2.1 A preview of calculus  (Page 2/13)

The accuracy of approximating the rate of change of the function with a secant line depends on how close x is to a . As we see in [link] , if x is closer to a , the slope of the secant line is a better measure of the rate of change of $f\left(x\right)$ at a .

The secant lines themselves approach a line that is called the tangent    to the function $f\left(x\right)$ at a ( [link] ). The slope of the tangent line to the graph at a measures the rate of change of the function at a . This value also represents the derivative of the function $f\left(x\right)$ at a , or the rate of change of the function at a . This derivative is denoted by $f^{\prime }\left(a\right).$ Differential calculus is the field of calculus concerned with the study of derivatives and their applications.

[link] illustrates how to find slopes of secant lines. These slopes estimate the slope of the tangent line or, equivalently, the rate of change of the function at the point at which the slopes are calculated.

Finding slopes of secant lines

Estimate the slope of the tangent line (rate of change) to $f\left(x\right)=x^2$ at $x=1$ by finding slopes of secant lines through $\left(1,1\right)$ and each of the following points on the graph of $f\left(x\right)=x^2.$

1. $\left(2,4\right)$
2. $\left(\frac{3}{2},\frac{9}{4}\right)$

Use the formula for the slope of a secant line from the definition.

1. $m_{\text{sec}}=\frac{4-1}{2-1}=3$
2. $m_{\text{sec}}=\frac{\frac{9}{4}-1}{\frac{3}{2}-1}=\frac{5}{2}=2.5$

The point in part b. is closer to the point $\left(1,1\right),$ so the slope of 2.5 is closer to the slope of the tangent line. A good estimate for the slope of the tangent would be in the range of 2 to 2.5 ( [link] ).

Got questions? Get instant answers now!

Got questions? Get instant answers now!

We continue our investigation by exploring a related question. Keeping in mind that velocity may be thought of as the rate of change of position, suppose that we have a function, $s\left(t\right),$ that gives the position of an object along a coordinate axis at any given time t . Can we use these same ideas to create a reasonable definition of the instantaneous velocity at a given time $t=a?$ We start by approximating the instantaneous velocity with an average velocity. First, recall that the speed of an object traveling at a constant rate is the ratio of the distance traveled to the length of time it has traveled. We define the average velocity    of an object over a time period to be the change in its position divided by the length of the time period.