Problem 14 - A Graphical Approach to Precalculus with Limits

Question

Find the slope of the tangent line to each curve when \(x\) has the given value. Do not use a calculator. $$f(x)=\sqrt{2 x} ; x=2$$

Step-by-Step Solution

Answer

The slope of the tangent line at \( x = 2 \) is \( \frac{1}{2} \).

Step 1: Understand the Function

The function given is \( f(x) = \sqrt{2x} \). This is a square root function, which generally has a curve and does not have constant slopes like linear functions.

Step 2: Find the Derivative

To find the slope of the tangent line to the curve at any point, we need to find the derivative of \( f(x) = \sqrt{2x} \). Recall the power rule for derivatives. The function can be rewritten as \( f(x) = (2x)^{1/2} \). Use the chain rule here. The derivative is \( f'(x) = \frac{1}{2}(2x)^{-1/2} \times 2 = \frac{1}{\sqrt{2x}} \).

Step 3: Evaluate the Derivative at x = 2

Now, find the slope of the tangent line at \( x = 2 \) by substituting \( x = 2 \) into the derivative \( f'(x) = \frac{1}{\sqrt{2x}} \). Substituting the value gives \( f'(2) = \frac{1}{\sqrt{4}} = \frac{1}{2} \).

Step 4: Interpret the Slope

The value \( \frac{1}{2} \) is the slope of the tangent line to the curve \( f(x) = \sqrt{2x} \) at \( x = 2 \). This means that at \( x = 2 \), the line just touching the curve (the tangent) rises 1 unit vertically for every 2 units it moves horizontally.

Key Concepts

Derivative

A derivative represents the rate at which a function is changing at any given point. It is a core concept in calculus and is often used to find the slope of the tangent line to a curve. In calculus, the derivative is like the speedometer of a moving car, showing how fast the car (or in this case, the function) is changing at any given moment. To find the derivative, we apply differentiation rules to the function.

• Power Rule: Used when differentiating expressions in the form of a power.
• Chain Rule: Used when differentiating composite functions.

By finding the derivative of a function, we learn how the function behaves locally, and we can assess whether it is increasing or decreasing at that point.

Slope of Tangent Line

The slope of a tangent line to a function at a given point shows how steep the curve is at that spot. Imagine you are skiing on a hill: the slope tells you how steep the hill is at any point. If the slope is 0, it is flat, like standing on the top of the hill. For a given function \( f(x) \), the slope of a tangent line at \( x = a \) is the value of its derivative, \( f'(a) \).

• A positive slope means the function is rising at that point.
• A negative slope means the function is falling at that point.
• A zero slope indicates a horizontal tangent line, suggesting a local maximum or minimum.

Understanding the slope of the tangent line helps characterize the behavior of a function at specific points.

Chain Rule

The chain rule is a fundamental tool in calculus for finding the derivative of composite functions. A composite function is essentially a function within another function, like a matryoshka doll. It allows us to "chain" together the derivatives of these nested functions. The chain rule is mathematically expressed as:\[\frac{d}{dx} [f(g(x))] = f'(g(x)) \cdot g'(x)\]When applying the chain rule, it's helpful to:

• Identify the outer function \( f \) and the inner function \( g \).
• Differentiate each function separately.
• Multiply the derivative of the outer function by the derivative of the inner function.

Using the chain rule simplifies finding the derivative of functions, such as \( \sqrt{2x} \), in our exercise.

Square Root Function

Square root functions are functions of the form \( f(x) = \sqrt{x} \), where the output is the square root of some input. Unlike linear functions with constant slopes, square root functions have a curve that becomes less steep as \( x \) gets larger. This means that the rate of change of the function decreases.

• The function \( f(x) = \sqrt{x} \) is only defined for \( x \geq 0 \) since the square root of a negative number is not a real number.
• Square root functions are part of the larger class of radical functions, which include cube roots and nth roots.

For example, with \( f(x) = \sqrt{2x} \), by differentiating, we use the chain rule and rewrite it as \( (2x)^{1/2} \). This helps determine the behavior, including the slope at any specific point.