Problem 36 - Calculus of a Single Variable

Question

In Exercises 31–38, find the slope of the graph of the function at the given point. Use the derivative feature of a graphing utility to confirm your results. $$ f(x)=2(x-4)^{2} \quad(2,8) $$

Step-by-Step Solution

Answer

The slope of the graph of the function \( f(x) = 2(x-4)^{2} \) at the point (2,8) is -8.

Step 1: Find the derivative

The first step is to find the derivative of \( f(x)=2(x-4)^{2} \). This is done by applying the power rule, which states that the derivative of \( x^{n} \) is \( n*x^{n-1} \). In this case, the derivative of \( (x-4)^{2} \) is \( 2*(x-4) \). So the derivative of \( f(x)=2(x-4)^{2} \) is \( f'(x) = 2*2*(x-4) = 4*(x-4) \).

Step 2: Substitute x into the derivative

Now, substitute the x-value of the given point (2,8) into the derived function to find the slope. Therefore, \( f'(2) = 4*(2-4) = 4*(-2) = -8 \). So, the slope of the function \( f(x) = 2(x-4)^{2} \) at the point (2,8) is -8.

Step 3: Confirm with a graphing utility

Use a graphing calculator to confirm the result. Plot the function \( f(x) = 2(x-4)^{2} \) and the point (2,8). The line tangent to the curve at the point (2,8) should have a slope of -8, confirming the previous calculation.

Key Concepts

Slope

The slope of a function's graph at a particular point tells you how steep the line is at that point. It represents how much the function's value changes for a small change in the input. More simply, the slope can describe the rate of change.

• If the slope is positive, the function is increasing.
• If the slope is negative, the function is decreasing.
• A slope of zero means the function is flat at that point.

To find the slope at a specific point on a curve, we use the derivative. The derivative can be thought of as a tool to give us the slope at any point on the graph of a function. It is an essential concept in calculus used to understand how functions behave. In our exercise, the slope we found was -8, indicating the function is decreasing at point (2,8).

Power Rule

The power rule is a key tool in calculus for finding derivatives. It provides a straightforward way to differentiate functions of the form \(x^n\). According to the power rule, if you have a function \(x^n\), its derivative is \(n \cdot x^{n-1}\). This makes calculating derivatives much simpler because you just bring the exponent down in front and lower the exponent by one.In the given exercise, the function was \((x-4)^2\). Applying the power rule:- Since we have an expression \((x-4)^2\), we see it's similar to \(x^2\) and bring down the 2 as a coefficient.- This turns into \(2 \cdot (x-4)\).Thus, applying the power rule makes finding derivatives faster and more systematic, especially useful for polynomials.

Tangent Line

A tangent line is a line that touches a curve at exactly one point without crossing over. It represents the instantaneous rate of change of the curve at that point. The equation of a tangent line at any given point can be found once we have the slope (derivative) and the specific point on the curve.To write the equation of the tangent line:- Use the point-slope form of a line: \(y - y_1 = m(x - x_1)\), where \(m\) is the slope and \((x_1, y_1)\) is the point.For our exercise:- The slope \(m\) is -8.- The point is (2,8).Inserting these into the formula, the equation of the tangent line is \(y - 8 = -8(x - 2)\). This line barely "skims" the curve at our desired point and mirrors the local slope.

Graphing Utility

A graphing utility is a powerful tool that helps visualize mathematical functions and their characteristics. These utilities range from graphing calculators to software programs like Desmos or GeoGebra.Using a graphing utility can confirm theoretical results and provides a visual intuition about the behavior of functions. For instance, in our exercise:- We plotted the function \(f(x) = 2(x-4)^2\) to see the curve.- By checking the point (2,8), we verified where the point lies on the curve.- Then, we checked the tangent line's slope, ensuring it matched the calculated slope of -8.A graphing utility proves invaluable when verifying analytical results and helps confirm that our derivative calculations and intuitive understanding align with the graphical representation.