How to evaluate difference quotient?
Understand the Problem
The question is asking how to evaluate the difference quotient, which is a concept in calculus used to find the slope of a secant line between two points on a function. To solve this, we will define the difference quotient and outline the steps to compute it, typically by using the formula (f(x+h) - f(x)) / h as h approaches 0.
Answer
The difference quotient evaluates to the derivative $f'(x)$ as $h$ approaches 0.
Answer for screen readers
The final answer will provide the value of the slope of the secant line as $h$ approaches 0, which is the derivative of the function at point $x$, represented as $f'(x)$.
Steps to Solve
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Define the Function and Points Identify the function $f(x)$ for which you want to evaluate the difference quotient and choose a specific point $x$.
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Apply the Difference Quotient Formula Use the difference quotient formula, which is given by
$$ \frac{f(x+h) - f(x)}{h} $$
Here, $h$ represents a small change in $x$.
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Calculate $f(x+h)$ and $f(x)$ Substitute the point $x$ and the increment $h$ into the function to compute values for $f(x+h)$ and $f(x)$.
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Plug into the Formula Insert the values of $f(x+h)$ and $f(x)$ into the difference quotient formula.
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Simplify the Expression Simplify the resulting expression from step 4 to eliminate $h$ from the denominator, if possible.
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Evaluate the Limit as $h$ Approaches 0 Finally, find the limit of the simplified expression as $h$ approaches 0 to determine the slope of the secant line.
The final answer will provide the value of the slope of the secant line as $h$ approaches 0, which is the derivative of the function at point $x$, represented as $f'(x)$.
More Information
The difference quotient is essential in calculus as it leads to the derivative, representing the instantaneous rate of change of the function at a particular point. Understanding how to evaluate it is fundamental for further studies in calculus.
Tips
- Forgetting to simplify the expression properly, which can lead to not finding the derivative correctly.
- Confusing the difference quotient with the average rate of change; be sure to clarify the distinction.
- Not applying the limit correctly as $h$ approaches 0, which is a crucial step in finding the derivative.
