Question

How do you find a tangent line parallel to secant line?

Answer 1

You can find a tangent line parallel to a secant line using the Mean Value Theorem.

The Mean Value Theorem states that if you have a continuous and differentiable function, then

`f'(x) = (f(b) - f(a))/(b - a)`

To use this formula, you need a function `f(x)`. I'll use `f(x) = -x^3` as an example.

I'll also use `a = -2` and `b = 2` for the interval for the secant line. This is the line that passes through the points `(-2, 8)` and `(2, -8)`.

So, we know that the slope of this line will be `(-8 - 8)/(2 - (-2)) = -4`.

To find the tangent lines parallel to this secant line, we will take the function's derivative, `f'(x)`, and set it equal to `-4`, then solve for `x`.

`-3x^2 = -4`

Solving this for `x` gives us: `x = ±sqrt(4/3)`.

So, the lines tangent to `y = -x^3` at `x = sqrt(4/3)` and `x = -sqrt(4/3)` must be parallel to the secant line passing through `x = 2` and `x = -2`.