AP Calculus – Final Review Sheet

These flashcards cover key concepts and definitions necessary for understanding the AP Calculus material.

Find the zeros

Set function = 0, factor or use quadratic equation if quadratic, graph to find zeros on calculator.

Even function

A function is even if f(−x) = f(x), making it symmetric to the y-axis.

Odd function

A function is odd if f(−x) = −f(x), indicating symmetry around the origin.

Limit exists

A limit exists at x = a if lim (x→a−) f(x) = lim (x→a+) f(x).

Find limit using calculator

Use a table to find y values for x-values close to a from left and right.

Find limit without calculator

Substitute x = a; limit is the value if b ≠ c.

Rationalize radicals

A technique often required when finding limits.

Known trig limits

Such as lim (x→0) (sin x)/x = 1 and lim (x→0) (cos x-1)/x = 0.

Horizontal asymptotes

Found by determining lim (x→∞) f(x) and lim (x→−∞) f(x).

Vertical asymptotes

Occurs where lim (x→a) f(x) = ±∞, often found by setting the denominator = 0.

Domain restrictions

Denominators cannot equal 0; square roots must be non-negative; logs require positive numbers.

Continuous function

A function is continuous at x = a if the limit exists, f(a) exists, and f(x) approaches f(a).

Tangent line slope

The slope at x = a is given by the derivative f′(a).

Normal line

The line perpendicular to the tangent line at a point.

Mean Value Theorem (MVT)

States there exists a c in [a,b] where f′(c) = (f(b) - f(a))/(b - a).

Intermediate Value Theorem (IVT)

If f is continuous on [a,b], then it takes every value between f(a) and f(b).

Instantaneous rate of change

Equal to f′(a) at x = a.

Critical points

Values where f′(x) = 0 or f′(x) is undefined.

Points of inflection

Where the second derivative changes sign.

Approximation of f(0.1)

Use the tangent line equation at x = 0.

Implications of increasing/decreasing functions

If f′(x) > 0, f is increasing; if f′(x) < 0, f is decreasing.

Second Derivative Test

Used to determine relative extrema based on the sign of f″(x).