Problem 25 - Calculus

Question

Compute \((x+h)^{3}-(x-h)^{3},\) divide by \(2 h,\) and set \(h=0\). Why divide by \(2 h\) to find this slope?

Step-by-Step Solution

Answer

The slope is 3x^2, found by dividing by 2h to get the average rate of change.

Step 1: Expand the first expression

Start by expanding \( (x+h)^3\) using the binomial theorem. This gives us: \[(x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3\]

Step 2: Expand the second expression

Next, expand \( (x-h)^3\) using the binomial theorem: \[(x-h)^3 = x^3 - 3x^2h + 3xh^2 - h^3\]

Step 3: Calculate the difference

Subtract the expansion of \( (x-h)^3\) from \( (x+h)^3\): \[(x+h)^3 - (x-h)^3 = (x^3 + 3x^2h + 3xh^2 + h^3) - (x^3 - 3x^2h + 3xh^2 - h^3)\] Simplifying, we get: \[= 6x^2h + 2h^3\]

Step 4: Divide by 2h

Divide the expression by \(2h\): \[\frac{6x^2h + 2h^3}{2h}\] Simplifying by canceling out \(h\) gives us: \[= 3x^2 + h^2\]

Step 5: Set h = 0

Substitute \(h = 0\) into the simplified expression: \[3x^2 + 0^2 = 3x^2\]

Step 6: Reason for dividing by 2h

Dividing by \(2h\) is necessary to normalize the difference quotient, as it represents the average rate of change or the slope of the secant line between the two points \(x+h\) and \(x-h\). At \(h = 0\), this gives the instantaneous rate of change, or the derivative.

Key Concepts

Binomial Theorem

The Binomial Theorem is a way to expand expressions that are raised to a power, like \((x+h)^3\). It is a useful formula in mathematics, especially when dealing with polynomials. For any positive integer \(n\), the binomial theorem states that:

• \((x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\)

Each term in the expansion is a combination of the coefficients \(\binom{n}{k}\) and the variables raised to appropriate powers.
In our example, we expanded \((x+h)^3\) and \((x-h)^3\) using this theorem. Notice how the terms for addition and subtraction are symmetric, which helps in simplifying when taking differences.

Average Rate of Change

The average rate of change represents how a quantity changes on average over a certain interval. Calculating this involves the expression of the form \(\frac{f(a+h)-f(a-h)}{2h}\).
This structure provides a meaningful measure of the rate of change around a point because it takes into account small increments both in the positive \((x+h)\) and negative directions \((x-h)\).

• For example, in an interval \([-h, h]\), the average rate of change tells us the average slope of the function over this span.
• It represents the slope of a secant line intersecting the curve at two points.

Computing the average rate of change is a preliminary step in finding derivatives, which measure instantaneous rates of change.

Secant Line

A secant line is a straight line connecting two points on the graph of a function. It allows us to approximate the slope between those two points, offering insight into the function's behavior.
In calculus, the difference quotient, which involves the difference of values at two points \((x+h)\) and \((x-h)\), divided by \(2h\), gives us the slope of this secant line.

• The formula for the slope of the secant line is \(\frac{f(x+h)-f(x-h)}{2h}\).
• If you minimize \(h\) (i.e., let \(h\to0\)), the secant line becomes tangent to the function at that point.

This transition from secant to tangent connects closely with the concept of the derivative, where \(h\) approaches zero.

Derivative

The derivative is a fundamental concept in calculus, symbolizing the instantaneous rate of change of a function at a given point. It is derived by taking the limit of the difference quotient as \(h\) approaches zero:

• \(\lim_{h \to 0} \frac{f(x+h)-f(x-h)}{2h}\)

This process captures the precise slope of the tangent line at a specific point, rather than the slope over an interval, as with the secant line.
Derivatives are useful in a wide range of applications, from understanding motion to optimizing functions in economics.
In summary, when you set \(h=0\) in the exercise, you transition from an average rate of change (secant line) to the instantaneous rate (derivative), which is more sensitive and precise.