Problem 77 - Brief Applied Calculus

Question

a. Find the derivative (by hand) of each function below, and observe that the derivative is undefined at \(x=0\). b. Find the derivative of each function below by using NDERIV on a graphing calculator and evaluate the derivative at \(x=0 .\) If your calculator gives you an answer, this is a "false value" for the derivative, since in part (a) you showed that the derivative is undefined at \(x=0\). [For an explanation, see the Graphing Calculator Exploration part (d) on page \(130 .\). $$ y=\frac{1}{x^{2}} $$

Step-by-Step Solution

Answer

The derivative \( \frac{dy}{dx} = -\frac{2}{x^3} \) is undefined at \( x=0 \).

Step 1: Understand the Function

The given function is \( y = \frac{1}{x^2} \). This function is expressed in terms of \( x \), and our goal is to find its derivative with respect to \( x \).

Step 2: Apply the Power Rule for Derivatives

The function can be rewritten using negative exponents as \( y = x^{-2} \). The power rule for derivatives states that \( \frac{d}{dx} x^n = nx^{n-1} \) for any real number \( n \).

Step 3: Differentiate the Function

Using the power rule, differentiate \( y = x^{-2} \):\[ \frac{dy}{dx} = -2x^{-3} = -\frac{2}{x^3} \].

Step 4: Check for Undefined Values at x=0

The expression for the derivative is \( -\frac{2}{x^3} \). Substituting \( x = 0 \) into this expression results in division by zero, which means the derivative is undefined at \( x = 0 \).

Step 5: Use a Graphing Calculator (NDERIV)

Use your graphing calculator's derivative function (NDERIV) to find the derivative at \( x = 0 \). Although it may provide an output, recognize that per analytical derivation, the derivative at \( x = 0 \) is undefined due to division by zero.

Key Concepts

Power Rule

The Power Rule is a fundamental concept in calculus that helps us easily find the derivative of functions of the form \( y = x^n \). By applying this rule, we can determine how a function changes as its input changes. In mathematical terms, the Power Rule states:

• If \( y = x^n \), then the derivative \( \frac{dy}{dx} = nx^{n-1} \).

This means we multiply the exponent by the base and decrease the power by one. However, understanding the Power Rule goes beyond just memorizing the formula. It's crucial to internalize why it works:

• Considered as a shortcut derived from first principles of calculus (limits and infinitesimals).
• Affects the way we tackle real-world problems by predicting the rate of change.

In our exercise, we applied the Power Rule to \( y = \frac{1}{x^2} \) by first rewriting it with a negative exponent: \( y = x^{-2} \). Using the rule, this becomes \( \frac{dy}{dx} = -2x^{-3} \). Remember, practice makes perfect when mastering this skill!
Try applying the Power Rule to different functions to become comfortable with derivatives.

Undefined Derivative

An undefined derivative occurs when a function's derivative does not exist at a certain point. In simple terms, this situation arises when something prevents the computation of the derivative. Common causes include:

• Division by zero in the derivative expression.
• Sharp corners or cusps in the graph of the function.
• Discontinuities or jumps in the function.
  In our exercise, the function \( y = \frac{1}{x^2} \) leads to a division by zero when calculating the derivative at \( x=0 \). The derivative expression \( -\frac{2}{x^3} \) becomes undefined because of the zero in the denominator. This tells us that the rate of change at \( x=0 \) cannot be computed. Recognizing these scenarios is crucial because it helps identify limits and boundaries within which mathematical models operate. Always check if expressions involve division or other operations that could be undefined at certain points.

Graphing Calculator

Graphing calculators are powerful tools that assist in visualizing and solving mathematical problems. They offer features like plotting functions, finding numerical derivatives, and integrals, which are incredibly helpful in calculus. However, when using a graphing calculator to evaluate derivatives, especially at points where they might be undefined, it's essential to stay cautious:

• Calculators use numerical approximations, often based on algorithms that can produce 'false values'.
• At \( x=0 \), the calculator might output a result for \( y = \frac{1}{x^2} \) due to its numerical nature, but this contradicts mathematical analysis.

While calculators extend our ability to compute quickly and handle complex equations, it's vital to understand the underlying math principles. This knowledge ensures that the results are interpreted correctly, especially when dealing with undefined behavior. Practice using both analytical strategies and technological tools to enhance accuracy in derivations.