Let f(x)=x^3−x. Using limits, find the equation of the line tangent to f (x) at x= -1 and x=2.

Question

If f(x)=x³-x, how would I find the equation of the tangent line for x={-1, 2} using the definition of limits? I can do it using the d/dx method, but my teacher doesn't want that. I know that at (-1, 0) the equation of the tangent line is y=2x+2 and at (2, 6) the equation is y=11x-16. I just don't know how to get to that point using the limit definition.

If anyone could help me out, I would greatly appreciate it. Thanks for your time!

Answer 1

Here is a Desmos graph that you can use to confirm your answers:

desmos.com/calculator/spglfme6io

Answer 2

Great question Noah! The process will be a combination of steps.

Plug the x value into the function to get the coordinate point.

Apply the alternate form of the derivative to find the tangent slope at that point.

f'(c) = lim_x→c [f(x) - f(c)]/(x - c)

You will have the function coordinate point and its tangent slope. Use point-slope form to create the tangent line equation.

y - y₁ = m(x - x₁)

Hope this helps you connect the dots on your question. You are most welcome!