1. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Functions, Limits, and the Derivative 2
Functions and Their Graphs
The Algebra of Functions
Functions and Mathematical Models
Limits
One-Sided Limits and Continuity
The Derivative
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2. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
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
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3. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
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
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4. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Function Notation
Example
Solution
Plug in –2
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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
The domain can be expressed as the intervals
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Domain of a Function
Ex. Find the domain of
Since the square root of a negative number is undefined we must have
The domain can be expressed as the interval
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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). y
f (1) = 2
(1, 2) x
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8. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
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.
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Example
Sketch the graph of the function:
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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.
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12. Composition of Functions
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Types of Functions
Polynomial Functions
n is a nonnegative integer, each
is a constant.
Ex.
Rational Functions
polynomials
Ex.
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Types of Functions
Power Functions
( r is any real number)
Ex.
Ex.
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15. Application Example 1 (Functions )
A shirt producer has a fixed monthly cost of $ If each shirt costs $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, or $3100.
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16. Application Example 2 (Functions)
A division of Chapman Corporation manufactures a
pager. The weekly fixed cost for the division is
$20,000, and the variable cost for producing x
pagers/week is
The company realizes a revenue of
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17. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
1. Find the total cost function.
The total cost function is the variable cost plus the fixed cost:
2. Find the total profit function.
The profit is the revenue minus the total cost
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What is the profit for the company if 2000 units are produced and sold each week?
Since the profit function is
we have
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19. Introduction to Calculus
There are two main areas of focus:
1. Finding the tangent line to a curve at a given point. y x
tangent line
2. Finding the area of a planar region bounded by a given curve. y
Area x
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20. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Velocity
Over any time interval
Average
If I travel 200 miles in 5 hours, my average velocity is 40 miles/hour.
As elapsed time approaches zero
Instantaneous
When I see the police officer, my instantaneous velocity is 60 miles/hour.
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21. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
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
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22. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
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. y L x a
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23. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Computing Limits
Ex. y 6
Note: f (-2) = 1
is not involved x 2
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Properties of Limits
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Computing Limits
Ex.
Ex.
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Indeterminate Form:
Ex.
Notice form
Factor and cancel common factors
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Limits at Infinity
For all n > 0,
provided that is defined.
Divide by
Ex.
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28. 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
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29. 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 M by taking x to be sufficiently close to the left of a. y M x a
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30. One-Sided Limit of a Function
Ex. Given
Find
Find
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31. Continuity of a Function
A function f is continuous at the point x = a if the following are true: y
f(a) x a
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32. Properties of Continuous Functions
The constant function f (x) is continuous everywhere.
Ex. f (x) = 10 is continuous everywhere.
The identity function f (x) = x is continuous everywhere.
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33. Properties of Continuous Functions
If f and g are continuous at x = a, then
A polynomial function y = P(x) is continuous at everywhere.
A rational function is continuous
at all x values in its domain.
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34. Intermediate Value Theorem
If f is a continuous function on a closed interval [a, b] and L is any number between f (a) and f (b), then there is at least one number c in [a, b] such that f(c) = L. y
f (b)
f (c) = L
f (a) x a c b
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35. Intermediate Value Theorem
Ex.
f (x) is continuous for all values of x and since
f (1) < 0 and f (2) > 0, by the Intermediate Value
Theorem, there exists a c on (1, 2) such that
f (c) = 0.
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36. Existence of Zeros of a Continuous Function
If f is a continuous function on a closed interval [a, b],
and f(a) and f(b) have opposite signs, then there is at least
one solution of the equation f(x) = 0 in the interval (a, b). y
f(b) x a b
f(a)
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37. Example (Existence of zeros of a continuous function)
Show that f(x) is a continuous function everywhere.
The function is a polynomial function and is therefore continuous everywhere.
2. Show that f(x) = 0 has at least one solution on the interval (0, 2)
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38. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Rates of Change
Average rate of change of f over the interval [x, x+h]
Slope of Secant Line
Instantaneous rate of change of f at x
Slope of the Tangent Line
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39. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
The Derivative
The derivative of a function f with respect to x is the function
given by
It is read “f prime of x.”
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40. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
The Derivative
Four-step process for finding
1. Compute
2. Find
3. Find
4. Compute
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41. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
The Derivative
Given 1. 2. 3. 4.
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42. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Example
Find the slope of the tangent line to the graph of at any point (x, f(x)).
Step 1.
Step 2.
Step 3.
Step 4.
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43. Differentiability and Continuity
If a function is differentiable at x = a, then it is continuous at x = a. y
Not Continuous x
Still Continuous
Not Differentiable
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44. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Example
The function is not differentiable at x = 0 but it is continuous everywhere. y x O
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