Derivative-Based Curve Sketching for AP Calculus BC (Unit 5)

Increasing (on an interval)

A function f is increasing on an interval if for any a<b in the interval, f(a)<f(b) (outputs rise as x moves left to right).

Decreasing (on an interval)

A function f is decreasing on an interval if for any af(b) (outputs fall as x moves left to right).

First derivative (f'(x))

Measures the instantaneous rate of change of f at x (the slope of the tangent line).

Derivative test for increasing

If $f'(x) > 0$ for all x in an interval, then f is increasing on that interval.

Derivative test for decreasing

If $f'(x) < 0$ for all x in an interval, then f is decreasing on that interval.

Critical number

A value c in the domain of f where either f'(c)=0 or f'(c) does not exist (but f(c) exists).

Critical point

A point (c, f(c)) where c is a critical number (often a possible location for changing behavior).

Horizontal tangent

Occurs where $f'(c) = 0$; the tangent line has slope 0 (this alone does not guarantee a max or min).

Sign chart method

Procedure to find intervals of increase/decrease by computing f', finding critical numbers, splitting into intervals, testing the sign of f' in each interval, and concluding where f increases/decreases.

Test interval

An interval between consecutive critical numbers (and/or domain restrictions) where you test a representative x-value to determine the sign of $f'(x)$ or $f''(x)$.

Domain restriction (in sign analysis)

A value where f is not defined; it can split the number line into separate intervals when determining increase/decrease or concavity.

Local maximum

At x=c, f(c) is larger than nearby values of f(x) (in some neighborhood around c).

Local minimum

At x=c, f(c) is smaller than nearby values of f(x) (in some neighborhood around c).

First Derivative Test

Classifies a critical number c by the sign change of f'(x): + to − gives a local max; − to + gives a local min; no sign change gives no local extremum.

Vertical tangent (as in f(x) = \root[3]{x} at $x=0$)

A point where the derivative does not exist (often due to unbounded slope), but the function value exists; it can be a critical number without being a turning point.

Second derivative (f''(x))

Measures the rate of change of the first derivative; describes how the slope f'(x) is changing.

Concave up

f is concave up on an interval if its slopes increase as x increases; equivalently, if $f''(x) > 0$ on that interval.

Concave down

f is concave down on an interval if its slopes decrease as x increases; equivalently, if $f''(x) < 0$ on that interval.

Inflection point

A point on the graph where concavity changes (concave up to concave down, or vice versa); requires verifying a sign change in f''.

Inflection point candidate

A value c where $f''(c) = 0$ or $f''(c)$ does not exist (you must still test for a sign change to confirm an inflection point).

Second Derivative Test

If $f'(c) = 0$ and $f''(c)$ exists: $f''(c) > 0$ implies local min; $f''(c) < 0$ implies local max; $f''(c) = 0$ makes the test inconclusive.

Inconclusive (Second Derivative Test)

When $f''(c) = 0$ at a critical point (with $f'(c) = 0$), the second-derivative test cannot classify the point; use another method such as the First Derivative Test.

Velocity (v(t))

If $s(t)$ is position, then $v(t) = s'(t)$ is velocity; $v(t) > 0$ means position increasing, and $v(t) < 0$ means position decreasing.

Acceleration (a(t))

If $s(t)$ is position and $v(t) = s'(t)$, then $a(t) = s''(t)$ is acceleration; $a(t) > 0$ means velocity increasing, $a(t) < 0$ means velocity decreasing.

Speeding up (motion interpretation)

A particle is speeding up when velocity and acceleration have the same sign (both positive or both negative).