Select the correct choice that completes the sentence below. The expression $\frac{f(x+h)-f(x)}{h}$ is called the $\square$ of f. A. correspondence B. image C. radicand D. difference quotient
Answer (2)
The expression h f ( x + h ) − f ( x ) represents the average rate of change of a function f over an interval and is called the difference quotient.
The expression calculates the average rate of change of f over the interval [ x , x + h ] .
This expression is fundamental in calculus for defining the derivative.
The derivative represents the instantaneous rate of change of the function.
The correct term for the expression is difference quotient .
Explanation
Understanding the Expression The expression h f ( x + h ) − f ( x ) is a fundamental concept in calculus and is used to define the derivative of a function. Let's break down what each part represents:
Components of the Expression
f ( x ) represents the value of the function f at the point x .
f ( x + h ) represents the value of the function f at the point x + h .
f ( x + h ) − f ( x ) represents the change in the value of the function as x changes to x + h .
h represents the change in the input variable x .
h f ( x + h ) − f ( x ) represents the average rate of change of the function f over the interval [ x , x + h ] .
Identifying the Correct Term This expression is known as the difference quotient, and it is used to find the derivative of a function, which represents the instantaneous rate of change of the function at a point.
Conclusion Therefore, the correct term to complete the sentence is 'difference quotient'.
Examples
In physics, the difference quotient can be used to calculate the average velocity of an object over a time interval. If f ( t ) represents the position of an object at time t , then h f ( t + h ) − f ( t ) gives the average velocity of the object over the time interval [ t , t + h ] . This concept is crucial for understanding motion and rates of change in various physical systems.
The expression h f ( x + h ) − f ( x ) is called the difference quotient. This expression is essential in calculus as it represents the average rate of change of a function and is the foundation for understanding derivatives. Therefore, the correct option to complete the sentence is 'difference quotient'.
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