Question

Question #ab2b2

Answer 1

The slope of the tangent line to the curve:

`y=3x^2-6x`

for `x=2` is `m=6`

Explanation:

By definition of the derivative:

`(df)/dx = lim_(h->0) (f(x+h)-f(x))/h`

So:

`d/dx (3x^2-6x) = lim_(h->0) (3(x+h)^2-6(x+h)-3x^2+6x)/h`

`d/dx (3x^2-6x) = lim_(h->0)( (3(x+h)^2-3x^2)/h -(6x+6h-6x)/h)`

`d/dx (3x^2-6x) = lim_(h->0)( (3x^2+6hx+3h^2-3x^2)/h - (6h)/h)`

`d/dx (3x^2-6x) = lim_(h->0)( (6hx+3h^2)/h - 6)`

`d/dx (3x^2-6x) = lim_(h->0)( 6x+3h - 6)`

`d/dx (3x^2-6x) = 6x - 6`

The general equation of the tangent line of a differentiable curve `y= f(x)` in the point `(x_0,y_0)` is:

`y(x) = y_0+f'(x_0)(x-x_0)`

having slope `f'(x_0)`, so the slope of the tangent line to the curve:

`y= 3x^2-6x`

for `x=2` is:

`f'(2) = [6x-6]_(x=2) = 6`

Answer 2

`m = 6`

Explanation:

We're asked to find the slope of the tangent line of a function at a certain value of the function.

To do this, we indeed differentiate

`d/(dx) [3x^2-6x]`

The definition of the first derivative is

`f'x = lim_(hrarr0)[(f(x+h) - f(x))/h]`

`= lim_(hrarr0)[(3(x+h)^2 - 6(x+h) - (3x^2 - 6x))/h]`

`= lim_(hrarr0)[(3(x^2 + 2hx + h^2) - 6x -6h - (3x^2 - 6x))/h]`

`= lim_(hrarr0)[(3x^2 + 6hx + 3h^2 - 6x -6h - 3x^2 + 6x)/h]`

`= lim_(hrarr0)[(6hx + 3h^2 - 6h)/h]`

`= lim_(hrarr0)[6x + 3h - 6]`

`= color(blue)(6x - 6`

This is the derivative of the given function.

To find the slope of the tangent line of the original function at some point, we plug in that point for `x` in the equation of the derivative of the function:

`"slope" = 6(2) - 6 = color(red)(6`

Answer 3

`6.`

Explanation:

Let, `y=f(x)=3x^2-6x.`

Knowing that the Slope of Tangent to a Curve, `y=f(x)` at

`x=a` is, `f'(a)=[dy/dx]_(x=a).`

By Definition, `f'(a)=[dy/dx]_(x=a)=lim_(x to a)(f(x)-f(a))/(x-a).`

`:. f'(2)=[dy/dx]_(x=2)=lim_(x to 2) {f(x)-f(2)}/(x-2).`

`=lim_(x to 2) {(3x^2-6x)-(3*2^2-6*2)}/(x-2),`

`=lim_(x to 2) {3x^2-6x-0}/(x-2),`

`=lim_(x to 2){3xcancel((x-2))}/cancel((x-2)),`

`=lim_(x to 2) 3x=3*2,`

`rArr f'(2)=[dy/dx]_(x=2)=6.`

Enjoy Maths.!