Answer:
Image 1: -6x -3h + 1
Image 2:
Image 3: 4x + 2h - 2
Images 4 & 5: 8
Explanation:
Image 1
f(x) = -3x^2 + x + 8
f(x+h) = -3(x+h)^2 + (x+h) + 8
= -3(x^2 + 2xh + h^2) + (x+h) + 8
= -3x^2 - 6xh - 3h^2 + x + h + 8
[f(x+h) - f(x) ] / h = [-3x^2 - 6xh - 3h^2 + x + h + 8 - (-3x^2 + x + 8) ] / h
= [-3x^2 - 6xh - 3h^2 + x + h + 8 + 3x^2 - x - 8) ] / h
= [-3x^2 - 6xh - 3h^2 + x + h + 8 + 3x^2 - x - 8 ] / h
= [- 6xh - 3h^2 + h] / h
= -6x -3h + 1
Image 3
f(x) = 2x^2 - 2x
f(x+h) = 2(x+h)^2 - 2(x+h)
= 2(x^2 + 2xh + h^2) - 2(x+h)
= 2x^2 + 4xh + 2h^2 - 2x - 2h
[f(x+h) - f(x) ] / h = [ 2x^2 + 4xh + 2h^2 - 2x - 2h - (2x^2 - 2x) ] / h
= [ 2x^2 + 4xh + 2h^2 - 2x - 2h - 2x^2 + 2x) ] / h
= [ 4xh + 2h^2 - 2h ]/ h
= 4x + 2h - 2
Images 4 & 5
[f(a+h) - f(a) ] / h = [ 8a + 8h - 4 - (8a - 4) ] / h
= [ 8a + 8h - 4 - 8a + 4) ] / h
= [ 8h ] / h
= 8