This document discusses average and instantaneous rates of change and limits of functions. It provides examples and definitions of: - Average rate of change, which represents total change over total change - Instantaneous rate of change, which measures change over infinitesimally small changes - Limits of functions as the input approaches a value, where the limit is the value the function approaches - One-sided limits, where the left and right limits may differ or agree It also distinguishes between average speed, which is distance over time, and average velocity, which is displacement over time.

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LIMITS
Calculus and Analytical Geometry

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Average and Instantaneous Change
What is the rate of change?
Rate of change is the change in one variable in relation to the change in another variable.
• The average rate of change represents the total change in one variable in relation to the total change of
another variable.
• Instantaneous rate of change, measures the rate of change of one variable in relation to infinitesimally
small change in the other variable.

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Average and Instantaneous Change

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Example 1: Total Cost
Suppose a company’s total cost in dollars to produce x units of its product is given by
Find the average rate of change of total cost for (a) the first 100 units produced (from to ) and (b) the second 100 units
produced.

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Figure below shows the percent of elderly men and of elderly women in the work force in selected census years from 1890
to 2000. For the years from 1950 to 2000, find and interpret the average rate of change of the percent of (a) elderly men in
the work force and (b) elderly women in the work force.
Example 2: Elderly in the Work Force

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Example 2: Elderly in the Work Force

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The Limit of a Function
Let’s investigate the behavior of the function f defined by
for values of x near 2.
The following table gives values of f(x) for values of x close to 2 but not equal to 2.
𝒇 (𝒙) = 𝒙𝟐 – 𝒙 + 𝟐

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From the previous table and the graph of f (a parabola) shown in Figure below we see that when x is close to
2 (on either side of 2), f(x) is close to 4.
• As x gets close to 2, f(x) gets close to 4
• The limit of the function 𝒇 (𝒙) = 𝒙𝟐
– 𝒙 + 𝟐 as x
approaches 2 is equal to 4
The Limit of a Function

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The Limit of a Function
• An alternative notation is f (x)  L as x  a which is usually read “f (x) approaches L as x approaches a.”
• Notice the phrase “but x # a” in the definition of limit. This means that in finding the limit of f (x) as x approaches a, it
does not matter what is happening at x = a.
• In fact, f (x) need not even be defined when x = a. The only thing that matters is how f is defined near a.

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in all three cases
f (a) is not defined
f (a) # L.
The Limit of a Function

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Example:
Solution:
The Limit of a Function
Graphically

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On the basis of the values in the tables, we can say
Numerically
The Limit of a Function

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One-Sided Limits
The Heaviside function H is defined by
• H(t) approaches 0 as t approaches 0 from the left and H(t) approaches 1 as t approaches 0 from the right.
• We indicate this situation symbolically by writing

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One-Sided Limits
Example:

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Solution:
(c) Since the left and right limits are different, we conclude from that limx2 g(x) does not exist
(f) This time the left and right limits are the same, so
One-Sided Limits

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Practice Problems

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Practice Problems

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Practice Problems

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Practice Problems

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Average speed is calculated as the distance traveled over the total time of travel.
In contrast, average velocity is defined as the change in position (or displacement) over the total time of travel
average speed of 40/30 = 1.33 m/s
Displacement is 0 meters. Therefore, average velocity, or displacement over time, would be 0 m/s
Average speed and Average Velocity