AP Calculus AB

• Average Rate of Change of f(x) on [a,b]

  
• Instantaneous Rate of Change at x=a

  
• If f(x) is increasing, then f'(x) is?

  f'(x) = Positive
• If f(x) is decreasing, then f'(x) is?

  f'(x) = Negative
• If f(x) is concave up, then f''(x) is?

  f''(x) = positive
• If f(x) is concave down, then f''(x) is?

  f''(x) = negative
• If f'(x) is increasing, then f''(x) is?

  f''(x) = positive
• If f'(x) is decreasing, then f''(x) is?

  f''(x) = negative
• Equation for the line tangent to f(x) at x=a


  y-f(a)=f'(a)(x-a)
• Slope of the line tangent to f(x) at x=a

  f'(a)
• A function is continuous if and only if;

  
• If a function is "differentiable" then,

  It is also continuous
• Derivative of e^x

  
• Derivative of a^x

  
• Derivative of sinx

  
• Derivative of cosx

  
• Derivative of tanx

  
• Derivative of lnx

  
• Derivative of arcsin(x)

  
• Derivative of arctan(x)

  
• Derivative of square root of x

  
• Derivative of x^n

  
• Derivative of f(x)g(x)

  
• Derivative of f(x)/g(x)

  
• Derivative of f(g(x))

  
• Derivative of the inverse of f or f^(-1)

  
• Mean Value Theorem

  
• Intermediate Value Theorem

  
• Extreme Value Theorem

  
• Critical Point of f(x)


  Where f'(x)=0 or f'(x) is undefined
• Local Minimum (First Derivative Test)


  Where f'(x) changes from negative to positive
• Local Maximum (First Derivative Test)


  Where f'(x) changes from positive to negative
• Local Minimum (Second Derivative Test)


  Where f'(a)=0 and f''(a)>0
• Local Maximum (Second Derivative Test)


  Where f'(a)=0 and f''(a)<0
• Inflection Point of f(x)


  Where f''(x) changes sign
• Candidates Test


  Plug x-coordinates of all closed end points and critical points back into the original function f(x) to determine global/absolute maximum and minimum.
• Global Maximum


  The largest y-value on the interval [a,b]. Based on the candidates test.
• Global Minimum


  The smallest y-value on the interval [a,b]. Based on the candidates test.
• L'Hospitals Rule

  
• ln(1)=

  0