|Introductory Calculus: Average Rate of Change, Equations of Lines
AVERAGE RATE OF CHANGE AND SLOPES OF SECANT LINES:
The average rate
of change of a function f
) over an interval between two points (a, f
(a)) and (b, f
(b)) is the slope
of the secant line
connecting the two points:
This means that the average of all the slopes of lines tangent to the graph
) between the points (0, –2) and (3, f(3)) is 10.
If we want the exact slope
of a tangent line
to this function
at the point
where x = 2, we would have to use other methods. The average slope
can be calculated using two points. The exact slope
at one point
defies our basic formula for slope
since we need to know TWO points, and this will be approached differently.
- For example: A man has driven a car 50 miles in one hour. Over the next three hours, he drives 140 miles. What is his average velocity (speed) over that next 3 hours? (We use t and d to represent time in hours and distance in miles).
We use the two points (1, 50) and (4, 190). Notice that 3 additional hours gives us a t value of 4 and the total number of miles is d = 50 + 140 = 190.
The average velocity
is the average rate
of change of this distance with respect to time. We have:
SLOPE AND THE EQUATION OF A LINE:
of a line
connecting two points is a ratio
of the “rise” to the “run,” which is a ratio
of the vertical distance between the points to the horizontal distance between the two points.
- A line passing through the points (2, 5) and (-3, 1) has a slope of
Since this is a positive number, the line
will appear to slope
upwards to the right when graphed.
How do we write an equation
of this line
which will describe ALL points on the line?
- The easiest way is to use point-slope form which is
For our example,
is the slope
and we have m
since you can use either point.
These equations look different, but they can both be rearranged to give us
which is the one line
4/5 and y-intercept
17/5. Generally speaking, do NOT rewrite this equation
unless you have to solve for y
to enter it into your calculator or you have specific instructions for rewriting.
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