Chapter 2: Differentiation
Section 2.1: What Is a Derivative?
Example 2.1.3
Show that at x=0 a unique slope cannot be assigned to the graph of fx=x1/3.
Does this curve have a tangent line at 0,0?
Solution
The sharp point the graph of f has at the origin in Figure 2.1.3(a) is an example of a cusp.
Consider secant lines through 0,0 and 0+h,f0+h=h,fh, the slopes of which are
f0+ h−00+h−0=fhh=h1/3h={1/h2/3h>0−1/h2/3h<0
The limiting values of these slopes are ±∞, the positive value when the limit is taken from the right; the negative, from the left.
Consequently, the limit of these slopes does not exist, so no unique slope can be assigned at x=0.
plot(abs(x)^(1/3),x=-1..1,scaling=constrained,tickmarks=[3,2],labels=[x,y]);
Figure 2.1.3(a) Graph of fx=x1/3
The following calculations verify the conclusion that there is no unique slope at x=0.
Define the function f
Control-drag fx=… Context Panel: Evaluate and Display Inline
fx=x1/3→assign as functionf
Limiting slopes of the secant lines
Expression palette: Limit operator Limit from the left
limh→0−fhh = −∞
Expression palette: Limit operator Limit from the right
limh→0+fhh = ∞
If the tangent line is a line whose slope is the limiting value of the slopes of secant lines, then this function has no tangent line at the origin. However, if a more geometric notion of a tangent line is invoked, then it could be said that the y-axis is a line tangent to the curve at the origin. Hence, the question about the existence of a tangent line hinges on perspective, either analytic or geometric.
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