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1. Rate of change
Rate of change describes how one quantity changes in relation to another quantity. In specific, rate of change describes the changes in the dependent variable with respect to the changes in the independent variable.
If y=f(x), then
| Rate of change=\({changes \text { in} \text { y}\over changes \text { in}\text { x}}\) |
2. Sign of rate of change
Rate of change is positive in two situations:
- when x increases, y also increases
- when x decreases, y also decreases
Rate of change is negative in two situations:
- when
- x increases but y decreases
- when x decreases but y increases
Example
Given the above table of values for
x and y, find the rate of change and plot the graph.
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\({22-10 \over 4-2} = {12\over2}=6\)
\({34-22 \over 6-4} = {12\over2}=6\)
\({34-10 \over 6-2} = {24\over4}=6\)
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So, rate of change is 6.
3. Average rate & instantaneous rate
Average rate is the slope of the line between two points.
Let
y=f(x) and let x=a and x=b be on the graph of y=f(x). Therefore, we have the two points or two sets of coordinates a,f(a) and b,f(b).
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\(Average \text{ rate}={f(b)-f(a) \over b-a}\)
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Instantaneous rate is the derivative at one point. The instantaneous rate at
x=a is f'(a).
| \(Instantaneous \text{ rate} = f'(a)\) |
Example
Let
\(fx=x^2\text{ and} \text{ x}=1 \text{ and } \text{x}=3 \text{ be on the curve y}=x^2.\)
1. Find the average rate of change between the two points.
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\({\text{Average rate} = {{f(3)-f(1)} \over 3-1}}\)
\(= {{3^2-1^2} \over 3-1}\)
\(= {{9-1} \over 2}\)
\(= 4\)
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2. Find the instantaneous rate of change at x=1.
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\(f'x=2x\)
Then, instantaneous rate at x=1
\(= f'(1)\)
\(=2(1)\)
\(=2\)
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Tag
Reflection
What is rate of change?
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What is positive rate of change?
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Given the following table of values for x and y, find the rate of change.
What is the average rate of change between and x=2 and x=4 for fx=x^2 - 1/x ?
Given f(x) = 3x^2 - 5x + 2, what is the instantaneous rate of change at x = 2 ?
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