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

2. 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

3. 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

4. Function Notation
Ex. Find
Plug in –2

5. 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:

6. 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:

7. 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)

8. 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.

9. 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.

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

12. Composition of Functions

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

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

15. Function Application
A shirt producer has a fixed monthly cost of $ If each shirt has a cost of $3 and sells for $12 find:
a. The cost function
Cost: C(x) = 3x 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 ) = 9x – 5000
P(900) = 9(900) – 5000 = $3100

16. 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

17. 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.

18. 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
Notice how elapsed time approaches zero
Answer: 12 ft/sec

19. 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

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

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

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

23. 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

24. 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

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

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

27. 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.

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

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

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

31. The Derivative
Given 1. 2. 3. 4.

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