Problem 58 - Calculus: Early Transcendentals

Question

(a) If \( g(x) = x^{2/3} \), show that \( g'(0) \) does not exist. (b) If \( a \neq 0 \), find \( g'(a) \). (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

(a) \( g'(0) \) does not exist. (b) \( g'(a) = \frac{2}{3}a^{-1/3} \). (c) There is a vertical tangent at \( (0, 0) \). (d) The graph shows a cusp at \( (0, 0) \).

Step 1: Evaluate the Derivative of g(x)

The function given is \( g(x) = x^{2/3} \). We'll first evaluate the derivative for any \( x eq 0 \) using the power rule. The derivative of \( f(x) = x^n \) is \( f'(x) = nx^{n-1} \). Applying this to our function: \( g'(x) = \frac{2}{3}x^{-1/3} \).

Step 2: Check the Derivative at x = 0

To find \( g'(0) \), substitute \( x = 0 \) into \( g'(x) = \frac{2}{3}x^{-1/3} \). However, \( x^{-1/3} \) becomes undefined at \( x = 0 \), indicating that \( g'(0) \) is not defined. Thus, the derivative does not exist at \( x = 0 \).

Step 3: Find the Derivative at a Non-zero Point

For \( a eq 0 \), the derivative is \( g'(a) = \frac{2}{3}a^{-1/3} \). Since \( a eq 0 \), this expression is defined and provides the slope of the tangent line at \( x = a \).

Step 4: Show Vertical Tangent at (0, 0)

A vertical tangent occurs where the derivative approaches infinity. Near \( x = 0 \), \( g'(x) = \frac{2}{3}x^{-1/3} \) tends to positive or negative infinity as \( x \) approaches zero from the right or left, respectively, demonstrating a vertical tangent at \( (0, 0) \).

Step 5: Graph y = x^{2/3} to Illustrate Vertical Tangent

Graphing \( y = x^{2/3} \), the curve will appear to have a cusp at \( (0, 0) \). The slopes of the tangents become steeper as the curve nears \( x = 0 \), indicating a vertical line, which visually represents the vertical tangent.

Key Concepts

Power rule in calculus

The power rule is a fundamental tool in calculus for finding the derivative of a function of the form \( f(x) = x^n \). The rule states that the derivative is \( f'(x) = nx^{n-1} \). This means you bring down the exponent as a coefficient and subtract one from the exponent.

• When applying it to a function like \( g(x) = x^{2/3} \), the derivative becomes \( g'(x) = \frac{2}{3}x^{-1/3} \).
• Notice the exponent \( -1/3 \) indicates that the derivative involves a reciprocal power, making it undefined at zero.

Using the power rule can highlight places where a function is not differentiable, such as \( x = 0 \) for \( g(x) \). Understanding how to apply and interpret the power rule helps in identifying behaviors like non-differentiability and can inform us about the nature of functions near points of interest.

Vertical tangents

Vertical tangents occur at points where the derivative of a function tends toward infinity, either positively or negatively. These are points where the instantaneous rate of change, or slope, is undefined but suggests an infinite steepness.To see this in action, consider our function \( y = x^{2/3} \):

• The derivative \( g'(x) = \frac{2}{3}x^{-1/3} \) becomes infinitely large as \( x \) approaches zero from either side.
• This behavior indicates a vertical tangent at the point \( (0, 0) \).

Vertical tangents reveal unique properties of functions, indicating points where graphs change direction sharply or where there is a form of a cusp. This understanding is especially crucial when analyzing graphs of non-differentiable functions where unexpected behaviors can occur.

Graphing functions

Graphing functions like \( y = x^{2/3} \) helps visually interpret mathematical concepts such as differentiability and vertical tangents. When you plot the graph of this function, you'll notice the following:

• The graph appears smoother than sharp-cornered functions but demonstrates a cusp at the origin \( (0, 0) \).
• Near this cusp, the slopes of tangent lines increase sharply as you move towards zero, signifying the presence of a vertical tangent.

Graphing not only aids in understanding theoretical calculus concepts but also provides a means to sense-check analytical results. It allows you to observe how changes in variables affect the shape and behavior of the function, offering deeper insights and making it easier to communicate complex mathematical ideas.