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Find and completely simplify the difference quotient (f(x+h)-f(x))/(h) for the functions. Be sure you answer the question. f(x)=3x^(2)-x+((5)/(2))^(4)

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The difference quotient is a measure of the average rate of change of the function over an interval h. To simplify the problem, we can follow these steps:

Step 1:
First we need to understand the function f(x)=3x^2-x+(5/2)^4. This function is a quadratic polynomial, where x is the variable.

Step 2:
We calculate f(x + h) by substituting (x + h) in place of x in the function:
f(x + h) = 3*(x + h)^2 - (x + h) + (5/2)^4

On simplifying this equation we get:
f(x + h) = 3(x^2 + 2*x*h + h^2) - x - h + (5/2)^4
=> f(x + h) = 3x^2 + 6xh + 3h^2 - x - h + (5/2)^4

Step 3:
Calculate the difference quotient. Subtract f(x) from f(x + h) and divide by h:

[f(x + h) - f(x)] / h

= [3x^2 + 6xh + 3h^2 - x - h + (5/2)^4 - (3x^2 -x + (5/2)^4)] / h

Simplify it further into:

= (6xh + 3h^2 - h) / h

Step 4:
Simplify by dividing each term by h:

= 6x + 3h - 1

Therefore, the simplified form of the difference quotient (f(x+h)-f(x))/(h) for the function f(x) = 3x^2 -x + (5/2)^4 is 3h + 6x - 1.

User Pieterjandesmedt
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