Simplify the difference quotient ƒ(x+h)-ƒ(x)/h
ƒ(x) = 10

Verified step by step guidance

1. Step 1: Understand the function given, \( f(x) = 10 \). This is a constant function, meaning it does not change with different values of \( x \).
2. Step 2: Substitute \( f(x) \) and \( f(x+h) \) into the difference quotient formula. Since \( f(x) = 10 \), we have \( f(x+h) = 10 \) as well.
3. Step 3: Write the difference quotient: \( \frac{f(x+h) - f(x)}{h} = \frac{10 - 10}{h} \).
4. Step 4: Simplify the expression: \( \frac{10 - 10}{h} = \frac{0}{h} \).
5. Step 5: Recognize that \( \frac{0}{h} = 0 \) for any non-zero \( h \). Therefore, the simplified difference quotient is 0.

Key Concepts

Difference Quotient

The difference quotient is a formula used to calculate the average rate of change of a function over an interval. It is expressed as (ƒ(x+h) - ƒ(x)) / h, where h represents a small change in x. This concept is fundamental in calculus as it leads to the definition of the derivative, which measures the instantaneous rate of change.

Function Evaluation

Function evaluation involves substituting a specific value into a function to determine its output. In this case, with ƒ(x) = 10, the function returns a constant value regardless of the input x. Understanding how to evaluate functions is crucial for simplifying expressions like the difference quotient.

Limit Concept

The limit concept is central to calculus, particularly in defining derivatives. As h approaches zero in the difference quotient, the expression converges to the derivative of the function at a point. This concept helps in understanding how functions behave as inputs change infinitesimally, which is essential for analyzing continuity and differentiability.