AP Calculus AB Cram Sheet

What is the definition of the derivative function?

f ' (x) = lim (h→0) [f(x + h) - f(x)] / h

What does the derivative at a point represent?

f ' (a) = lim (h→0) [f(a + h) - f(a)] / h, resulting in a number.

What does f ' (a) represent in terms of a function's behavior?

The instantaneous rate of change of f at x = a, the slope of the tangent line to the graph of f at x = a.

What is the derivative of a constant multiplied by a function?

d/dx (k·f(x)) = k·f ' (x) where k is a constant.

What is the derivative of the sum of two functions?

d/dx [f(x) ± g(x)] = f ' (x) ± g ' (x).

What is the product rule for derivatives?

d/dx [f(x)·g(x)] = f(x)·g ' (x) + g(x)·f ' (x).

What is the derivative of sin(f(x))?

d/dx [sin(f(x))] = cos(f(x))·f ' (x).

What is the derivative of cos(f(x))?

d/dx [cos(f(x))] = -sin(f(x))·f ' (x).

What is the derivative of tan(f(x))?

d/dx [tan(f(x))] = sec²(f(x))·f ' (x).

What is L'Hôpital's Rule used for?

To evaluate limits of the form 0/0 or ∞/∞ by taking the limit of the derivatives.

What is a critical point in calculus?

Any c in the domain of f such that f ' (c) = 0 or f ' (c) is undefined.

What is the equation of the tangent line to the curve y = f(x) at x = a?

y - f(a) = f ' (a)(x - a).

What does it mean for a function to be increasing or decreasing?

A function y = f(x) is increasing on an interval if its derivative is positive; decreasing if its derivative is negative.

What indicates a local minimum or maximum in a function?

A local minimum occurs where the first derivative changes from negative to positive; a local maximum occurs where it changes from positive to negative.

What is a point of inflection?

The point where the concavity of the function changes.

What is the Fundamental Theorem of Calculus?

It connects differentiation and integration, stating that if f is continuous on [a, b] and F' = f, then F(b) - F(a) = ∫[a to b] f(x) dx.

What is the area approximation method using rectangles?

The area under a curve can be approximated by dividing the interval into n strips and summing the areas of rectangles.

What is the trapezoidal rule?

An approximation of the area under a curve that averages the left and right endpoint sums.

What is the antiderivative of a function?

The antiderivative or indefinite integral of f(x) is a function F(x) whose derivative is f(x), represented as F(x) + C.

What is the derivative of ln(f(x))?

d/dx [ln(f(x))] = (1/f(x))·f ' (x).

What is the formula for the volume of a solid of revolution using disks?

Volume = ∫[a to b] π[r(x)]² dx.

What is the formula for arc length of a curve y = f(x)?

s = ∫[a to b] √(1 + (f ' (x))²) dx.

What does it mean for a function to be concave upward?

A function is concave upward on an interval if its second derivative is positive.

What does it mean for a function to be concave downward?

A function is concave downward on an interval if its second derivative is negative.

What is the significance of global maximum or minimum values?

A global maximum occurs at x = c if f(c) is greater than all y values on the interval; a global minimum occurs if f(c) is less.

What is the derivative of e^(f(x))?

d/dx [e^(f(x))] = e^(f(x))·f ' (x).

What is the derivative of a^f(x)?

d/dx [a^(f(x))] = a^(f(x))·ln(a)·f ' (x).