AP Calculus AB Cram Sheet

Definition of Derivatives

• Derivative Function:
  • $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$
• Derivative at a Point:
  • $f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$
  • Represents the instantaneous rate of change, slope of tangent line, and slope of curve at $x = a$.

Derivative Formulas

• Constant Rule: $\frac{d}{dx}(k) = 0$
• Constant Multiple Rule: $\frac{d}{dx}(kf(x)) = kf'(x)$
• Power Rule: $\frac{d}{dx}(x^n) = nx^{n-1}$
• Sum/Difference Rule: $\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)$
• Product Rule: $\frac{d}{dx}[f(x)g(x)] = f(x)g'(x) + g(x)f'(x)$
• Quotient Rule: $\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{g(x)f'(x) - f(x)g'(x)}{(g(x))^2}$
• Trigonometric Derivatives:
  • $\frac{d}{dx}\sin(f(x)) = \cos(f(x))f'(x)$
  • $\frac{d}{dx}\cos(f(x)) = -\sin(f(x))f'(x)$
  • $\frac{d}{dx}\tan(f(x)) = \sec^2(f(x))f'(x)$
• Exponential and Logarithmic Derivatives:
  • $\frac{d}{dx}e^{f(x)} = e^{f(x)}f'(x)$
  • $\frac{d}{dx}\ln(f(x)) = \frac{1}{f(x)}f'(x)$
  • $\frac{d}{dx}a^{f(x)} = a^{f(x)}\ln(a)f'(x)$

Critical Points

• A critical point occurs where $f'(c) = 0$ or $f'(c)$ is undefined.

Tangents and Normals

• Tangent Line: $y - f(a) = f'(a)(x - a)$
• Normal Line: $y - f(a) = -\frac{1}{f'(a)}(x - a)$

Increasing/Decreasing Functions

• Increasing if $f'(x) > 0$ on interval.
• Decreasing if $f'(x) < 0$ on interval.

Maxima, Minima, and Inflection Points

• Local Minimum: $f'(x)$ changes from negative to positive.
• Local Maximum: $f'(x)$ changes from positive to negative.
• Concavity:
  • Concave up if $f''(x) > 0$
  • Concave down if $f''(x) < 0$
• Point of Inflection: $f''(x)$ changes sign.

Related Rates

• Differentiate related variables with respect to time.

Approximating Areas

• Definite Integral: Approximated by dividing the area into strips (rectangles/trapezoids) and summing the areas.
• Methods:
  • Left Sum
  • Right Sum
  • Midpoint Sum
  • Trapezoidal Rule: Average of left and right sum approximations.

Antiderivatives

• Indefinite Integral: $F(x) + C$ where $F'(x) = f(x)$.

Integration Formulas

• Power Rule: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$
• Trigonometric Integrals:
  • $\int \cos(x) dx = \sin(x) + C$
  • $\int \sin(x) dx = -\cos(x) + C$
• Exponential and Logarithmic Integrals:
  • $\int e^x dx = e^x + C$
  • $\int a^x dx = \frac{a^x}{\ln(a)} + C$

Fundamental Theorems of Calculus

• First Theorem: Relates definite integral to antiderivative.
• Second Theorem: Derivative of integral function returns the original function.

Definite Integral Properties

• Properties:
  • $\int_a^a f(x) dx = 0$
  • $\int_a^b f(x) dx = -\int_b^a f(x) dx$

Areas and Volumes

• Area Between Curves: $\int_a^b (f(x) - g(x)) dx$
• Volume of Revolution (Disks/Washers):
  • Disks: $\pi \int_a^b [f(x)]^2 dx$
  • Washers: $\pi \int_a^b ([R(x)]^2 - [r(x)]^2) dx$

Arc Length

• Arc length of $y = f(x)$ from $x = a$ to $x = b$: $\int_a^b \sqrt{1 + [f'(x)]^2} dx$