Problem 21 - Calculus

Question

Find the slope of \(y=\sqrt{x}\) by algebra (then \(h \rightarrow 0\) ): $$ \frac{\Delta y}{h}=\frac{\sqrt{x+h}-\sqrt{x}}{h}=\frac{\sqrt{x+h}-\sqrt{x}}{h} \frac{\sqrt{x+h}+\sqrt{x}}{\sqrt{x+h}+\sqrt{x}}. $$

Step-by-Step Solution

Answer

The slope is \( \frac{1}{2\sqrt{x}} \).

Step 1: Setup the expression

We need to find the slope of the function \( y = \sqrt{x} \) by evaluating \( \frac{\Delta y}{h} = \frac{\sqrt{x+h} - \sqrt{x}}{h} \). Our goal is to find its limit as \( h \rightarrow 0 \).

Step 2: Rationalize the numerator

To handle the square roots, multiply both the numerator and denominator by the conjugate \( \sqrt{x+h} + \sqrt{x} \). This gives:\[ \frac{(\sqrt{x+h} - \sqrt{x})(\sqrt{x+h} + \sqrt{x})}{h(\sqrt{x+h} + \sqrt{x})} \]

Step 3: Simplify the numerator

Use the difference of squares formula \((a-b)(a+b) = a^2 - b^2\) to simplify the numerator:\[ \sqrt{x+h}^2 - \sqrt{x}^2 = (x+h) - x = h.\]So the fraction becomes:\[ \frac{h}{h(\sqrt{x+h} + \sqrt{x})}.\]

Step 4: Cancel terms and simplify

Since \( h \) is in both the numerator and denominator, cancel it out:\[ \frac{h}{h(\sqrt{x+h} + \sqrt{x})} = \frac{1}{\sqrt{x+h} + \sqrt{x}}.\]

Step 5: Evaluate the limit as h approaches 0

As \( h \rightarrow 0 \), the term \( \sqrt{x+h} \) approaches \( \sqrt{x} \). Substitute \( h = 0 \) in the simplified expression to find the limit:\[ \lim_{{h \to 0}} \frac{1}{\sqrt{x+h} + \sqrt{x}} = \frac{1}{\sqrt{x} + \sqrt{x}} = \frac{1}{2\sqrt{x}}.\]

Step 6: Conclusion: Identify the slope

The slope of the function \( y = \sqrt{x} \) at any point \( x \) is \( \frac{1}{2\sqrt{x}} \).

Key Concepts

Limit

When we talk about limits, we are referring to the value that a function approaches as the input approaches some value. In the context of finding the slope of a curve, such as the function \( y = \sqrt{x} \), we often look for the slope as one point on the curve approaches another. This typically involves letting \( h \rightarrow 0 \). Here, \( h \) is a small change in \( x \), and our job is to see what happens to the quotient \( \frac{\Delta y}{h} \) as \( h \) shrinks towards zero. This process helps us compute the derivative, which represents the slope of the tangent line to the curve at a specific point. It's like zooming in infinitely close to the curve until it looks straight. Understanding limits is crucial for evaluating expressions like \( \frac{\sqrt{x+h} - \sqrt{x}}{h} \) in a precise manner.

Difference of squares

The difference of squares is a specific algebraic identity that shows how to multiply two binomials with the same terms but opposite operations. It is expressed as:

• \((a - b)(a + b) = a^2 - b^2\)

This formula is particularly useful for simplifying expressions that involve radicals, like in our exercise. When we multiplied \( (\sqrt{x+h} - \sqrt{x}) \) by \( (\sqrt{x+h} + \sqrt{x}) \), the result was:

• \( (\sqrt{x+h})^2 - (\sqrt{x})^2 \)

This simplifies to \( (x+h) - x = h \). Using this identity allows us to eliminate the square roots and simplify the rational expression effectively, paving the way to finding the limit and, subsequently, the slope.

Rationalization

Rationalization is a technique used to remove radicals from the denominator or to simplify expressions containing radicals. In our exercise, rationalization helps in dealing with the fraction \( \frac{\sqrt{x+h} - \sqrt{x}}{h} \). Square roots in the numerator can make it difficult to simplify further or evaluate limits, so rationalization comes into play. By multiplying the numerator and denominator by the conjugate \( \sqrt{x+h} + \sqrt{x} \), we employ a method to convert a complex fraction into a simpler form. Rationalization serves to manipulate the expression such that the numerator forms a recognizable pattern, like the difference of squares, which can be further simplified.

Conjugate multiplication

Multiplying by the conjugate is a strategic method used when simplifying expressions with radicals. A conjugate is formed by changing the operator between two terms. If you have \( a - b \), its conjugate is \( a + b \). In this exercise, we multiply \( \sqrt{x+h} - \sqrt{x} \) by its conjugate \( \sqrt{x+h} + \sqrt{x} \). This approach not only simplifies the expression but also enables us to eliminate the root terms. The multiplication results in a difference of squares, transforming the complex radical expression into a polynomial form. This step is essential for further simplification, especially when trying to find the limit, as it aligns the expression with easily cancelable components.