AP Calculus AB

AP Calculus AB

1. Derivatives

   • Rules
     • Definition lim((f(a+h) - f(a))/h))
     • Constant f(x)=C then f'(x)=0
     • Power Term f(x)=ax^n then f'(x)=anx^n-1
     • Sum of Terms f(x)=u±v then f'(x)=u'±v'
     • Product y=uv then y'=uv'+vu'
     • Quotient y=u/v then y'=(vu'-uv')/v^2
     • y = f(g(x)) then g’(x) • f’(g(x))
   • Applications
     • Implicit Differentiation
       • dy/dx is derivative of y with respect to x
       • Related Rates
         • Change in something over time
       • Second Deivative d^2y/dx^2
     • Extremas
       • Optimization
         • Real-world Problems
         • Use 2 equations
       • f'(x)
       • Changes in Direction
         • Abrupt
           • Vertical Tangent
         • Gradual
           • Horizontal Tangent
       • Types of EVPs
         • Maximum and Minimum
           • Relative
           • Absolute
       • Critical Numbers
     • Concavity
       • Types
         • Up
         • Down
       • f''(x)
       • Point of Inflection
     • Absolute Values
       • Piecewise Functions
       • Compare Slopes
       • f'(x)
     • Special Lines
       • tangent
       • normal
       • secant
     • Physics Applications
       • Position
         • s(t)
       • Velocity
         • v(t) = s'(t)
       • Acceleration
         • a(t) = v'(t) = s''(t)
     • Business and Economics
       • C(x)=Cost
       • R(x)=Revenue
       • P(x)=Profit
       • Marginal=derivative

2. Limits

   • Approaches
     • Graphically
       • Limit = DNE
         • Osciallations
         • One Sided Functions
           • Piecewise Functions
           • Step Functions
           • Vertical Asymptotes
         • Infinities
       • Limit Theoroms
         • If the limit of L = limit of R, the limit exists
         • If the limit of L ≠ limit of R, limit DNE
         • L=left R=right
     • Algebraically
       • Substitute x=c into L=f(c)
   • Limit Properties
     • Constant Term lim b = b
     • Limit of x limx = c
     • Power term limx^n = c^n
     • Scalar Multiple lim(b•f(x)) = b • L
     • Product lim(f(x)g(x)) = L(K)
     • Sum/Difference lim(f(x)±g(x)) = L±K
     • Quotient (f(x)/g(x)) = L/K
   • Notation lim f(x) = L
   • Indeterminates
     • Functions
       • Extended Function
         • Simplified Parent Function
         • Used to find holes
       • Parent Function
     • Algebraic Techniques
       • Factor Theorom
       • Synthetic Division
       • Rationalizing Numerator
       • Simplifying Complex Fractions
   • Infinity
     • Coefficient of the Equation
       • Same degree
     • HA is the x-axis
       • Numerator<Denominator
     • No HA exists
       • Numerator>Denominator
   • Continuity/Discontinuity
     • Continuity Rules
       • f(c) is defined
       • limit exists
       • f(c) = L
     • Discontinuities
       • Removeable
         • Holes
       • Non-removable
         • Steps
         • Vertical Asymptotes
     • Intermediate Value Theorom
   • Rates
     • Average
       • Change in Quantity/Change in Time
     • Instantaneous
       • lim((f(c+h) - f(c))/h))