>{x \to c^+} f(x)$
• Derivative Rules:
  • $\frac{d}{dx}[c] = 0$ (Constant Rule)
  • $\frac{d}{dx}[x^n] = nx^{n-1}$ (Power Rule)
  • $\frac{d}{dx}[cf(x)] = c f'(x)$ (Constant Multiple Rule)
  • Product Rule
  • Quotient Rule

Derivatives of Trig Functions

• $\frac{d}{dx}[sin x] = cos x$
• $\frac{d}{dx}[cos x] = -sin x$
• $\frac{d}{dx}[tan x] = sec^2 x$
• $\frac{d}{dx}[cot x] = -csc^2 x$
• $\frac{d}{dx}[sec x] = sec x tan x$
• $\frac{d}{dx}[csc x] = -csc x cot x$
• General Power Rule with Chain Rule: $\frac{d}{dx}[u^n] = nu^{n-1} \frac{du}{dx}$

Applications of Derivatives

• Mean Value Theorem: $f'(c) = \frac{f(b) - f(a)}{b - a}$
• Critical Numbers: x = c if $f'(c) = 0$ or $f'$ is undefined.
• Absolute Extrema on [a, b]: Candidates are critical points and endpoints.
• Relative Extrema on (a, b)
• Points of Inflection: x = poi if $f'' = 0$ or $f''$ is undefined and concavity changes.
• First Derivative Test: Analyzing increasing/decreasing behavior and relative max/min.
• Second Derivative Test: Evaluate $f''$ at critical numbers to determine relative max/min.
• Equation of Tangent Line: $y = f'(c)(x - c) + f(c)$
• Differential Notation: $dy = f'(x) dx$

Basic Integration

• Riemann Sums: Approximating area using left endpoints, right endpoints, midpoint sum, trapezoidal sum.
• Indefinite Integration: Power Rule (remember + C).
• Fundamental Theorem of Calculus: $\int_a^b f(x) dx = F(b) - F(a)$
• Net Change Theorem: $\int_a^b F'(x) dx = F(b) - F(a)$
• Particle Motion: x(t) = position, v(t) = velocity, a(t) = acceleration.
  • Displacement: $\int_a^b v(t) dt = x(b) - x(a)$
  • Total Distance: $\int_a^b |v(t)| dt$
• Mean Value Theorem for Integrals/Average Value: $\frac{1}{b-a} \int_a^b f(t) dt$

Transcendental Functions

• Properties of Logarithms:
  • $ln(1) = 0$
  • $ln(e) = 1$
  • $ln(ab) = ln(a) + ln(b)$
  • $ln(\frac{a}{b}) = ln(a) - ln(b)$
  • $ln(a^n) = n ln(a)$
• Derivatives:
  • $\frac{d}{dx}[ln x] = \frac{1}{x}$
  • $\frac{d}{dx}[e^x] = e^x$
  • $\frac{d}{dx}[a^x] = a^x ln a$
• Integrals:
  • $\int \frac{1}{x} dx = ln|x| + C$
  • $\int e^x dx = e^x + C$
  • $\int a^x dx = \frac{a^x}{ln a} + C$

Integrals of Trig Functions

• $\int cos x dx = sin x + C$
• $\int sin x dx = -cos x + C$
• $\int sec x tan x dx = sec x + C$
• $\int csc x cot x dx = -csc x + C$
• $\int csc^2 x dx = -cot x + C$

Differential Equations

• Exponential Growth/Decay: $\frac{dy}{dt} = ky \implies y = Ce^{kt}$

Area and Volume

• Area between curves: $\int<em>a^b (top - bottom) dx$ or $\int</em>c^d (right - left) dy$.
• Disk Method
• Washer Method
• Volume of Solid with Known Cross-Sections: $\int A(x) dx$