Question

Coordinates, gradient and tangent work (see image).

Are my answers correct? -
a. P= (1.9,1.9) or (2,2) Q=(2,3)
b. (rise over run) 3-1.9/2-1.9 = 11
... and can anyone solve q.c -
c.

Answer 1

a)
`P(x,2x^2-5),\ Q(x+h,2(x+h)^2-5)`

b)
`4x+2h`

c)
`4x`

Explanation:

Given the curve

`y=2x^2-5`

a) If the x-coordinate of P is
`x`, then the y-coordinate is
`2x^2-5,` so point P has coordinates
`(x,2x^2-5)`

If the x-coordinate of Q is
`x+h`, then the y-coordinate is
`2(x+h)^2-5` so point Q has coordinates
`(x+h,2(x+h)^2-5)`

b) The gradient of the secant RQ is

`(y_Q-y_P)/(x_Q-x_P)\\ \\=((2(x+h)^2-5)-(2x^2-5))/((x+h)-x)\\ \\=(2(x+h)^2-5-x^2+5)/(x+h-x)\\ \\=(2(x+h)^2-2x^2)/(h)\\ \\=(2x^2+4xh+2h^2-2x^2)/(h)\\ \\=(4xh+2h^2)/(h)\\ \\=4x+2h`

c) If
`h\rightarrow 0,` then the gradient
`4x+2h\rightarrow 4x`