AP CalculusReview of Chapter 4

To approximate the change in a function f For a small value of delta x

The exact value of the change in f

Linear Approximation

To approximate f (x) at x = c (x is close to c )

Linearization

Find the linearization for the function f(t) = 32t – 4t2, a = 2. Use this to approximate f(2.1).

L(t) approximates f(2.1) as 49.6. Actual value is 49.56

Example

Local Extrema a function f (x) has a: Local Minimum at x = c if f (c) is the

minimum value of f on some open interval (in the domain of f containing c.

Local Maximum at x = c if f (c) is the maximum value of f on some open interval containing c.

Extrema

Local Max and Min

Local Min

Local Max

Local Min

Definition of Critical Points A number c in the domain of f is called a

critical point if either f’ (c) = 0 or f’ (c) is undefined.

Critical Points

Find the extreme values of g(x) = sin x cos x on [0, π]

Example

Critical points: g’(x) = cos 2 x – sin 2 x g’(x) = 0, x = π/4, 3π/4 g(π/4) = ½ , max g(3π/4) = -1/2 , min Endpoints (0, 0), (π, 0)

Example

Assume f (x) is continuous on [a, b] and differentiable on (a, b). If f (a) = f (b) then there exists a number c between a and b such that f’(c) = 0

Rolle’s Theorem

f(a) f(b)

a bc

f(c)

Assume that f is continuous on [a, b] and differentiable on (a, b). Then there exists at least one value c in (a, b) such that

Mean Value Theorem

If f’(x) > 0, for x in (a, b) then f is increasing on (a, b).

If f’(x) < 0 for x in (a, b), then f is decreasing on (a, b)

If f’(x) changes from + to – at x = c, f(c) is local maximum

If f’(x) changes from – to + at x = c, f(c) is a local minimum

Increasing/Decreasing Behavior

Concavity f is concave up if f’(x) is increasing on (a, b) f is concave down if f’(x) is decreasing on

(a, b)

Shape of the Graph

If f’’(x) > 0 for x in (a, b), the f is concave up on (a, b).

If f’’(x) < 0 for x in (a, b), the f is concave down on (a, b).

Inflection Points – If f’’(c) = 0 and f’’(x) changes sign at x = c then f(x) has a point of inflection at x = c.

Testing Concavity

F’’(c) > 0, then f (c) is a local minimum F’’(c) < 0, f (c) is a local maximum F’’(c) = 0, inconclusive…may be local max,

min or neither

Second Derivative Test for CP

For a Function f (x): Domain/Range of the Function Intercepts (x and y) Horizontal/Vertical Asymptotes End Behavior (Polynomials) Period, frequency, amplitude, shifts

(Trigonometric)

Curve Sketching

For the Derivative f’ (x): Critical points Increasing/Decreasing Intervals Extrema

Curve Sketching

For the Second Derivative f’’ (x): Points of Inflection Concavity

Curve Sketching

Application of the Derivative involving finding a maximum or minimum.

Example problems on page 227 (#41) and page 228 (#43)

Optimization

Method for finding approximations for zeros of a function.

Uses a number of iterations to locate the zero.

Newton’s Method