For the function g(x) = 2x^2-3x determine the following expression/quantity.

a. g(x+h) -g(x-h) / 2h =

Using the results of Part A

b. g(3.1)-g(2.9) / 0.1 =

a. To determine the expression g(x+h) - g(x-h) / 2h, we need to substitute the given function g(x) = 2x^2 - 3x into the formula.

Step 1: Replace x with (x+h) in the given function.
g(x+h) = 2(x+h)^2 - 3(x+h)

Step 2: Simplify the equation.
g(x+h) = 2(x^2 + 2xh + h^2) - 3(x+h)
= 2x^2 + 4xh + 2h^2 - 3x - 3h

Step 3: Replace x with (x-h) in the given function.
g(x-h) = 2(x-h)^2 - 3(x-h)

Step 4: Simplify the equation.
g(x-h) = 2(x^2 - 2xh + h^2) - 3(x-h)
= 2x^2 - 4xh + 2h^2 - 3x + 3h

Step 5: Subtract g(x-h) from g(x+h).
g(x+h) - g(x-h) = (2x^2 + 4xh + 2h^2 - 3x - 3h) - (2x^2 - 4xh + 2h^2 - 3x + 3h)
= 2x^2 + 4xh + 2h^2 - 3x - 3h - 2x^2 + 4xh - 2h^2 + 3x - 3h
= 8xh

Step 6: Divide the result by 2h.
(g(x+h) - g(x-h)) / 2h = 8xh / 2h
= 4x

Therefore, the expression g(x+h) - g(x-h) / 2h simplifies to 4x.

b. To find g(3.1) - g(2.9) / 0.1, we need to substitute the given values into the function g(x).

Step 1: Replace x with 3.1 in the given function.
g(3.1) = 2(3.1)^2 - 3(3.1)
= 2(9.61) - 9.3
= 19.22 - 9.3
= 9.92

Step 2: Replace x with 2.9 in the given function.
g(2.9) = 2(2.9)^2 - 3(2.9)
= 2(8.41) - 8.7
= 16.82 - 8.7
= 8.12

Step 3: Subtract g(2.9) from g(3.1) and divide by 0.1.
(g(3.1) - g(2.9)) / 0.1 = (9.92 - 8.12) / 0.1
= 1.8 / 0.1
= 18

Therefore, g(3.1) - g(2.9) / 0.1 evaluates to 18.