The expression $\frac{f(x+h)-f(x)}{h}, h \neq 0$, is called the $\square$ of the function f. We find this expression by replacing x with $\square$ each time x appears in the function's equation. Then we subtract $\square$. After simplifying, we factor $\square$ from the numerator and divide out identical factors of $\square$ in the numerator and denominator.
Answer (2)
The expression h f ( x + h ) − f ( x ) is called the difference quotient .
We replace x with x + h in the function's equation.
Then we subtract f ( x ) .
After simplifying, we factor h from the numerator.
The expression h f ( x + h ) − f ( x ) , h = 0 , is called the difference quotient of the function f . We find this expression by replacing x with x + h each time x appears in the function's equation. Then we subtract f ( x ) . After simplifying, we factor h from the numerator and divide out identical factors of h in the numerator and denominator.
Explanation
Understanding the Difference Quotient The expression h f ( x + h ) − f ( x ) , h = 0 , is a fundamental concept in calculus. It represents the average rate of change of the function f over an interval of length h . As h approaches 0, this expression becomes the derivative of the function f . Let's break down the process of finding and simplifying this expression.
Filling in the Blanks The expression h f ( x + h ) − f ( x ) is called the difference quotient of the function f . We find this expression by replacing x with x + h each time x appears in the function's equation. Then we subtract f ( x ) . After simplifying, we factor h from the numerator and divide out identical factors of h in the numerator and denominator.
Completed Statement Therefore, the completed statement is: The expression h f ( x + h ) − f ( x ) , h = 0 , is called the difference quotient of the function f . We find this expression by replacing x with x + h each time x appears in the function's equation. Then we subtract f ( x ) . After simplifying, we factor h from the numerator and divide out identical factors of h in the numerator and denominator.
Importance of the Difference Quotient The expression h f ( x + h ) − f ( x ) is a crucial concept because it leads to the definition of the derivative, which measures the instantaneous rate of change of a function. Understanding how to manipulate and simplify this expression is essential for mastering calculus.
Examples
In physics, the difference quotient can be used to calculate the average velocity of an object over a time interval. If f ( x ) represents the position of an object at time x , then h f ( x + h ) − f ( x ) gives the average velocity of the object between times x and x + h . As h approaches 0, this average velocity approaches the instantaneous velocity at time x , which is the derivative of the position function.
The expression h f ( x + h ) − f ( x ) is known as the difference quotient of the function f . It helps find the average rate of change of the function over an interval and leads to the concept of the derivative as h approaches 0. Understanding this process is essential for studying calculus.
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