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Magazine Chapter 1: Problem 37
In Exercises \(29-44,\) find the difference quotient \(\frac{f(x+h)-f(x)}{h}\) for each function. $$f(x)=x^{3}+x^{2}$$
Short Answer
Expert verified
The difference quotient is \(3x^2 + 3xh + h^2 + 2x + h\).
Step by step solution
01
Understand the function
Identify the given function. We are given the function \(f(x) = x^3 + x^2\). Our task is to find the difference quotient for this function, which is the expression \(\frac{f(x+h)-f(x)}{h}\).
02
Find \(f(x+h)\)
Substitute \(x+h\) into the function \(f(x) = x^3 + x^2\). This gives:\[f(x+h) = (x+h)^3 + (x+h)^2\]Calculate each component separately. Start with \((x+h)^3\) and \((x+h)^2\).
03
Expand \((x+h)^3\)
Apply the binomial theorem to expand \((x+h)^3\):\[(x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3\]
04
Expand \((x+h)^2\)
Apply the binomial theorem to expand \((x+h)^2\):\[(x+h)^2 = x^2 + 2xh + h^2\]
05
Write the expression for \(f(x+h)\)
Combine the expanded expressions from Steps 3 and 4:\[f(x+h) = (x^3 + 3x^2h + 3xh^2 + h^3) + (x^2 + 2xh + h^2)\]Combine like terms to get:\[f(x+h) = x^3 + x^2 + 3x^2h + 3xh^2 + h^3 + 2xh + h^2\]
06
Compute \(f(x+h) - f(x)\)
Subtract \(f(x)\) from \(f(x+h)\):\[f(x+h) - f(x) = (x^3 + 3x^2h + 3xh^2 + h^3 + x^2 + 2xh + h^2) - (x^3 + x^2)\]This simplifies to:\[f(x+h) - f(x) = 3x^2h + 3xh^2 + h^3 + 2xh + h^2\]
07
Simplify the Difference Quotient
Form the difference quotient \(\frac{f(x+h) - f(x)}{h}\):\[\frac{3x^2h + 3xh^2 + h^3 + 2xh + h^2}{h}\]Factor out \(h\) from the numerator:\[\frac{h(3x^2 + 3xh + h^2 + 2x + h)}{h}\]Cancel \(h\) in the numerator and denominator:\[3x^2 + 3xh + h^2 + 2x + h\]
08
Conclusion
The difference quotient simplifies to:\[3x^2 + 3xh + h^2 + 2x + h\].
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Binomial Theorem
The binomial theorem is a powerful tool in algebra that helps us expand expressions raised to a power. This theorem is particularly useful when dealing with expressions like
- \((a + b)^n\), where \(a\) and \(b\) are any numbers, and \(n\) is a positive integer.
- \((x + h)^3\), it's expanded into \(x^3 + 3x^2h + 3xh^2 + h^3\).
- Similarly, for \((x + h)^2\), we get \(x^2 + 2xh + h^2\).
Function Expansion
Function expansion involves expressing a function in a different form to simplify calculations and understand its behavior better. In the context of our problem, function expansion allows us to express \(f(x + h)\) by substituting \(x + h\) into the original function \(f(x) = x^3 + x^2\).We start by finding expanded forms for the terms
- \((x + h)^3\) gives us \(x^3 + 3x^2h + 3xh^2 + h^3\),
- and \((x + h)^2\) results in \(x^2 + 2xh + h^2\).
Difference Quotient Simplification
The difference quotient is a key concept in calculus used to find the derivative of a function. It is expressed as \(\frac{f(x+h) - f(x)}{h}\), where \(f(x+h)\) and \(f(x)\) represent the function values at \(x+h\) and \(x\), respectively.To simplify the difference quotient:
- We first compute \(f(x+h) - f(x)\) by subtracting the original function \(f(x)\) from our expanded \(f(x+h)\).
- The expanded form contains terms that cancel or combine in a standardized way, reducing to a simpler expression in terms of \(h\).
- The next step is to factor out \(h\) from the numerator, leaving us with an expression \(\frac{h(3x^2 + 3xh + h^2 + 2x + h)}{h}\).
- Finally, we cancel \(h\) from the numerator and denominator, yielding the simplified result: \(3x^2 + 3xh + h^2 + 2x + h\).
