Find the difference quotient g(c) =5x^2+9x-5
The difference quotient of g(c) = 5x^2 + 9x - 5 is found by evaluating g(x + Δx) - g(x) and then dividing by Δx.
g(x + Δx) = 5(x + Δx)^2 + 9(x + Δx) - 5 = 5(x^2 + 2xΔx + Δx^2) + 9x + 9Δx - 5 = 5x^2 + 10xΔx + 5Δx^2 + 9x + 9Δx - 5
g(x) = 5x^2 + 9x - 5
g(x + Δx) - g(x) = (5x^2 + 10xΔx + 5Δx^2 + 9x + 9Δx - 5) - (5x^2 + 9x - 5)
= 5x^2 + 10xΔx + 5Δx^2 + 9x + 9Δx - 5 - 5x^2 - 9x + 5
= 10xΔx + 5Δx^2 + 9Δx
Now we divide this difference by Δx:
(10xΔx + 5Δx^2 + 9Δx) / Δx = 10x + 5Δx + 9
Therefore, the difference quotient g(c) is equal to 10c + 5Δc + 9.