Problem 48 - Calculus

Question

The slope of \(y=x^{3}\) comes from this identity: $$\frac{(x+h)^{3}-x^{3}}{h}=(x+h)^{2}+(x+h) x+x^{2}.$$ (a) Check the algebra. Find \(d y / d x\) as \(h \rightarrow 0\). (b) Write a similar identity for \(y=x^{4}\).

Step-by-Step Solution

Answer

(a) Algebra is verified; \( \frac{dy}{dx} = 3x^2 \). (b) Similar identity for \( y = x^4: \frac{(x+h)^4 - x^4}{h} = 4x^3 + 6x^2h + 4xh^2 + h^3 \).

Step 1: Expand the Cubic Expression

To check the given identity, let's expand \((x+h)^3\) step by step. We have \((x+h)^3 = (x+h)(x+h)(x+h)\). First, expand \((x+h)(x+h) = x^2 + 2xh + h^2\). Then multiply this result by \((x+h)\) again to get: \((x^2 + 2xh + h^2)(x+h) = x^3 + 3x^2h + 3xh^2 + h^3\).

Step 2: Simplify the Fraction

Substitute \((x+h)^3 - x^3\) with the expanded form found in Step 1 into the left side of the given identity: \[\frac{(x^3 + 3x^2h + 3xh^2 + h^3) - x^3}{h}\]. On simplifying, this becomes \[\frac{3x^2h + 3xh^2 + h^3}{h}\].

Step 3: Cancel the Common Factor

Cancel the factor of \(h\) from the numerator and the denominator: \[\frac{3x^2h + 3xh^2 + h^3}{h} = 3x^2 + 3xh + h^2\]. This shows the algebra is correct and matches the right-hand side of the given identity \((x+h)^2 + (x+h)x + x^2\).

Step 4: Find the Derivative as h Approaches Zero

Find the derivative \(\frac{dy}{dx}\) by taking the limit of \(3x^2 + 3xh + h^2\) as \(h \to 0\). This gives: \[\lim_{h \to 0} (3x^2 + 3xh + h^2) = 3x^2\].

Step 5: Generalize for y=x^4

Write an identity for \(y = x^4\) starting from the difference quotient: \[\frac{(x+h)^4 - x^4}{h}\]. Expand \((x+h)^4 = (x+h)^2(x+h)^2\), and simplify like previous steps to get the associated identity.

Key Concepts

Difference Quotient

The difference quotient is a fundamental concept used in calculus to represent the slope of the tangent line to the curve at any given point. It is a way of finding the average rate of change of the function, which is valuable when calculating derivatives. For a function \( y = f(x) \), the difference quotient is written as:

• \( \frac{f(x+h) - f(x)}{h} \).

This formula gives the change in \( y \) divided by the change in \( x \), representing the slope of the secant line connecting the points \( x \) and \( x+h \). In the original exercise, this concept was used to find the slope of the function \( y = x^3 \) by expanding \((x+h)^3\) and subtracting \(x^3\), which results in a polynomial that includes \( h \) in the denominator. As \( h \) approaches zero, this quotient can be used to find the derivative.

Limit Process

The limit process is essential in calculus as it allows us to find the instantaneous rate of change, or the derivative, by analyzing the behavior of a function as a particular input approaches a specific value. In our context, to determine the derivative \( \frac{dy}{dx} \) of \( y = x^3 \), we apply the limit:

• \( \lim_{h \to 0} \left( 3x^2 + 3xh + h^2 \right) \).

As \( h \) approaches zero, terms including \( h \) vanish, simplifying our expression to \( 3x^2 \). This step is crucial in confirming the function’s behavior at an infinitesimally small scale and gives us the slope of the tangent line. Understanding limits helps in localizing change at a very precise point, thus leading to a deeper grasp of how derivatives function.

Polynomial Differentiation

Polynomial differentiation refers to the process of finding the derivative of polynomial functions. These functions are expressed in the form of a polynomial, such as \( ax^n + bx^{n-1} + \ldots + c \). The key rule for differentiating any term \( ax^n \), where \( n \) is a power and \( a \) is a coefficient, is given by bringing the power down and reducing it by one:

• \( \frac{d}{dx}(ax^n) = anx^{n-1} \).

In the exercise, we differentiated \( y = x^3 \) using polynomial rules, while applying limits, resulting directly in the derivative \( 3x^2 \). Each term in a polynomial can be differentiated separately, which makes the process straightforward once you understand the foundational rules.

Algebraic Expansion

Algebraic expansion is a technique used to open up expressions written in a compact or factored form by applying the distributive property. For example, expanding \((x+h)^3\) involves multiplying it out to obtain each term separately:

• \((x+h)^3 = (x+h)(x+h)(x+h)\).
• Expand step by step to reach: \( x^3 + 3x^2h + 3xh^2 + h^3 \).

This is crucial in the exercise, as expanding polynomials allows us to simplify and compare expressions, particularly when forming difference quotients. Understanding how to perform algebraic expansions is invaluable in solving complex equations and finding derivatives. This skill is broadly applicable across various areas of mathematics and serves as a cornerstone for mastering calculus.