Calculus I, Chapter 3: The Average Rate of Change and Limits
For the Previous Sections, see Helpful Tools for Calculus: Chapter 2
Slope
The study of Calculus begins with attempting to identify how to find a line tangent to a curve. It is important to note that calculus is primarily focused on curved lines, as a linear equation provides little to search for. This problem, of finding tangent lines, can arise in a number of ways. First let us provide some motivation for this question, then we can explain what it means for a line to be tangent to the curve and how to solve for one.
In linear equations we are obsessed with the idea of slope. How fast is it changing over time? In linear equations, this is easy. The slope of a linear equation is represented by the m in y=mx+b and can be found by solving for the change in y divided by the change in x between a couple of points. This idea is important, we want to know how to predict future outcomes or to understand previous ones. However, with curved lines this method does not work.
It doesn’t take long to realize that it does not work because the slope is changing. So how can we study the change in a curved line? How can we study the change in a function, in general?
Average Rate of Change
One option is to simply calculate an average rate of change over some interval. That is to say, if we want to know how a function is changing over a certain set of inputs, we can just find the slope between the ends and that should describe the overall change of the function. This is no different than what was done in algebra to find a slope.
The equation that describes this calculation is average rate of change= (f(x_2 )-f(x_1))/(x_2-x_1 ) for a function f(x) over the interval [x_1,x_2]. It is helpful, perhaps, to acknowledge that this is identical to the equation for slope from algebra: m=(y_2-y_1)/(x_2-x_1 ).
Example 1: Find the Average Rate of Change for the function y=x^2 over the interval x∈[0,5]. Draw a sketch of the graph and draw a line connecting the endpoints.
Solution:
(f(x_2 )-f(x_1 ))/(x_2-x_1 )=((5)^2-(0)^2)/(5-0)
=25/5
=5
Example 2: Find the Average Rate of Change for the function y=2 sin(x)+x over the interval x∈[ 0,π/2]. Draw a sketch of the graph and draw a line connecting the endpoints.
Solution:
(f(x_2 )-f(x_1 ))/(x_2-x_1 )=([2 sin(π/2)+π/2]-[2 sin(0)+0])/(π/2-0)
=(2+π/2)/(π/2)
=(4/2+π/2)∙2/π
=(4+π)/2∙2/π
=(4+π)/π
Secant Lines
In the first example we are able to show that a parabolic function has an average rate of change of 5 over the interval from 0 to 5. The line that was drawn to connect the endpoints is called a secant line. A secant line is a line that touches a curve at two spots and can represent a linear approximation of the function between two points. So, when we find the average rate of change, we are actually finding the slope of a linear approximation. Let’s take a moment to note that that is not what we want to do! We want to find the tangent line of a curve, not a secant line! However, let’s be satisfied with this for now as we begin to perfect our method.
Velocity, Acceleration, and Physics
An example of average rate of change can be applied to basic physics. In studying motion we can define the average velocity as the average speed along a given path. This can be calculated as Average Velocity=(change in distance)/(change in time). This is very similar to the average rate of change, precisely because that is exactly what it is.
Example 3: Find the average velocity of ball that is dropped from a 30foot building between 1 and 1.5 seconds if the equation for distance travelled is given by x(t)=30-12t^2.
Solution:
Average Velocity=(x(t_2 )-x(t_1 ))/(t_2-t_1 )
= ([30-12(1.5)^2 ]-[30-12(1)^2 ] feet)/(1.5-1 seconds)
=(30-12(2.25)-30+12 feet)/(0.5 seconds)
=(27+12)/0.5 ft/s
=39/0.5 ft/s
=78ft/s
Example 4: Find the average acceleration of a car whose velocity is given as v(t)=12+0.7t ln(t) m/s on the interval t∈[1,3].
Solution:
Average Acceleration=(v(t_2 )-v(t_1 ))/(t_2-t_1 )
=([12+0.7(3)ln(3)]-[12+0.7(1)ln(1)] meters/second)/(3-1 seconds)
=(12+2.1ln(3)-12)/2 m/s^2
=(2.1 ln(3))/2 m/s^2
=1.05ln(3)m/s^2
= 1.1535429031 m/s^2
Review
The Average Rate of Change calculates the slope of a linear line that approximates the curve between two values
A straight line that connects the endpoints of an average rate of change on a graph is called a secant line
The average rate of change of displacement is called velocity
The average rate of change of velocity is called acceleration
Equations
average rate of change= (f(x_2 )-f(x_1))/(x_2-x_1 )