Question

Question #6523f

Answer 1

`(y - 6) = (-1) (x - 2)` in the Point-Slope Form.

If you prefer Slope-Intercept Form [ `y = mx + b` ] then we get

`y = -x +8`

Explanation:

We are given `f(x) = x^3 - 4x^2 + 3x + 8`

`color(red)(Step.1)`

Find the First Derivative:

`f'(x) = 3x^ 2 - 8x +3`

`color(red)(Step.2)`

We are given a value for x; x = 2

Hence, `f'(2) = 3(2)^2 - 8(2) + 3`

That is, `f'(2) = 3(4) - 16 +3`

We get `f'(2) = (-1)` on simplification.

This is our Slope value. That is `m = (-1)`

`color(red)(Step.3)`

To find the y-coordinate value, we must find `f(2)`

using our original equation `f(x) = x^3 - 4x^2 + 3x + 8`

`f(x) = x^3 - 4x^2 + 3x + 8`

`f(2) = 2^3 - 4(2^2) + 3(2) + 8`

On simplification we get

`f(2) = 8 - 16 + 6 + 8`

Therefore,

`f(2) = 16 - 16 + 6`

which gives us

`f(2) = 6`

Now, we have the following values - x-coordinate, y-coordinate and Slope values:

`x = 2, y = 6 and m(Slope) = (-1)`

`color(red)(Step.4)`

Equation of the Slope (in Point-Slope form) for a line (tangent) is given by

`y - y_1 = m(x - x_1)`

Next, we substitute the values that we know, in this equation

Our `(x_1 , y_1)` values are `(2, 6)`

Hence, we get

`(y - 6) = ( -1 ) (x - 2)` is our required Equation in the Point-Slope Form.

`color(red)(Step.5)` (Optional)

If you need the equation in Slope-Intercept Form `y = mx + b`, then we can write the equation as as

`(y -6) = (-1)(x - 2)`

This equation can be simplified to

`y = -x + 8`

Answer 2

See below.

Explanation:

First you need to find the gradient function of `y=x^3-4x^2+3x+8`

This is often called the derivative. We can then use this gradient function to find the gradient at some value of `x`

We can find the gradient function in the following way:

Using the power rule:

`d/dx ax^n=na^(n-1)`

We can do this with each term to find the derivative:

`dy/dx (x^3-4x^2+3x+8)=color(blue)(3x^2-8x+3)`

We now plug `x=2` into this derived function:

`3(2)^2-8(2)+3=-1`

So gradient at `x=2` is `-1`

Hope this helps.