Problem 52 - Precalculus with Limits

Question

In Exercises 51-54, use a graphing utility to graph \(f\) over the interval \([-2, 2]\) and complete the table. Compare the value of the first derivative with a visual approximation of the slope of the graph. $$ f(x) = \frac{1}{4}x^3 $$

Step-by-Step Solution

Answer

The first derivative of \(f(x) = \frac{1}{4}x^3 \) is \( f'(x) = \frac{3}{4}x^2 \). So, at any point on the graph of \(f(x)\), the slope should match \(f'(x)\) when \(x\) is the x-coordinate of that point.

Step 1: Define the function

Define the given function. Here, the function is \(f(x) = \frac{1}{4}x^3 .\)

Step 2: Calculate the derivative

Calculate the derivative of the function using the power rule. The first derivative of \(f(x) = \frac{1}{4}x^3 \) is \( f'(x) = \frac{3}{4}x^2 .\) This gives the rate of change or slope at any point \(x\) of the graph of the function.

Step 3: Plot the function and observe the slope

Use a graphing tool to plot \(f(x) = \frac{1}{4}x^3 .\) Observe the slope at points on the graph and verify that the approximated slopes align with the values calculated using the derivative. The slope of the curve at a particular point is equal to the value of the derivative at that point. So comparing the value of the first derivative with a visual approximation of the slope will help to verify the correctness of the derivative.

Step 4: Compare the derivative with the graph slope

Compare the calculated derivative with the slope of the graph at various intervals within the range \([-2, 2]\). Ensure they match to finalize the exercise.

Key Concepts

Power Rule Differentiation

Understanding the power rule for differentiation is fundamental in calculus. It simply states that if you have a function of the form \(f(x) = ax^n\), where \(a\) and \(n\) are constants, the derivative of this function with respect to \(x\) is \(f'(x) = nax^{n-1}\).

Applying this rule makes calculating derivatives for polynomial functions straightforward. For example, if we have the function \(f(x) = \frac{1}{4}x^3\), we can find its derivative by bringing down the exponent and multiplying it by the coefficient, then decreasing the exponent by one. So, the derivative is \(f'(x) = \frac{3}{4}x^2\). This derivative function tells us how the slope of the original function changes at any given point on its curve.

First Derivative Test

The first derivative test is a useful tool when analyzing the graph of a function, especially for identifying local maximums and minimums. After finding the first derivative of the function, set it equal to zero and solve for \(x\) to find critical points.

Interpreting the First Derivative Test

By testing intervals around those critical points, you can determine whether the function is increasing or decreasing within those intervals. If the function changes from increasing to decreasing at a critical point, you have a local maximum. Conversely, if it changes from decreasing to increasing, that indicates a local minimum. This test gives us a deeper understanding of the function's behavior on a graph.

Slope of a Curve

In calculus, the slope of a curve at any given point is the value of its derivative at that point. This concept is what connects algebra to calculus; where in algebra we discuss the slope of a straight line, calculus allows us to find the slope of curves, which can be variable and not constant.

Visual Intuition of Slope

Graphically, the slope tells us how steep the curve is at a particular point. A larger slope means a steeper curve, while a slope of zero indicates a horizontal tangent, which could be at a peak, trough, or inflection point. Understanding the slope of a curve is crucial, as it provides insight into the rate of change of the function at any point.

Graphing Polynomial Functions

Graphing polynomial functions provides a visual representation of the behavior and characteristics of the function. With graphing tools or software, we can plot the function over a specific interval to examine key features such as intercepts, turning points, and end behavior.

Steps for Graphing

• Identify the polynomial's degree to predict its general shape.
• Calculate the function's roots or x-intercepts by setting the function equal to zero and solving for \(x\).
• Analyze the leading coefficient to determine the end behavior of the graph.
• Use the first and second derivatives for finding maxima, minima, and concavity.

Regular plotting alongside the analytical process validates the algebraic findings and develops a stronger intuition regarding the functions' behavior.