AP Calculus AB Flashcards

• d/dx (cot−1 x)

  
• d/dx (sec−1 x)

  
• d/dx (csc−1 x)

  
• d/dx (tan−1 x)

  1/1 + x^2
• d/dx (cos−1 x)

  -1/√(1 − x^2)
• d/dx (sin−1 x)


  1/√ (1 − x^2)
• implicit differentiation process

  1. Differentiate both sides of the equation with respect to x

  2. Collect the terms with dy/dx on one side of the equation

  3. Factor out dy/dx

  4. Solve for dy/dx
• chain rule

  If f is differentiable at the point u = g(x), and g is differentiable at x,

  
  

  • then the composite function y = f(u) for u = g(x) is differentiable at x and

  dy/dx = dy/du ∗du/dx

  
  

  • then the composite function (fog)(x) = f(g(x)) is differentiable at x and

  (fog)'(x) = f′(g(x)) ∗ g′(x)
• d/dx cscx

  -cscxcotx
• d/dx cotx

  -csc^2x
• d/dx secx

  secxtanx
• d/dx (tan(x))

  sec^2x
• d/dx cos(x)

  -sin(x)
• d/dx sin(x)

  cos(x)
• the quotient rule

  At a point where v ≠ 0, the quotient y =

  u/v of two differentiable functions is differentiable, and

  
• the product rule


  The product of two differentiable functions u and v is differentiable, and

  
• derivative of e^x

  e^x
• derivative of a^x

  
• general equation for displacement

  
• the sum and difference rule

  If u and v are differentiable functions of x, then their sum and difference are differentiable at every point where u and v are differentiable. At such points:

  
• the constant multiple rule

  If u is a differentiable function of x and c is a constant, the

  
• power rule for positive integer powers of x

  If n is a positive integer, then

  
• derivative of a constant Function

  If f is the function with the constant value c, then

  
• implications of differentiability

  1. local linearity

  2. continuity
• f'(a) might fail to exist

  A function will not have a derivative at a point P(a, f(a)) where the slopes of the secant lines [f(x)−f(a)] / (x−a) fail to approach a limit as x approaches a.

  
  

  ex.

  1. corner

  2. cusp

  3. vertical tangent

  4. discontinuity
• one sided derivatives and differentiability

  A function y = f(x) is differentiable on a closed interval [a,b] if it has a derivative at every interior point of the interval, and if the limits: ... exist at endpoints

  
• The Derivative


  The function f whose value is f′

  (x) = ... provided the limit exists

  
• The derivative of a function f at the point x=a

  
• Slope of a Curve at a Point


  The slope of a curve y = f(x) at the point P(a, ff(a)) is the number:

  
• intermediate value theorem


  If f(x) is a continuous function and a < b and there is a value w such that n is between f(a) and f(b), then there is a number c such that a < c < b and f(c) = w
• jump discontinuity


  A function has a jump discontinuity at x = c if the function has a left- and right- hand limit, but they do not equal each other.
• infinite discontinuity


  A function has an infinite discontinuity at x = c if the function value increases or decreases indefinitely as x approaches c from the left and right.
• removable discontinuity


  A function has a removable discontinuity if the function is continuous everywhere except for a "hole" at x = c
• continuity test

  
• end behavior model


  The function g(x) is

  (a) A right-end behavior model for f if and only if limx→∞

  f(x)/g(x) = 1

  (b) A left-end behavior model for f if and only if lim x→−∞

  f(x)/g(x)

  
• squeeze theorem


  If f(x) ≤ g(x) ≤ h(x) for all x ̸= a and limx→a f(x) = limx→a h(x) = L, then

  limx→a g(x) = L
• limits


  a function f(x) has a limit as x approaches c if and only if the right hand and the left hand limits at c exist and are equal.

  
• instantaneous rate of change

  
• average rate of change on an interval [a, b]