Chapter 2: Differentiation
Section 2.1: What Is a Derivative?
Example 2.1.4
At x=1, what can be said about the slope of the graph of fx={x2x≤1x 3−xx>1 ?
Solution
The graph of f in Figure 2.1.4(a) shows that this function has a jump discontinuity at x=1.
Secant lines through 1,1 for the curve to the left of x=1 also pass through 1−h,1−h2,h>0. The limiting value of the slopes of these secants is 2.
Secant lines through 1,2 for the curve to the right of x=1 also pass though 1+h,1+h3−1+h, h>0. The limiting value of the slopes of these secants is 1.
The portion of the graph to the left of x=1 has a tangent line with slope 2; but the portion to the right has a tangent line with slope 1.
plot(piecewise(x<=1,x^2,x*(3-x)),x=-2..4,discont=true,view=[-2..4,-2..3],scaling=constrained);
Figure 2.1.4(a) Graph of f with its jump discontinuity
The following calculations verify that the limiting slopes of secant lines to either side of x=1 differ.
Limiting slopes
To the left of x=1
To the right of x=1
flx=x2→assign as functionfl
frx=x 3−x→assign as functionfr
limh→0−fl1+h−1h = 2
limh→0+fr1+h−2h = 1
On the closed interval −2,1, the function gx=x2 would be said to have slope 2 at x=1 by considering just a one-sided limit at x=1. However, because of the jump at x=1 in fx, it is probably best to say that there is no unique slope for this function at this point.
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