Simplify the difference quotient f(x)-f(a)/x-a for the given function. f(x)=18x^3
thanks
Do you mean
[ f(x)-f(a)] / [x-a ] ? That would be the slope
[ 18 x^3 - 18 a^3 ] / (x-a)
= 18 [ x^3-a^3 ] / (x-a) = 18 (x^2 + x a + a^2)
note as a ---->x it is 18 (3 x^2) which is the derivative
To simplify the difference quotient for the function f(x) = 18x^3, we need to substitute the values of f(x) and f(a) into the formula.
The difference quotient is given by the formula: [f(x) - f(a)] / [x - a]
Step 1: Substitute f(x) = 18x^3 and f(a) = 18a^3 into the formula:
[f(x) - f(a)] / [x - a] = [18x^3 - 18a^3] / [x - a]
Step 2: Factor out the common factor of 18 from the numerator:
[f(x) - f(a)] / [x - a] = 18(x^3 - a^3) / [x - a]
Step 3: Factor the difference of cubes in the numerator:
[f(x) - f(a)] / [x - a] = 18(x - a)(x^2 + ax + a^2) / [x - a]
Step 4: Cancel out the (x - a) terms:
[f(x) - f(a)] / [x - a] = 18(x^2 + ax + a^2)
Therefore, the simplified difference quotient for the given function f(x) = 18x^3 is 18(x^2 + ax + a^2).