Problem 27 - Calculus

Question

For \(f(x)=x^{2}+x^{3},\) write out \(f(x+\Delta x)\) and \(\Delta f / \Delta x .\) What is the limit at \(\Delta x=0\) and what rule about sums is confirmed?

Step-by-Step Solution

Answer

Limit is \(2x + 3x^2\), confirming the sum rule for differentiation.

Step 1: Substitute into f(x)

We need to calculate \(f(x + \Delta x)\). Start by substituting \(x + \Delta x\) into the function \(f(x) = x^2 + x^3\). So,\[f(x + \Delta x) = (x + \Delta x)^2 + (x + \Delta x)^3.\]

Step 2: Expand the Square Term

Expand the first term \((x + \Delta x)^2\):\[(x + \Delta x)^2 = x^2 + 2x\Delta x + (\Delta x)^2.\]

Step 3: Expand the Cube Term

Now expand the second term \((x + \Delta x)^3\):\[x^3 + 3x^2\Delta x + 3x(\Delta x)^2 + (\Delta x)^3.\]

Step 4: Combine the Expanded Terms

Combine the results of Step 2 and Step 3:\[f(x + \Delta x) = x^2 + 2x \Delta x + (\Delta x)^2 + x^3 + 3x^2\Delta x + 3x(\Delta x)^2 + (\Delta x)^3.\]

Step 5: Calculate \(\Delta f\)

Find \(\Delta f\) by subtracting \(f(x)\) from \(f(x + \Delta x)\):\[\Delta f = (f(x + \Delta x) - f(x)) = (x^2 + 2x\Delta x + (\Delta x)^2 + x^3 + 3x^2\Delta x + 3x(\Delta x)^2 + (\Delta x)^3) - (x^2 + x^3).\]

Step 6: Simplify \(\Delta f\)

Collect like terms after subtraction:\[\Delta f = (2x\Delta x + (\Delta x)^2 + 3x^2\Delta x + 3x(\Delta x)^2 + (\Delta x)^3).\]

Step 7: Find \(\frac{\Delta f}{\Delta x}\)

Divide \(\Delta f\) by \(\Delta x\) to find \(\frac{\Delta f}{\Delta x}\):\[\frac{\Delta f}{\Delta x} = \frac{(2x\Delta x + (\Delta x)^2 + 3x^2\Delta x + 3x(\Delta x)^2 + (\Delta x)^3)}{\Delta x}.\] Simplify it:\[= 2x + \Delta x + 3x^2 + 3x\Delta x + (\Delta x)^2.\]

Step 8: Take the Limit \(\Delta x \to 0\)

Taking the limit as \(\Delta x\) approaches 0:\[\lim_{\Delta x \to 0} (2x + \Delta x + 3x^2 + 3x\Delta x + (\Delta x)^2) = 2x + 3x^2.\]

Step 9: Confirm the Rule for Sums

Notice that we separated \(f(x) = x^2 + x^3\) into parts. Each differentiated term added up to the derivative of \(x^2 + x^3\). This confirms that the derivative of the sum is the sum of the derivatives.

Key Concepts

Derivative of a polynomial

A polynomial is a mathematical expression consisting of variables and coefficients, involving only addition, subtraction, multiplication, and non-negative integer exponents of variables. To find the derivative of a polynomial, we apply the differentiation rules to each term separately.
For instance, consider the function \( f(x) = x^2 + x^3 \). The derivative of this function is the sum of the derivatives of each term: \( x^2 \) and \( x^3 \).
The power rule of differentiation tells us that \( \frac{d}{dx}[x^n] = nx^{n-1} \).
This rule simplifies the process of finding the derivative of each term.

• For \( x^2 \), the derivative is \( 2x \).
• For \( x^3 \), the derivative is \( 3x^2 \).

The comprehensive result shows that the derivative of \( f(x) = x^2 + x^3 \) is \( 2x + 3x^2 \). Differentiating polynomials term-by-term is straightforward and forms the basis for solving more complex problems involving polynomials.

Limit definition of derivative

The limit definition of a derivative is a fundamental concept in calculus. It provides a precise way to compute the rate of change of a function. The limit definition of the derivative of a function \( f(x) \) at a point \( x \) is given by:\[\lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}.\]
This expression describes how the function \( f(x) \) changes as \( x \) changes by a small increment \( \Delta x \).
By applying it to our function \( f(x) = x^2 + x^3 \), we get a formula to find its instantaneous rate of change at any point.
We calculate \( f(x + \Delta x) \) by expanding and simplifying the polynomial, and then find \( \Delta f \), the change in \( f \), by subtracting \( f(x) \).
Dividing \( \Delta f \) by \( \Delta x \) gives us \( \frac{\Delta f}{\Delta x} \).
Finally, by taking the limit as \( \Delta x \to 0 \), we eliminate the terms involving \( \Delta x \), leaving \( 2x + 3x^2 \) as the derivative.

Sum rule for derivatives

When dealing with derivatives, one of the convenient rules is the sum rule. It states that the derivative of the sum of two functions is the sum of their derivatives.
This rule simplifies the differentiation process, especially when working with polynomials. In the example \( f(x) = x^2 + x^3 \), we apply the sum rule to find the derivative efficiently.
The sum rule can be mathematically expressed as:\[\frac{d}{dx}[u(x) + v(x)] = \frac{d}{dx}[u(x)] + \frac{d}{dx}[v(x)].\]
In practice, this means:

• If \( u(x) = x^2 \) and \( v(x) = x^3 \), then \( \frac{d}{dx}[x^2] = 2x \) and \( \frac{d}{dx}[x^3] = 3x^2 \).
• Thus, the overall derivative is \( 2x + 3x^2 \).

The sum rule simplifies our calculations by allowing us to differentiate each term separately, then adding the results to get the final derivative of the entire polynomial.