Problem 23 - Calculus an Applied Approach

Question

Use the limit definition to find the slope of the tangent line to the graph of \(f\) at the given point. $$ f(x)=2 \sqrt{x} ;(4,4) $$

Step-by-Step Solution

Answer

The slope of the tangent line to the graph of function at the point (4,4) is \( \frac{1}{2} \).

Step 1: Write down the limit definition of the derivative

The derivative of a function at a certain point gives us the slope of the tangent line at that point. The limit definition of the derivative can be expressed as: \[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]. This step requires understanding of limits and derivatives.

Step 2: Substitute \(f(x)\) and \(x\) values

Substitute \(f(x)=2 \sqrt{x}\) and \(x=4\) (as the given point is (4, 4)) into the function and simplify. \[\lim_{h \to 0} \frac{2\sqrt{4+h} - 2\sqrt{4}}{h}\] This results in: \[\lim_{h \to 0} \frac{2(\sqrt{4+h} - \sqrt{4})}{h}\]

Step 3: Rationalize the numerator

Simplify the expression further by multiplying the numerator and denominator with the conjugate of the numerator to get rid of the square root in the numerator: \[\lim_{h \to 0} \frac{2(\sqrt{4+h} - \sqrt{4})(\sqrt{4+h} + \sqrt{4})}{h(\sqrt{4+h} + \sqrt{4})}\] After simplifying, we get: \[\lim_{h \to 0} \frac{2(4 + h - 4)}{h(\sqrt{4+h} + 2)}\] Further simplifying we get: \[\lim_{h \to 0} \frac{2h}{h(\sqrt{4+h} + 2)}\] After canceling h from numerator and denominator, we get: \[\lim_{h \to 0} \frac{2}{(\sqrt{4+h} + 2)}\].

Step 4: Compute the limit as \(h\) approaches \(0\)

Substitute \(h = 0\) in the expression \(\frac{2}{(\sqrt{4+h} + 2)}\) to get the slope of the tangent line. \(\frac{2}{(\sqrt{4+0} + 2)} = \frac{1}{2}\)

Key Concepts

Limit Definition of Derivative

The limit definition of a derivative is foundational in calculus. It helps us determine the slope of the tangent line to a function at a particular point. This slope, essentially, is the instantaneous rate of change of the function at that point. To find a derivative using limits, we apply the formula:

• \( f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \)

This formula requires calculating the limit of the average rate of change of the function as the interval \( h \) approaches zero. Here, \( x \) is the point at which you're finding the derivative. Begin by substituting the given function and point values into the formula. This sets the stage for simplifying and evaluating the expression further.

Rationalizing the Numerator

Rationalizing the numerator simplifies expressions involving square roots, which is crucial for finding derivatives using limits. When the expression includes \( \sqrt{x} \), simplifying can sometimes be tricky due to fractional forms. Here's how rationalization usually works:

• Multiply both the numerator and denominator by the conjugate of the numerator.
• The conjugate changes the sign between two terms (e.g., from \( \sqrt{a} - \sqrt{b} \) to \( \sqrt{a} + \sqrt{b} \)).

In the step-by-step solution, rationalizing the numerator involved the expression \(2(\sqrt{4+h} - \sqrt{4})\). By multiplying by its conjugate \((\sqrt{4+h} + \sqrt{4})\), the square roots cancel within the expression, simplifying the process and aiding in removing the zero denominator problem.

Derivative Calculation

Calculating derivatives involves simplifying expressions to reach a point where limits can be evaluated. Once the numerator is rationalized, the next step is simplification:

• After multiplication by the conjugate, expand the terms in the numerator and denominator.
• Simplify to cancel out common terms, particularly \( h \) in this instance.

Once simplified, you are often left with an expression where \( h \) is isolated or easily removable, allowing for straightforward evaluation of the limit as \( h \to 0 \). In our example, after cancelling \( h \) from both the numerator and denominator, the expression is further reduced, nearing the final calculation stages.

Function Evaluation

Evaluating the function is the concluding step in determining the derivative at a specific point. Once the expression is simplified, replacing \( h \) with 0 gives the exact slope of the tangent line.

• Substitute \( h = 0 \) into the simplified expression.
• Solve the expression to find the value of the derivative.

For instance, inserting \( h = 0 \) into \( \frac{2}{(\sqrt{4+h} + 2)} \) results in \( \frac{2}{4} = \frac{1}{2} \). This slope is the slope of the tangent line to the graph of \( f(x) = 2 \sqrt{x} \) at the point (4,4). Understanding this final step helps in interpreting how instantaneous change (or slope) relates directly to the derivative function.