Functions Limits and the Derivative Functions Graphs Algebra

Functions, Limits, and the Derivative • Functions (Graphs, Algebra, Models) • Limits (One-sided, Continuity) • The Derivative

A rule that assigns to each element in a set A (the domain), one and only one element in a set B (the range) Range Domain -1 1 3 -4 1 -6

is a function, with values of x as the domain and values of y as the range. We write in place of y. This is read “f of x. ” So NOTE: It is not f times x

Ex. Find Plug in – 2

The domain of a function is the set of values for x for which f (x) is a real number. Ex. Find the domain of Since division by zero is undefined we must have So which can be expressed as the intervals:

Ex. Find the domain of Since the square root of a negative number is undefined we must have So which can be expressed as the interval:

The graph of a function is the set of all points (x, y) such that x is in the domain of f and y = f (x). Given the graph of y = f (x), find f (1) = 2 (1, 2)

Vertical Line Test: The graph of a function can be crossed at most once by any vertical line. Function Not a Function It is crossed more than once.

Domain: Domain of f intersected with the domain of g with the exclusion of all values of x, such that g(x) = 0.

Domain: all values of x, such that f(x) lies in the domain of g(x) = 0.

Polynomial Functions n is a nonnegative integer, each is a constant. Ex. Rational Functions polynomials Ex.

Power Functions ( r is any real number) Ex.

A shirt producer has a fixed monthly cost of $5000. If each shirt has a cost of $3 and sells for $12 find: a. The cost function Cost: C(x) = 3 x + 5000 where x is the number of shirts produced. b. The revenue function Revenue: R(x) = 12 x where x is the number of shirts sold. c. The profit from 900 shirts Profit: P(x) = Revenue – Cost = 12 x – (3 x + 5000) = 9 x – 5000 P(900) = 9(900) – 5000 = $3100

There are two main areas of focus: 1. Finding the tangent line to a curve at a given point. tangent line 2. Finding the area of a planar region bounded by a given curve. Area

Average Over any time interval If I travel 200 km in 5 hours my average velocity is 40 km/hour. Instantaneous As elapsed time approaches zero When I see the police officer, my instantaneous velocity is 60 km/hour.

Ex. Given the position function where t is in seconds and s(t) is measured in feet, find: a. b. The average velocity for t = 1 to t = 3. The instantaneous velocity at t = 1. t Notice how elapsed time approaches zero 1. 1 1. 001 Average velocity 12. 1 12. 001 Answer: 12 ft/sec

The limit of f (x), as x approaches a, equals L written: if we can make the value f (x) arbitrarily close to L by taking x to be sufficiently close to a. L a

Ex. 6 Note: f (-2) = 1 is not involved -2

Ex. Notice form Factor and cancel common factors Copyright (c) 2003 Brooks/Cole, a division

For all n > 0, provided that Ex. is defined. Divide by

The right-hand limit of f (x), as x approaches a, equals L written: if we can make the value f (x) arbitrarily close to L by taking x to be sufficiently close to the right of a. L a

The left-hand limit of f (x), as x approaches a, equals M written: if we can make the value f (x) arbitrarily close to L by taking x to be sufficiently close to the left of a. M a

A function f is continuous at the point x = a if the following are true: f(a) a

If f and g are continuous at x = a, then A polynomial function y = P(x) is continuous at every point x. A rational function at every point x in its domain. is continuous

Average rate of change of f over the interval [x, x+h] Instantaneous rate of change of f at x

The derivative of a function f with respect to x given by is the function

Four-step process for finding 1. Compute 2. Form 3. Form 4. Compute

Given 1. 2. 3. 4.

If a function is differentiable at x = a, then it is continuous at x = a. Not Continuous Not Differentiable Still Continuous