Find the Difference Quotient f(x)=-5 square root of x+3
The given problem involves calculating the difference quotient of a function, which is a way to measure the average rate of change of the function over a small interval. The function provided is f(x) = -5√(x + 3), and the difference quotient formula is typically given by [f(x + h) - f(x)] / h, where h is a non-zero increment. The problem requires simplification of the difference quotient expression by substituting and simplifying the function with x + h in place of x and then subtracting the original function value f(x), all divided by the increment h. This calculation is a fundamental concept in calculus and is closely related to the concept of a derivative.
$f \left(\right. x \left.\right) = - 5 \sqrt{x + 3}$
Answer
Solution:
Step 1:
Start with the difference quotient formula: $\frac{f(x + h) - f(x)}{h}$
Step 2:
Determine the function values needed for the formula.
Step 2.1:
Calculate $f(x + h)$ by substituting $x$ with $x + h$ in the function: $f(x + h) = -5\sqrt{(x + h) + 3}$
Step 2.1.1:
Expand the expression to simplify: $f(x + h) = -5\sqrt{x + h + 3}$
Step 2.1.2:
This simplification yields the expression: $-5\sqrt{x + h + 3}$
Step 2.2:
Determine $f(x)$ and $f(x + h)$ using the given function: $f(x + h) = -5\sqrt{x + h + 3}$ and $f(x) = -5\sqrt{x + 3}$
Step 3:
Insert the calculated function values into the difference quotient formula: $\frac{f(x + h) - f(x)}{h} = \frac{-5\sqrt{x + h + 3} - (-5\sqrt{x + 3})}{h}$
Step 4:
Proceed to simplify the expression.
Step 4.1:
Focus on simplifying the numerator first.
Step 4.1.1:
Extract a factor of $-1$ from the terms: $\frac{-5\sqrt{x + h + 3} - (-5\sqrt{x + 3})}{h}$
Step 4.1.2:
Extract a common factor of $5$ from the terms: $\frac{-5(\sqrt{x + h + 3} - \sqrt{x + 3})}{h}$
Step 4.1.3:
Combine the multiplication of $-1$ and $5$: $\frac{-5(\sqrt{x + h + 3} - \sqrt{x + 3})}{h}$
Step 4.2:
Position the negative sign in front of the fraction: $-\frac{5(\sqrt{x + h + 3} - \sqrt{x + 3})}{h}$
Step 5:
The process is complete, and the difference quotient is simplified.
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 the function value at $x$, $f(x + h)$ is the function value at $x + h$, and $h$ is the difference between the two x-values.
To find the difference quotient for a given function, you need to:
Evaluate the function at $x + h$ and at $x$.
Substitute these values into the difference quotient formula.
Simplify the resulting expression, if possible.
In this problem, the function given is $f(x) = -5\sqrt{x + 3}$. When finding the difference quotient, we substitute $x$ with $x + h$ in the function to get $f(x + h)$. We then simplify the expression by combining like terms and factoring out common factors. The goal is to simplify the expression as much as possible, which often involves combining like terms, factoring, and reducing fractions.
