Question

I dont get this.. Can anyone help on this... I appreciate it.

Answer 1

7.
`(f(4 + h) - f(4))/(h) = 8`

`h((f(4 + h) - f(4))/(h)) = h(8) \\f(4 + h) - f(4) = 8h`

`(h + 4)^(2) + (4)^(2) = 8h \\(h + 4)(h + 4) + 16 = 8h \\h(h + 4) + 4(h + 4) + 16 = 8h`

`h(h) + h(4) + 4(h) + 4(4) + 16 = 8h \\h^(2) + 4h + 4h + 16 + 16 = 8h`

`h^(2) + 8h + 32 = 8h \\h^(2) + 32 = 0`

`h = \frac{-(0) \± \sqrt{(0)^(2) - 4(1)(32)}}{2(1)}`

`h = \frac{0 \± \sqrt{(0)^(2) - 4(1)(32)}}{2}`

`h = (0 \± √(-128))/(2)`

`h = (0 \± 8i√(2))/(2)`

`h =(8i√(2))/(2)`

`h = 4i√(2)`

8.
`f(x) = (x^(2) + 4x - 32)/(x - 4) \\f(x) = (x^(2) + 8x - 4x - 32)/(x - 4) \\f(x) = (x(x) + x(8) - 4(x) - 4(8))/(x - 4) \\f(x) = (x(x + 8) - 4(x + 8))/(x - 4) \\f(x) = ((x - 4)(x + 8))/(x - 4) \\f(x) = x + 8`


`g(x) = x(2x - 5) - 2(x^(2) - 3x - 4) \\g(x) = x(2x) - x(5) - 2(x^(2)) + 2(3x) + 2(4) \\g(x) = 2x^(2) - 5x - 2x^(2) + 6x + 8 \\g(x) = 2x^(2) - 2x^(2) - 5x + 6x + 8 \\g(x) = x + 8`

The functions are equivalent.

9.
`Slope: 3 \\Derivative: -8x + 11 \\3 = -8x = x + 11 \\-9x = 11 \\(-9x)/(-9) = (11)/(-9) \\x = -1(2)/(9) \\f(x) = -4x^(2) + 11x - 2 \\ f(x) = -4(-1(2)/(9)) + 11(-1(2)/(9)) - 2 \\f(x) = 4(8)/(9) - 13(4)/(9) - 2 \\f(x) = -8(5)/(9) - 2 \\f(x) = -10(5)/(9) \\y - y_(1) = m(x - x_(1)) \\y - (-11(4)/(9)) = 3(x - (-1(2)/(9)) \\y + 11(4)/(9) = 3(x + 1(2)/(9)) \\y + 10(5)/(9) = 3(x) + 3(1(2)/(9)) \\y + 10(5)/(9) = 3x + 3(2)/(3) \\y = 3x - 6(8)/(9)`