Question

a continuous curve y = f(x) has a vertical tangent line at the point where x = x₀ if lim(h→0) (f(x₀ + h) - f(x₀))/h = ∞ or -∞.
a. graph the curve y = 2x^(3/5). where does the graph appear to have vertical tangent lines?
b. confirm your findings in part (a) with limit calculations.
determine where, if any, there are vertical tangent lines. select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. using the graph of the function, there appears to be vertical tangent line(s) at x =
(use a comma to separate answers as needed.)
b. there is no vertical tangent line.

Explanation:

Step1: Recall the derivative - limit definition

The derivative of \(y = f(x)=2x^{\frac{3}{5}}\) using the limit formula \(\lim_{h
ightarrow0}\frac{f(x_0 + h)-f(x_0)}{h}\). First, find \(f(x_0 + h)=2(x_0 + h)^{\frac{3}{5}}\) and \(f(x_0)=2x_0^{\frac{3}{5}}\). Then \(\frac{f(x_0 + h)-f(x_0)}{h}=\frac{2(x_0 + h)^{\frac{3}{5}}-2x_0^{\frac{3}{5}}}{h}\).

Step2: Analyze the graph visually

For the function \(y = 2x^{\frac{3}{5}}\), when we consider the graph, we know that the power - rule for differentiation of \(y = ax^n\) is \(y^\prime=anx^{n - 1}\). Here \(a = 2\) and \(n=\frac{3}{5}\), so \(y^\prime=\frac{6}{5}x^{-\frac{2}{5}}=\frac{6}{5x^{\frac{2}{5}}}\). The function \(y = 2x^{\frac{3}{5}}\) is a continuous function. By looking at the graph of \(y = 2x^{\frac{3}{5}}\), we can see that the slope of the tangent line becomes infinite at \(x = 0\).

Step3: Calculate the limit

We want to find \(\lim_{h
ightarrow0}\frac{2(x+h)^{\frac{3}{5}}-2x^{\frac{3}{5}}}{h}\). Let \(x = 0\), then \(\lim_{h
ightarrow0}\frac{2(0 + h)^{\frac{3}{5}}-2(0)^{\frac{3}{5}}}{h}=\lim_{h
ightarrow0}\frac{2h^{\frac{3}{5}}}{h}=\lim_{h
ightarrow0}2h^{-\frac{2}{5}}=\lim_{h
ightarrow0}\frac{2}{h^{\frac{2}{5}}}\). As \(h
ightarrow0^+\), \(\lim_{h
ightarrow0^+}\frac{2}{h^{\frac{2}{5}}}=\infty\), and as \(h
ightarrow0^-\), \(\lim_{h
ightarrow0^-}\frac{2}{h^{\frac{2}{5}}}=\infty\).

Answer:

A. Using the graph of the function, there appears to be vertical tangent line(s) at \(x = 0\)