AP Calculus Ch. 5 Vocab

Absolute Extreme Values

Let f(x) be a function defined on an interval [a,b].

• Absolute maximum value:
  f(c) is an absolute maximum on [a,b] if

  f(c)≥f(x) for all x in [a,b].

• Absolute minimum value:
  f(c) is an absolute minimum on [a,b] if

  f(c)≤f(x) for all x in [a,b].

Antiderivative

A function F(x) is an antiderivative of a function f(x) if F’(x)=f(x) for all x in the domain of f

Concavity Test

Uses the second derivative of a function to determine whether the graph is concave up or concave down on an interval

Critical Point

A value x=c in the domain of f where either

f’(c) = 0

f’(c) does not exist

Decreasing Function

Let f be a function defined on an interval I. Then f decreases on I of, for any two points x₁ and x₂ in I,

x₁ < x₂ → f(x₁) > f(x₂)

Differential

If y=f(x) is a differentiable function, the differential dx is an independent variable and the differential dy is dy = f’(x) dx

The Extreme Value Theorem

If f is continuous on a closed interval [a, b], then f has both a maximum value and a minimum value on the interval

First Derivative Test

Increasing Function

Let f be a function defined on an interval I. Then f increases I if, for any two points x₁ and x ₂ in I,

x₁ < x₂ → f(x₁) < f(x₂)

Linearization

The approximating function L(x) = f(a) + f’(a)(x-a) when f is differentiable at x=a

Local Extreme Value Theorem

If a function f has a local maximum value or a local minimum value at an interior point c of its domain, and if f’ exists at c, then

f’(c) = 0

Maximum Profit

Maximum profit (if any) occurs at a production level at which marginal revenue equals marginal cost

Mean Value Theorem

If y = f(x) is continuous at every point of the closed interval [a, b] and differentiable at every point of its interior (a, b), then there is at least one point c in (a, b) at which

f’(c)= (f(b)-f(a)) / (b-a)

Newton’s Method (Formula)

x_n+1 = x_n - f(x_n)/f’(x_n)

Point of Inflection

A point where the graph of a function has a tangent line and the concavity changes

Second Derivative Test

If f’(c) = 0 and f’’(c) < 0, then f has a local maximum at x=c

If f’(c) = 0 and f’’(c) > 0, then f has a local minimum at x=c

Stationary Point

A point in the interior of the domain of a function f at which f’ = 0