Flashcards - AP Calc AB

Mean Value Theorem

If a function is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one c in (a, b) such that the derivative at c is equal to the average rate of change over [a, b].

Visual/Wording for Mean Value Theorem

There is at least one point where the tangent to the curve has the same slope as the average slope between two endpoints.

Extreme Value Theorem

If a function is continuous on a closed interval [a, b], then it has both a maximum and a minimum value on that interval.

Visual/Wording for Extreme Value Theorem

Continuous functions will reach their highest and lowest points on a closed interval.

Squeeze Theorem

If f(x) ≤ g(x) ≤ h(x) for all x in some interval, and if lim x→c f(x) = lim x→c h(x) = L, then lim x→c g(x) = L.

Visual/Wording for Squeeze Theorem

If a function is 'squeezed' between two others that converge to the same limit, it must converge to that limit as well.

Candidate’s Test

Used to find critical points by evaluating the function's derivative and testing values around those points to determine if they are local extrema.

Visual/Wording for Candidate’s Test

Check sign changes of derivative to identify local maxima or minima at critical points.

Intermediate Value Theorem

If f is continuous on [a, b] and N is any number between f(a) and f(b), then there exists at least one c in (a, b) such that f(c) = N.

Visual/Wording for Intermediate Value Theorem

A continuous line must cross any value between its endpoints.

First Derivative Test

If the derivative changes from positive to negative at a critical point, then the function has a local maximum there; if it changes from negative to positive, then it has a local minimum.

Visual/Wording for First Derivative Test

Look for changes in slope to find local highs and lows.

L’Hopital’s Rule

If the limit of f(x)/g(x) results in an indeterminate form (0/0 or ±∞/±∞), then the limit can be found by taking the derivative of f and g.

Visual/Wording for L’Hopital’s Rule

Differentiate numerator and denominator to resolve limits that 'don't work'.

Second Derivative Test

Used to determine the concavity of a function and find local extrema; if f''(c) > 0, f has a local minimum at c; if f''(c) < 0, f has a local maximum at c.

Visual/Wording for Second Derivative Test

Check the second derivative to see if the curve is 'smiling' (min) or 'frowning' (max).

Indeterminate form

Occurs in calculus when limits result in forms like 0/0 or ∞/∞ that do not provide enough information for evaluation.

Visual/Wording for Indeterminate form

Limits that need additional work because they aren’t straightforward to evaluate.

Fundamental Theorem of Calculus (Pt 1)

If f is continuous on [a, b], then the function F defined by F(x) = ∫_a^x f(t) dt is continuous on [a, b], differentiable on (a, b), and F'(x) = f(x).

Visual/Wording for Fundamental Theorem of Calculus (Pt 1)

The area under a curve gives us a function that can be differentiated back to the original function.

Fundamental Theorem of Calculus (Pt 2)

If F is an antiderivative of f on [a, b], then ∫_a^b f(x) dx = F(b) - F(a).

Visual/Wording for Fundamental Theorem of Calculus (Pt 2)

Evaluate the antiderivative at the endpoints to find the net area under the curve.