AP Calculus AB Cram Sheet

Definition of the Derivative Function

• 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 average rate of change from ( x = a ) to ( x = a + h )

Interpretations of the Derivative

• Instantaneous rate of change at ( x = a )
• Slope of the tangent line to the graph at ( x = a )
• Slope of the curve at ( x = a )

Derivative Formulas

• ( \frac{d}{dx} (k) = 0 ) where ( k ) is constant
• ( \frac{d}{dx} (kf(x)) = kf'(x) )
• ( \frac{d}{dx} (f(x) \pm g(x)) = f'(x) \pm g'(x) )
• ( \frac{d}{dx} (f(x)g(x)) = f(x)g'(x) + g(x)f'(x) )
• ( \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) )
• Logarithmic and exponential derivatives:
  • ( \frac{d}{dx} \ln(f(x)) = \frac{1}{f(x)}f'(x) )
  • ( \frac{d}{dx} e^{f(x)} = e^{f(x)}f'(x) )
  • ( \frac{d}{dx} a^{f(x)} = a^{f(x)} \ln(a)f'(x) )

L'Hopital's Rule

• Rule: If ( \lim_{x \to a} \frac{f(x)}{g(x)} ) forms ( \frac{0}{0} ) or ( \frac{\infty}{\infty} ), then ( \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)} )_

Critical Points

• Critical Points: ( f'(c) = 0 ) or ( f'(c) ) is undefined

Tangents and Normals

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

Increasing and Decreasing Functions

• Increasing: ( f'(x) > 0 )
• Decreasing: ( f'(x) < 0 )

Maximum, Minimum, and Inflection Points

• Local Minimum: ( f' ) changes from negative to positive
• Local Maximum: ( f' ) changes from positive to negative
• Concavity:
  • Upward: ( f''(x) > 0 )
  • Downward: ( f''(x) < 0 )
• Inflection Point: ( f''(x) = 0 ) and changes sign

Related Rates

• Differentiate with respect to time ( t ) to relate rates of change

Approximating Areas

• Definite Integral: ( \int_{a}^{b} f(x) , dx )
• Methods:
  • Left Sum
  • Right Sum
  • Midpoint Sum
  • Trapezoidal Rule_

Antiderivatives and Integration

• Antiderivative: ( F(x) ) such that ( F'(x) = f(x) )
• Integration Formulas:
  • ( \int kf(x) , dx = k\int f(x) , dx )
  • ( \int (f(x) \pm g(x)) , dx = \int f(x) , dx \pm \int g(x) , dx )

Fundamental Theorems of Calculus

• First Theorem: ( \int_{a}^{b} f(x) , dx = F(b) - F(a) )
• Second Theorem: ( F(x) = \int_{a}^{x} f(t) , dt ) implies ( F'(x) = f(x) )

Definite Integral Properties

1. ( \int_{a}^{a} f(x) , dx = 0 )
2. ( \int_{a}^{b} f(x) , dx = -\int_{b}^{a} f(x) , dx )_

Volumes and Arc Length

• Volumes: Using disks, washers, or known cross-sections
• Arc Length: ( \int_{a}^{b} \sqrt{1 + (f'(x))^2} , dx )_