Question

Tangent and parallel?

find an equation of the line that is tangent to the graph of f and parallel to the given line.

function: `f(x) = 2x^2`

Line: 2x - y + 5= 0

Answer 1

`y = 2x -1/2`

Explanation:

The gradient of the tangent to a curve at any particular point is given by the derivative of the curve at that point. (If needed, then the normal is perpendicular to the tangent so the product of their gradients is `-1`).

We have:

`f(x) = 2x^2`

Differentiating wrt `x` we have:

`f'(x)=4x`

Now if we take the given line equation and put it into standard form `y=mx+c`:

`2x - y + 5 = 0 => y = 2x-5`

So the gradient of this line is:

`m=2`

If we want our tangent equation to be parallel to this line then it must have the same gradient, thus we want:

`f'(x) =2 => 4x = 2`
`:. x=1/2`

When `x=1/2 => f(x) = 2*1/4 = 1/2`

So the tangent passes through `(1/2,1/2)` and has gradient `m_T=2`, so using the point/slope form `y-y_1=m(x-x_1)` the tangent equation we seek is;

`y - 1/2 = 2(x-1/2)`
`:. y - 1/2 = 2x-1`
`:. y = 2x -1/2`

We can verify this graphically: