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Introductory Calculus: Average Rate of Change, Equations of Lines
online assistance  AVERAGE RATE OF CHANGE AND SLOPES OF SECANT LINES: The average rate of change of a function f(x) over an interval between two points (a, f(a)) and (b, f(b)) is the slope of the secant line connecting the two points:
 
 
 
    • For example, to calculate the average rate of change between the points:
     
        (0, -2) = (0, f(0)) and (3, 28) = (3, f(3))
         
      where f(x) = 3x2 + x – 2 we would:
 

This means that the average of all the slopes of lines tangent to the graph of f(x) 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:
 
   
ExamplesExamples:
Let . What is the average rate of change of f(x) between the points (1, 3) and (3, 15)?  
Let . What is the average rate of change of g(x) between times t = 50 and t = 500?


online assistance  SLOPE AND THE EQUATION OF A LINE: The slope 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,
 
 
where m is the slope and we have m = 4/5.
 
Our equation is
 
 
 
since you can use either point.
 
These equations look different, but they can both be rearranged to give us
 
 
which is the one line with slope 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.
     
ExamplesExamples:
What is an equation of the secant line connecting the points (1, 3) and (3, 15) in problem 1 above []?
What is an equation of the line through points ?
A line has slope 4. It passes through point (1, 3). Write an equation for this line.
A horizontal line passes through the point (4, -3). Write an equation of this line.
A vertical line passes through the point (3, 41.395). Write an equation.
A car rental agency charges $0.47 per mile and $125 to rent a car. Using m for miles and C for cost, write an equation describing the cost.
Suppose that  . What is the slope of a line containing points ? Simplify your answer.



M Ransom

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