AP Calculus Chapter 5

Definition Let f be defined on an interval, and let x 1 and x 2 denote numbers in that interval f is increasing on the interval if f(x 1 ) < f(x 2 ) whenever x 1 < x 2. f is decreasing on the interval if f(x 1 ) > f(x 2 ) whenever x 1 > x 2. f is constant on the interval if f(x 1 )= f(x 2 ) for all x 1 and x 2.

Concavity (First Derivative) If f is differentiable on an open interval I, then: f is said to be concave up on I if f ´ is increasing on I and, f is said to be concave down on I if f ´ is decreasing on I

Concavity (Second Derivative) Let f be twice differentiable on an open interval I If f´´(x ) > 0 on I, then f is concave up on I If f ´´ (x) < 0 on I, then f is concave down on I

Inflection Points Inflection points mark the places on the curve y = f(x) where the rate of change of y with respect to x changes from increasing to decreasing, or vice versa

Relative Extrema Suppose that f is a function defined on an open interval containing the number x 0. If f has a relative extremum at x = x 0, then either f ´ (x) = 0 or f is not differentiable at x 0 If a function is defined on an open interval, its relative extrema on the interval, if any, occur at critical numbers(i.e. values where f ´ (x) = 0 or f ´ (x) is undefined). Stationary points occur only when f ´ (x) = 0

Second Derivative Test Suppose that f is twice differentiable at x 0. If f ´ (x 0 ) = 0 and f ´´ (x 0 ) > 0 then f has a relative minimum at x 0. If f ´ (x 0 ) = 0 and f ´´ (x 0 ) < 0 then f has a relative maximum at x 0. If f ´ (x 0 ) = 0 and f ´´ (x 0 ) = 0 then the test is inconclusive; that is, f may have a relative maximum, a relative minimum, or neither at x 0.

For a differentiable function y = f(x), the rate of change of y with respect to x will have a relative extremum at any inflection point of f. That is, an inflection point identifies a place on the graph of y = f(x) where the graph is steepest in the vicinity of the point.

Extreme Value Theorem If a function f is continuous on a finite closed interval [a,b], then f has both an absolute maximum and an absolute minimum on [a,b]. If f has an absolute extremum on an open interval (a,b), then it must occur at a critical point of f. Corollary

A Procedure for Finding the Absolute Extrema of a Continuous Function f on a Finite Closed Interval [a,b] Step 1. Find the critical numbers of f in [a,b] Step 2. Evaluate f at all critical numbers and at the endpoints a and b. Step 3. The largest of the values in Step 2 is the absolute maximum value of f on [a,b] and the smallest value is the absolute minimum

Absolute Extrema Suppose that f is continuous and has exactly one relative extremum on an interval I, say at x 0. If f has a relative minimum at x 0, then f(x 0 ) is the absolute minimum of f on I. If f has a relative maximum at x 0, then f(x 0 ) is the absolute maximum of f on I.

Instantaneous Velocity If s(t) is the position function of a particle moving on a coordinate line, then the instantaneous velocity of the particle t is defined by: v(t) = s ´ (t) = (ds/dt)

Instantaneous Speed The instantaneous speed of a particle is the absolute value of its instantaneous velocity instantaneous speed at = І v(t) І = І (ds/dt) І time t

Instantaneous Acceleration If s(t) is the position function of a particle moving on a coordinate line, then the instantaneous acceleration of the particle at time t is defined by: a(t) = v ´ (t) = (dv/dt) Or a(t) = s ´´ (t) = (d 2 s/dt²) since v(t) = s ´ (t)

Speeding Up/Slowing Down A particle in rectilinear motion is speeding up when velocity and acceleration have the same sign and slowing down when they have opposite signs.