How to find the difference quotient of a function?
Understand the Problem
The question is asking how to calculate the difference quotient of a function, which is typically done using the formula (f(x+h) - f(x))/h. This is a fundamental concept in calculus used to find the slope of the secant line between two points on the function's graph.
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, determine the function $f(x)$ for which you need to calculate the difference quotient. For example, let's say $f(x) = x^2$. -
Calculate $f(x+h)$
Substitute $x + h$ into the function to find $f(x+h)$. For our example:
$$ f(x+h) = (x+h)^2 = x^2 + 2xh + h^2 $$ -
Apply the difference quotient formula
Now, use the difference quotient formula:
$$ \text{Difference Quotient} = \frac{f(x+h) - f(x)}{h} $$
Plug in the expressions we have:
$$ \text{Difference Quotient} = \frac{(x^2 + 2xh + h^2) - x^2}{h} $$ -
Simplify the expression
Simplify the numerator:
$$ = \frac{2xh + h^2}{h} $$
Then, divide each term by $h$:
$$ = 2x + h $$ -
Final expression
This gives us the final result for the difference quotient:
$$ \text{Difference Quotient} = 2x + h $$
The difference quotient for the function $f(x) = x^2$ is given by $2x + h$.
More Information
The difference quotient is a key component in calculus, particularly in understanding the derivative of a function. It represents the average rate of change of the function between two points, which approximates the slope of the tangent line as $h$ approaches 0.
Tips
- Forgetting to simplify the expression after substituting into the difference quotient formula.
- Not properly substituting $x + h$ into the function.
- Confusing the difference quotient with the derivative formula.
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