Question

In Example 2 of this section we showed that the derivative of \(y=\sqrt{x}\) is a function with domain \((0, \infty) .\) However, the function \(y=\sqrt{x}\) itself has domain \([0, \infty),\) so it could have a right-hand derivative at \(x=0 .\) Prove that it does not.

Short answer

The function \(y=\sqrt{x}\) does not have a right-hand derivative at \(x=0\) because the limit used to define the derivative does not exist at this point. This limit becomes infinite as h approaches 0 from the right.

Step-by-step solution

Definition of the right-hand derivative

The right-hand derivative of a function \(f\) at a certain point \(c\) is defined as: \[f'_{+}(c) = \lim_{{h \to 0^{+}}} \frac{f(c+h) - f(c)}{h}\] Apply this definition to the function \(f(x)=\sqrt{x}\) at the point \(x=0\). This requires calculating the limit: \[\lim_{{h \to 0^{+}}} \frac{1}{h}\left(\sqrt{0+h} - \sqrt{0}\right)\]

Simplification

Simplify the limit expression by substituting in the known values and reducing what we can: \[\lim_{{h \to 0^{+}}} \frac{1}{h}\left(\sqrt{h}\right) = \lim_{{h \to 0^{+}}} \frac{\sqrt{h}}{h}\]

Compute the limit

Next, simplify the fraction further to compute the limit: \[\lim_{{h \to 0^{+}}} \frac{1}{\sqrt{h}}\] Unfortunately, this limit does not exist as h approaches 0 from the right because the function \(\frac{1}{\sqrt{h}}\) becomes infinitely large as h becomes very small.

Key concepts

Understanding the Limit of a Function

The notion of a limit is a foundational concept in calculus. In simple terms, it's about figuring out what value a function approaches as the input (often denoted by the variable 'x') gets ever closer to a certain number or even infinity. When we say \( \lim_{{x \to a}} f(x) \), we're asking, 'what value does \( f(x) \)' approach as \( x \) gets very, very close to \( a \) without necessarily reaching it?

Think of limits as the mathematical equivalent of 'getting close enough to see the detail without touching.' It's not about where you are, but where you're headed as you get closer to a point. In our exercise, we are concerned with what happens to the function \( f(x) = \sqrt{x} \) as \( x \) approaches 0 from the right. This idea is crucial to understanding how functions behave near specific points and is a building block for finding derivatives and integrals, which are central to calculus.

The Quotient \(\frac{1}{\sqrt{h}}\) as \( h \) Approaches 0

When considering the limit of \( \frac{1}{\sqrt{h}} \) as \( h \) approaches 0 from the right, we're really looking at how the values of the function change as \( h \) gets smaller and smaller. To put it in visual terms, imagine \( h \) is the distance you are from a wall. As you take steps closer to the wall, each half the length of the previous step, the function's value represents the inverse of this reducing distance squared. These values will surge upwards toward infinity as you draw infinitesimally close to the wall.

This arises from the fact that squaring a small number (like \( h \) when it's close to 0) gives an even smaller number, and taking the reciprocal of a very small number creates a very large number. Hence, \( \frac{1}{\sqrt{h}} \) grows without bound as \( h \) approaches 0.

Derivative of \( \sqrt{x} \) and Right-Hand Limits

The derivative of a function is a measure of its instantaneous rate of change at any given point. For the function \( f(x) = \sqrt{x} \) the derivative tells us how fast the square root is changing with respect to a small change in \( x \). Understanding right-hand derivatives specifically helps us deal with functions that might not be defined or smoothly changeable to the left of a given point, particularly at boundaries of their domains.

In the context of \( \sqrt{x} \) at \( x=0 \), we seek to discern such a rate of change from the right. The concept of a right-hand derivative adds nuance to our understanding of derivatives, especially for functions defined piecewise or having distinct behavior at the borders of their domains.

The Importance of Calculus Education

Calculus is often considered the language of change, offering powerful tools to describe and understand the dynamics of natural phenomena, economics, engineering, and beyond. Its education is thus a critical component of modern scientific and mathematical training.

An essential aspect of this education is enabling students to grasp abstract concepts such as limits, derivatives, and the behavior of functions. The right approach can demystify these topics, rendering them not only understandable but also intriguing as students see the connections between mathematics and the real world. It's this bridge between theory and practicality that underlines the significance of calculus in the curriculum.