Problem 79 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Prove the power rule (Rule 2) for the special case \(n=3\). Hint: Compute \(\lim _{h \rightarrow 0}\left[\frac{(x+h)^{3}-x^{3}}{h}\right]\).

Step-by-Step Solution

Answer

We are given the expression \(\frac{(x+h)^3-x^3}{h}\) and need to find its limit as \(h\) approaches 0. Simplifying the expression, we first expand \((x+h)^3\) to get \(x^3 + 3x^2h + 3xh^2 + h^3\). Then, we substitute the expanded term into the expression and simplify, resulting in \((3x^2 + 3xh + h^2)\). Finally, we find the limit as \(h\) approaches 0, which is \(3x^2\). This result proves the power rule for the special case when \(n=3\).

Step 1: Expand the expression #(x+h)^3#

To find the limit of the given expression, we first need to expand the term (x+h)^3 using the binomial theorem or by simply expanding the expression. Expanding the expression, we get: (x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3

Step 2: Substitute the expanded term into the expression

Now that we have expanded the term (x+h)^3, we can substitute it back into the given expression: \(\frac{(x+h)^3-x^3}{h}=\frac{(x^3 + 3x^2h + 3xh^2 + h^3)- x^3}{h}\)

Step 3: Simplify the expression

We can now simplify the expression by cancellation: \(\frac{(x^3 + 3x^2h + 3xh^2 + h^3)- x^3}{h} = \frac{3x^2h + 3xh^2 + h^3}{h}\) We can factor out an h from the expression: \(\frac{3x^2h + 3xh^2 + h^3}{h} = h(3x^2 + 3xh + h^2)\) Now, we can cancel h from the numerator and the denominator: h(3x^2 + 3xh + h^2) / h = (3x^2 + 3xh + h^2)

Step 4: Find the limit as h approaches 0

We have simplified the expression, and now we can find the limit as h approaches 0: \(\lim_{h \rightarrow 0} (3x^2 + 3xh + h^2)\) As h approaches 0, the terms containing h will approach 0, so the limit will be: \(\lim_{h \rightarrow 0} (3x^2 + 3xh + h^2) = 3x^2\) This result shows that the derivative of the function \(x^3\) with respect to x is \(3x^2\), proving the power rule for the special case when n=3.

Key Concepts

Limit Calculations

Limit calculations help us understand the behavior of a function as its input approaches a certain value. In our case, we're looking at what happens to \( \lim_{h \rightarrow 0} \left[\frac{(x+h)^{3}-x^{3}}{h}\right] \). To solve this, we need to simplify the expression first.
We do this by expanding \((x+h)^3\) and simplifying the resulting terms. Limits are crucial in calculus because they allow us to find derivatives. For the special case when \(n=3\), calculating this limit proves the power rule.
It shows that despite small changes in \(h\), the function behaves predictably, allowing us to understand its slope at any point. The calculation involves finding the expression within the limit and carefully reducing it as \(h\) becomes insignificant, letting us conclude with the function's derivative. Understanding limit calculations is key to grasping derivative concepts.

Binomial Expansion

Binomial expansion is a technique used to expand expressions raised to a power. It's especially helpful when dealing with powers like \((x+h)^3\), as in our original exercise. By using the binomial theorem, we can easily break down the expression.

• Start with \((x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3 \).
• This shows how each term in the expansion comes together.
• The coefficients in the expression derive from binomial coefficients.

By expanding first, we can substitute back into the expression we need to evaluate. This simplifies limit calculations and makes it easier to see how each part contributes to finding the derivative.
Understanding binomial expansion allows us to manage complex algebraic manipulations with ease.

Derivative Proof

A derivative proof is a method of showing that a particular derivative formula is true. In our case, we're proving the power rule for \( n=3 \), which states that the derivative of \( x^n \) is \( nx^{n-1} \). Using our limit result,

• we confirm that \( \lim_{h \rightarrow 0} \frac{(x+h)^3 - x^3}{h} = 3x^2 \).
• This mathematical process relies on simplification and careful limit evaluation.
• By reaching \(3x^2\), we verify that the power rule holds for this specific power of \(3\).

Derivative proofs require step-by-step logic that links limit calculations and algebraic simplification directly to the core rules of calculus. Mastering these proofs enhances comprehension of why calculus works the way it does.

Special Case Analysis

Special case analysis involves breaking down a problem with specific parameters to understand broader principles. Here, we focus on \( n=3 \) as our special case. By handling this specific instance, we explore the nuances of the power rule in action.

• Analyzing for \(n=3\) shows how \(3x^2\) emerges from our limit calculation.
• This direct application demonstrates the principle behind the derivative of \(x^n\).
• Understanding special cases helps to reinforce and validate general mathematical rules.

By isolating and solving these particular scenarios, students can see broader patterns.
Engaging with special cases builds a foundational understanding, which supports tackling more complex, varied functions. Proving a rule for specific situations like this solidifies confidence in applying conceptual knowledge across different mathematical problems.