Find the Difference Quotient f(x)=95

Solution:

Step 1:

Consider the difference quotient given by the formula: $\frac{f(x+h)-f(x)}{h}$.

Step 2:

Determine the function values required by the formula.

Step 2.1:

Calculate $f(x+h)$.

Step 2.1.1:

Substitute $x$ with $x+h$ in $f(x)$. Thus, $f(x+h)=95$.

Step 2.1.2:

Since the function value is constant, $f(x+h)=95$.

Step 2.2:

Identify $f(x)$.

Step 2.2.1:

By definition, $f(x)=95$.

Step 3:

Insert the function values into the difference quotient formula: $\frac{f(x+h)-f(x)}{h}=\frac{95-95}{h}$.

Step 4:

Simplify the expression.

Step 4.1:

Reduce the numerator.

Step 4.1.1:

Calculate the difference $95-95$.

Step 4.1.2:

The result is $\frac{0}{h}$.

Step 4.2:

Divide the numerator by $h$ to get the final answer: $0$.

Step 5:

The difference quotient for the constant function $f(x)=95$ is $0$.

Knowledge Notes:

The difference quotient is a formula used in calculus to determine the slope of the secant line between two points on a graph of a function. It is given by:

$$\frac{f(x+h)-f(x)}{h}$$

where $f(x)$ is a function, $x$ is a point on the domain of $f$, and $h$ is the difference between $x$ and another point in the domain. The difference quotient is used as the basis for the definition of the derivative, which is the limit of the difference quotient as $h$ approaches zero.

In this specific problem, the function $f(x)=95$ is a constant function, which means that for any value of $x$, the function's value is always $95$. When evaluating the difference quotient for a constant function, the numerator will always be zero because $f(x+h)-f(x)=95-95=0$. Since the numerator is zero, the entire difference quotient is zero, regardless of the value of $h$ (provided $h\ne 0$). This indicates that the slope of the secant line, and hence the derivative, of a constant function is zero, reflecting the fact that the graph of a constant function is a horizontal line with a slope of zero.