Rate of Change and Slope
Section 5-1
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Determining Constant and Variable Rates of Change from Data, Lecture notes of Calculus
Instructions on how to find rates of change and slope from tables. It includes examples and exercises to help students understand the concept of constant and variable rates of change. The document also explains how to identify dependent and independent variables and graph the data to visualize the rates of change.
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2021/2022
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Rate of Change and Slope
Section 5-
Goals
Goal
- To find rates of change from tables.
- To find slope.
Definition
- Rate of Change – a ratio that compares the amount
of change in a dependent variable to the amount of
change in an independent variable.
- The rates of change for a set of data may vary or
they may be constant.
Determine whether the rates of change are constant or variable. A. Find the difference between consecutive data points.
x 0 1 3 5 8
y 0 2 6 10 16
+1 +2 +2 + +2 +4 +4 + Find each ratio of change in y to change in x. 2 1
The rates of change are constant.
Example:
Determine whether the rates of change are constant or variable. A. Find the difference between consecutive data points.
x 0 1 2 3 4
y 1 3 5 7 11
+1 +1 +1 + +2 +2 +2 + Find each ratio of change in y to change in x. 2 1
The rates of change are variable.
Your Turn:
Determine whether the rates of change are constant or variable. B. Find the difference between consecutive data points.
x 0 1 4 6 9
y 0 2 8 12 18
+1 +3 +2 + +2 +6 +4 + Find each ratio of change in y to change in x. 2 1
The rates of change are constant.
Your Turn:
Step 2 Find the rates of change. The temperature increased at the greatest rate from month 5 to month 7. 3 to 5 5 to 7 7 to 8 2 to 3
The table shows the balance of a bank account on different days of the month. Find the rate of change during each time interval. During which time interval did the balance decrease at the greatest rate? Step 1 Identify the dependent and independent variables.
dependent: balance independent: day
Your Turn:
To show rates of change on a graph, plot the data
points and connect them with line segments.
Data in a graph is always read from left to right. Remember! Rates of Change on a Graph
Graph the data from the first Example and show the rates of change. Graph the ordered pairs. The vertical segments show the changes in the dependent variable, and the horizontal segments show the changes in the independent variable. Notice that the greatest rate of change is represented by the steepest of the red line segments. Example: Rates of Change from a Graph
Graph the data from the last your turn and show the rates of change. Graph the ordered pairs. The vertical segments show the changes in the dependent variable, and the horizontal segments show the changes in the independent variable. Notice that the greatest rate of change is represented by the steepest of the red line segments.
Your Turn:
Also notice that between days 16 to 22, when the balance did not change, the line segment is horizontal.
Your Turn: Continued
1 to 6 6 to 16 16 to 22 22 to 30
Slope
slo pe =
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vertical ch
al chang
ange r
e n
ise
ru
rise run