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.