How to evaluate the difference quotient?
Understand the Problem
The question is asking how to compute the difference quotient, which is a fundamental concept in calculus that involves finding the average rate of change of a function over a specific interval. The difference quotient is typically expressed as (f(x+h) - f(x)) / h.
Answer
The difference quotient for the function $f(x) = x^2$ is $2x + h$.
Answer for screen readers
The difference quotient for the function $f(x) = x^2$ is given by $2x + h$.
Steps to Solve
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Identify the Function
First, you need to know what the function $f(x)$ is for which you want to compute the difference quotient. For example, let’s say $f(x) = x^2$.
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Calculate $f(x+h)$
Now, calculate $f(x+h)$. For our function, this means substituting $x+h$ into the function:
$$ f(x+h) = (x+h)^2 $$
Expanding this gives:
$$ f(x+h) = x^2 + 2xh + h^2 $$
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Compute the Difference Quotient
Now, use the difference quotient formula $(f(x+h) - f(x)) / h$. Substitute $f(x+h)$ and $f(x)$ into the formula:
$$ \frac{f(x+h) - f(x)}{h} = \frac{(x^2 + 2xh + h^2) - x^2}{h} $$
Simplifying the numerator:
$$ \frac{2xh + h^2}{h} $$
Now divide each term by $h$ (assuming $h \neq 0$):
$$ 2x + h $$
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Final Expression
The final expression for the difference quotient is:
$$ 2x + h $$
The difference quotient for the function $f(x) = x^2$ is given by $2x + h$.
More Information
The difference quotient is a crucial part of calculus as it lays the groundwork for understanding derivatives, which measure how a function changes at a specific point. When $h$ approaches 0, this expression approaches the derivative of the function.
Tips
- Forgetting to simplify the expression before dividing by $h$ can lead to incorrect results. Always simplify $f(x+h) - f(x)$ fully before performing the division.
