Problem 48 - Calculus and its applications

Question

Find the simplified difference quotient for each function listed. $$ f(x)=a x^{3}+b x^{2} $$

Step-by-Step Solution

Answer

The simplified difference quotient is \(3ax^2 + 3axh + ah^2 + 2bx + bh\).

Step 1: Understand the Difference Quotient Formula

The difference quotient for a function \(f(x)\) is given by the formula \( \frac{f(x+h) - f(x)}{h} \), where \(h eq 0\). This formula is used to find the average rate of change of the function over the interval \([x, x+h]\).

Step 2: Determine \(f(x+h)\) for the Function

Substitute \(x + h\) into the function \(f(x) = ax^3 + bx^2\):\[f(x+h) = a(x+h)^3 + b(x+h)^2.\]

Step 3: Expand \((x+h)^3\) and \((x+h)^2\)

Use the binomial expansion to expand the terms:\[(x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3,\]\[(x+h)^2 = x^2 + 2xh + h^2.\]

Step 4: Substitute Expansions Back into \(f(x+h)\)

Replace the expansions into \(f(x+h)\):\[f(x+h) = a(x^3 + 3x^2h + 3xh^2 + h^3) + b(x^2 + 2xh + h^2).\]

Step 5: Simplify \(f(x+h)\)

Distribute the \(a\) and \(b\) across their respective terms:\[f(x+h) = ax^3 + 3ax^2h + 3axh^2 + ah^3 + bx^2 + 2bxh + bh^2.\]

Step 6: Calculate \(f(x+h) - f(x)\)

Subtract \(f(x) = ax^3 + bx^2\) from the expanded \(f(x+h)\):\[f(x+h) - f(x) = (ax^3 + 3ax^2h + 3axh^2 + ah^3 + bx^2 + 2bxh + bh^2) - (ax^3 + bx^2).\]This simplifies to:\[3ax^2h + 3axh^2 + ah^3 + 2bxh + bh^2.\]

Step 7: Form the Difference Quotient

Insert the expression for \(f(x+h) - f(x)\) into the difference quotient formula:\[\frac{3ax^2h + 3axh^2 + ah^3 + 2bxh + bh^2}{h}.\]

Step 8: Factor Out \(h\) and Simplify

Factor \(h\) out of the numerator:\[\frac{h(3ax^2 + 3axh + ah^2 + 2bx + bh)}{h}.\]Cancel the \(h\) in the numerator and denominator:\[3ax^2 + 3axh + ah^2 + 2bx + bh.\]

Step 9: Result of Simplified Difference Quotient

The simplified difference quotient is:\[3ax^2 + 3axh + ah^2 + 2bx + bh.\]

Key Concepts

Binomial Expansion

When faced with expressions like \((x+h)^3\) and \((x+h)^2\) in a polynomial function, the binomial expansion method becomes immensely useful. Binomial expansion helps to simplify these expressions by expanding them into a series of terms. This process is based on the Binomial Theorem, which states that:

• \((x+h)^n\) can be expanded as a sum involving terms of the form \(\binom{n}{k}x^{n-k}h^k\).
• For our problem specifically, with powers of 3 and 2, it breaks down into: \((x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3\) and \((x+h)^2 = x^2 + 2xh + h^2\).

Understanding this principle allows for the transformation of these expressions into forms that are more conducive to algebraic manipulation, particularly when calculating difference quotients or derivatives. Binomial expansion simplifies the complexity involved in working with powers of sums, enabling further mathematical operations with ease.

Polynomial Functions

Polynomial functions, such as the one given in our exercise \(f(x) = ax^3 + bx^2\), are algebraic expressions consisting of variables raised to whole number powers and multiplied by coefficients. These functions have various features that make them crucial in calculus and algebra. Polynomial functions are characterized by:

• The degree of the polynomial, which is the highest power of the variable. In our case, the degree is 3, indicating a cubic function.
• Coefficients like \(a\) and \(b\), which scale the impact of each term in the function.

Such functions are significant because they provide models for a variety of real-world phenomena and can describe everything from basic growth trends to complex curvature patterns in data. Understanding how to manipulate and simplify these functions, particularly using techniques such as binomial expansion and calculus, is key to exploring their properties and applications.

Rate of Change

The rate of change is a fundamental concept in calculus, aptly depicted through the difference quotient. Essentially, it measures how a quantity changes with respect to another, commonly time. The difference quotient formula \(\frac{f(x+h) - f(x)}{h}\) reveals the average rate of change of a function over a small interval \([x, x+h]\).To understand the rate of change in our function \(f(x) = ax^3 + bx^2\),:

• Calculate the function values at \(x\) and \(x+h\), then find their difference.
• Simplify the result, dividing by \(h\), and assess how the function behaves, i.e., how steeply or gradually it changes as \(x\) changes.

The rate of change leads to deeper concepts like derivatives, which provide instantaneous rates of change and form the basis of differential calculus. Mastery of these concepts is crucial for understanding dynamics in physical systems, economics, biology, and more.