find and simplify the difference quotient of f(x) = √3x
difference quotient
= (f(x+h) - f(x))/h
= (√3(x+h) - √3x)/h
= (√3x + √3h - √3x)/h
= √3h/h
= √3
Oh, I see you're fond of a square root of 3 times x. Well, let's have some fun with the difference quotient, shall we? The difference quotient of f(x) = √3x is simply the difference between f(x) and f(a), divided by the difference between x and a. In equation form, it looks like this:
[f(x) - f(a)] / [x - a]
Now, if we insert f(x) = √3x into the equation, we get:
[√3x - √3a] / [x - a]
To simplify this further, we could multiply the top and bottom by the conjugate, which is:
[√3x + √3a].
That gives us:
[√3x - √3a] * [√3x + √3a] / [x - a] * [√3x + √3a]
See, we just created some clownish algebra magic! Now, the numerator simplifies to:
(3x - 3a)
The denominator remains the same:
(x - a)
And voila, there you have it! The simplified difference quotient of f(x) = √3x is:
(3x - 3a) / (x - a)
Remember, math can be as entertaining as a clown juggling knives. Enjoy!
The difference quotient of a function is a measure of the average rate of change of the function over a given interval. To find the difference quotient for the function f(x) = √3x, we need to evaluate [f(x+h) - f(x)] / h.
Step 1: Substitute f(x) = √3x into the expression for the difference quotient:
[f(x+h) - f(x)] / h = [√3(x+h) - √3x] / h
Step 2: Simplify the numerator using the distributive property of square roots:
√3(x+h) - √3x = √3x + √3h - √3x
Step 3: Combine like terms in the numerator:
√3x - √3x + √3h = √3h
Step 4: Divide the numerator by h:
(√3h) / h = √3
Therefore, the difference quotient for the function f(x) = √3x is √3.
To find the difference quotient of a function, we need to determine the rate of change of the function as the difference in the input (x-values) approaches zero.
The difference quotient for a function f(x) is given by:
[f(x + h) - f(x)] / h
Let's apply this concept to the given function f(x) = √3x.
Step 1: Substitute f(x + h) and f(x) into the difference quotient formula:
[f(x + h) - f(x)] / h = [(√3(x + h)) - √3x] / h
Step 2: Simplify the numerator:
[(√3x + √3h) - √3x] / h
Step 3: Simplify further by removing the common term (√3x):
√3h / h
Step 4: Cancel out the h terms:
√3
Therefore, the difference quotient of f(x) = √3x is √3.