Problem 50 - Calculus Early Transcendentals

Question

(a) If \(g(x)=x^{2 / 3}\) , show that \(g^{\prime}(0)\) does not exist. (b) If \(a \neq 0,\) find \(g^{\prime}(a).\) (c) Show that \(y=x^{2 / 3}\) has a vertical tangent line at \((0,0).\) (d) Illustrate part (c) by graphing y \(=x^{2 / 3}\).

Step-by-Step Solution

Answer

The derivative at 0 does not exist, \(g'(a) = \frac{2}{3a^{1/3}}\) for \(a \neq 0\), and the function has a vertical tangent at (0,0).

Step 1: Differentiate the function for a general x

The function is given as \(g(x) = x^{2/3}\). To find the derivative \(g'(x)\), use the power rule \(\frac{d}{dx} x^n = nx^{n-1}\). Applying this rule:\[g'(x) = \frac{2}{3}x^{-1/3}.\]

Step 2: Evaluate the derivative at x = 0 to show non-existence

Calculate \(g'(0)\) using the expression for \(g'(x)\):\[g'(0) = \frac{2}{3}(0)^{-1/3}\]However, \(0^{-1/3}\) is undefined because division by zero is not allowed, making \(g'(0)\) undefined. Therefore, \(g'(0)\) does not exist.

Step 3: Find the derivative for \(x = a\) where \(a \neq 0\)

Substitute \(x = a\) into the general derivative formula:\[g'(a) = \frac{2}{3}a^{-1/3}\]For \(a eq 0\), \(g'(a)\) is defined and simplifies to \(\frac{2}{3a^{1/3}}\).

Step 4: Interpret the derivative for a vertical tangent line at (0,0)

The derivative \(g'(x) = \frac{2}{3}x^{-1/3}\) becomes undefined at \(x = 0\), indicating an infinite slope, meaning there's a vertical tangent line. Thus, \(y = x^{2/3}\) has a vertical tangent at \((0,0)\).

Step 5: Graph y = x^{2/3} to illustrate the vertical tangent

To graph \(y = x^{2/3}\), note that as \(x\) approaches 0, the slope of the tangent line increases infinitely, effectively becoming vertical. Plotting this function shows a cusp at the origin \((0,0)\) with an obvious vertical tangent at this point.

Key Concepts

Understanding Derivatives

In calculus, a derivative is an important concept that measures how a function changes as its input changes. More precisely, it represents the slope or the steepness of the graph at any given point. Derivatives can tell us how fast or slow a function is changing.
To find the derivative of a function, we use the concept of limits, which helps us understand the function’s rate of change at specific points.
For the function given in the exercise, we have to explore how the derivative works for special situations, like at zero. By taking derivatives, we can analyze not just straight lines, but also curves, making it very powerful in calculus.

Explaining Vertical Tangents

A vertical tangent occurs when a function's derivative at a certain point is undefined or tends to infinity. In simpler terms, this means that the slope of the tangent to the curve at that point is so steep, it becomes perpendicular to the x-axis.
In the example function, this situation happens at the point (0,0). Here, the slope approached by the tangent line is vertical because the change in the y-values with respect to the x-values becomes infinitely large.
Vertical tangents are a fascinating aspect of calculus, helping us understand the behavior of curves at points where other traditional rules don’t apply.

Using the Power Rule

The power rule is a basic yet powerful tool in calculus, used to find derivatives of polynomial functions quickly. The rule states: if you have a power of x, such as \(x^n\), the derivative is \(nx^{n-1}\). Simply put, you multiply the function by the current power and then decrease the power by one.
This rule was applied to the function \(g(x) = x^{2/3}\), resulting in the derivative \(g'(x) = \frac{2}{3}x^{-1/3}\).
The power rule simplifies the process of differentiation, making it easier to find how curves behave as they change, which is extremely helpful for understanding the dynamics of functions.

What is an Undefined Slope?

An undefined slope happens when a vertical line occurs, meaning the change in the horizontal direction is zero. In mathematical terms, an undefined slope signals that the denominator in the slope formula, which measures how far right or left the line moves, is zero.
When the derivative of a function at a point is undefined, as seen at \(x = 0\) for \(y = x^{2/3}\), this indicates the slope can’t be measured using traditional methods. Thus, it reveals the presence of a vertical tangent at that point.
Understanding this concept is crucial since it highlights where normal calculus methods might not apply directly, calling for a deeper analysis of the function's behavior at specific points.