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

Function 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 -6 1 3 -4

Function Notation 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

Function Notation Ex. Find Plug in –2

Domain of a Function 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:

Domain of a Function 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:

Graph of a Function 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). f (1) = 2 (1, 2)

Graph of a Function 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.

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

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

Composition of Functions

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

Types of Functions Power Functions ( r is any real number) Ex. Ex.

Function Application 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) = 3x + 5000 where x is the number of shirts produced. b. The revenue function Revenue: R(x) = 12x where x is the number of shirts sold. c. The profit from 900 shirts Profit: P(x) = Revenue – Cost = 12x – (3x + 5000) = 9x – 5000 P(900) = 9(900) – 5000 = $3100

Introduction to Calculus 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

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

Velocity Ex. Given the position function where t is in seconds and s(t) is measured in feet, find: a. The average velocity for t = 1 to t = 3. b. The instantaneous velocity at t = 1. t Average velocity 1.1 12.1 Notice how elapsed time approaches zero 1.01 12.01 1.001 12.001 Answer: 12 ft/sec

Limit of a Function 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

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

Indeterminate Forms: Ex. Notice form Factor and cancel common factors Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.

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

One-Sided Limit of a Function 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

One-Sided Limit of a Function 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

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

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

Continuous Functions 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 is continuous at every point x in its domain.

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

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

The Derivative Four-step process for finding 1. Compute 2. Form

The Derivative Given 1. 2. 3. 4.

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