How to find the difference quotient?
Understand the Problem
The question is asking for the method to calculate the difference quotient, which is a concept used in calculus to find the average rate of change of a function over an interval. The difference quotient is defined as (f(x + h) - f(x)) / h, where f is a function, x is a point in the domain, and h is a small increment. To solve it, we will replace f with a specific function and simplify the expression based on this definition.
Answer
The final result depends on the function used.
Answer for screen readers
The final result after simplification will vary based on the specific function you are using. The steps provided form a general method to find the difference quotient.
Steps to Solve
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Substitute the increment variable into the function
Replace $x$ with $(x + h)$ in the function $f$. This gives you $f(x + h)$.
- Plug the values into the difference quotient formula
Use the definition of the difference quotient: $$ \frac{f(x + h) - f(x)}{h} $$
- Simplify the expression
Perform the necessary algebraic operations to simplify the expression by combining like terms or factoring if necessary. This will give you the simplified form of the difference quotient.
The final result after simplification will vary based on the specific function you are using. The steps provided form a general method to find the difference quotient.
More Information
The difference quotient is the foundation for finding the derivative of a function at a point. In essence, it measures the average rate of change over an interval and forms the basis of more advanced calculus techniques.
Tips
A common mistake is neglecting to simplify the numerator $f(x + h) - f(x)$ correctly. Ensure that all terms are combined appropriately, and always check your algebra.
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