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

| May 27, 2018

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 = Âħsqrt(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##.

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