Problem 11 - Brief Applied Calculus

Question

a. Show that the definition of the derivative applied to the function \(f(x)=\sqrt{x}\) at \(x=0\) gives \(f^{\prime}(0)=\lim _{h \rightarrow 0} \frac{\sqrt{h}}{h}\). b. Use a calculator to evaluate the difference quotient \(\frac{\sqrt{h}}{h}\) for the following values of \(h\) : \(0.1,0.001\), and \(0.00001 .\) [Hint: Enter the calculation into your calculator with \(h\) replaced by \(0.1\), and then change the value of \(h\) by inserting zeros.] c. From your answers to part (b), does the limit exist? Does the derivative of \(f(x)=\sqrt{x}\) at \(x=0\) exist? d. Graph \(f(x)=\sqrt{x}\) on the window \([0,1]\) by \([0,1]\). Do you see why the slope at \(x=0\) does not exist?

Step-by-Step Solution

Answer

The derivative does not exist at x=0 as the limit does not exist.

Step 1: Understanding the Derivative Definition

The derivative of a function \( f(x) \) at a point \( x = a \) is defined as \( f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \). For \( f(x) = \sqrt{x} \), we want to find \( f'(0) \). Substitute \( a = 0 \) into the definition, which gives us the expression \( \lim_{h \to 0} \frac{\sqrt{h} - \sqrt{0}}{h} = \lim_{h \to 0} \frac{\sqrt{h}}{h} \).

Step 2: Calculate the Difference Quotient for Given Values

Evaluate \( \frac{\sqrt{h}}{h} \) using a calculator for \( h = 0.1, 0.001, \) and \( 0.00001 \). For \( h = 0.1 \), \( \frac{\sqrt{0.1}}{0.1} \approx 3.1623 \). For \( h = 0.001 \), \( \frac{\sqrt{0.001}}{0.001} \approx 31.6228 \). For \( h = 0.00001 \), \( \frac{\sqrt{0.00001}}{0.00001} \approx 316.2278 \).

Step 3: Assessing the Limit

Observe that as \( h \) approaches zero, the values of \( \frac{\sqrt{h}}{h} \) increase without bound. This suggests that \( \lim_{h \to 0} \frac{\sqrt{h}}{h} \) does not exist because it approaches infinity. Since the limit does not exist, \( f'(0) \) does not exist.

Step 4: Graphical Interpretation

Graph \( f(x) = \sqrt{x} \) from \([0, 1]\) to \([0, 1]\). Notice that near \( x = 0 \), the curve becomes vertical, indicating that the slope (derivative) at \( x = 0 \) does not exist. This visual confirms that there is no defined tangent line or derivative at \( x = 0 \).

Key Concepts

Limit of a Function

The limit of a function helps us understand the behavior of functions as they approach specific points. In terms of calculus, it's crucial for determining a derivative. When we say we're finding the limit, it means we're examining what happens to a function's value as the variable approaches a specific number. For instance, the expression \( \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \) represents the process of finding the derivative, i.e., the slope of the tangent to the function at \( x = a \).

• This limit must exist for the derivative to be defined at that point.
• If the limit approaches a finite number, the function is differentiable at that point.
• If not, as in the case of \( f(x) = \sqrt{x} \) at \( x = 0 \), the function does not have a derivative at that point.

Understanding limits is foundational since they indicate continuity and potential differentiability of functions.

Difference Quotient

The difference quotient is a key tool in calculus for finding a function's derivative. It represents the average rate of change of the function over a small interval. For a function \( f(x) \), it's expressed as \( \frac{f(a+h) - f(a)}{h} \). By calculating this as \( h \) approaches zero, we can approximate the derivative, which tells us the function's instantaneous rate of change at \( x = a \).

In the given exercise:

• The difference quotient \( \frac{\sqrt{h}}{h} \) was evaluated for different small values of \( h \).
• We found that as \( h \) becomes smaller, the values increase significantly, which is a hint that the limit doesn't exist at \( x = 0 \) for this function.

Thus, the difference quotient is essential for practically assessing how a function behaves around specific points and whether a derivative can be defined.

Graphical Interpretation of Derivative

Graphs offer tremendous insights into derivatives. They help visualize how a function behaves at certain points and the nature of its slope. The derivative of a function at a particular point is the slope of the tangent to the curve at that point. Graphically, if the tangent is vertical, this signifies an undefined or non-existent derivative.

• In our example with \( f(x) = \sqrt{x} \), when graphed between \([0, 1]\), the curve becomes steeper as it approaches \( x = 0 \).
• The slope tends to infinity at the origin, thus illustrating why no finite derivative exists there.
• This visual tool is invaluable in confirming analytical findings, especially in cases where calculated limits suggest non-existence.

Non-Existence of Derivative

A function may fail to have a derivative at certain points due to specific characteristics in its graph. Such omissions in a derivative can occur when:

• The function makes a sharp turn or a cusp.
• The tangent line at the point is vertical, as is the case with \( f(x) = \sqrt{x} \) at \( x = 0 \).
• There is discontinuity or a break in the graph.

In our exercise, the derivative of \( f(x) = \sqrt{x} \) does not exist at \( x = 0 \) due to the vertical tendency of the curve at origin. This indicates that functions approaching infinite slope at any point will not have a well-defined derivative there. Understanding when and why derivatives do not exist enhances the comprehension of function behavior comprehensively.