Write the equation of the line tangent to the graph of the function at the indicated point. As a check, graph both the function and the tangent line you found to see whether it looks correct.
(1/x^3 -x)^2 at x= 2
y'(x) = 2(1/x^3 - x) * (-3/x^4 - 1)
y'(2) = 2(1/8 - 2)(-3/16 - 1)
= 2(-15/8)(-19/16)
= 285/64
To find the equation of the tangent line to the graph of the function at the indicated point, we will follow these steps:
1. Differentiate the function to find the derivative.
2. Plug the given x-coordinate into the derivative to find the slope of the tangent line.
3. Use the point-slope form of a linear equation to write the equation of the tangent line.
Let's go through each step:
1. Differentiate the function:
To find the derivative of the function f(x), we will use the chain rule. The function is given as f(x) = (1/x^3 - x)^2.
Using the chain rule:
d/dx (f(x)) = 2(1/x^3 - x) * d/dx (1/x^3 - x)
= 2(1/x^3 - x) * ((d/dx (1/x^3)) - (d/dx (x)))
= 2(1/x^3 - x) * (-3/x^4 - 1)
= -6(1/x^3 - x) / x^4 - 2(1/x^3 - x)
Therefore, the derivative is:
f'(x) = -6(1/x^3 - x) / x^4 - 2(1/x^3 - x)
2. Plug in x = 2 to find the slope of the tangent line:
To find the slope of the tangent line, we will plug in the given x-coordinate, x = 2, into the derivative.
Slope (m) = f'(2) = -6(1/2^3 - 2) / 2^4 - 2(1/2^3 - 2)
= -6(1/8 - 2) / 16 - 2(1/8 - 2)
= -6(-15/8) / 16 - 2(-15/8)
= 90/8 / 16 + 30/8
= 90/8 / 16/1 + 30/8
= 90/8 * 1/16 + 30/8
= 90 / 8 * 1 / 16 + 30/8
= 90/128 + 30/8
= 45/64 + 15/4
= 45/64 + 240/64
= 285/64
So, the slope of the tangent line is 285/64.
3. Write the equation of the tangent line using point-slope form:
The equation of a line in point-slope form is given by:
y - y1 = m(x - x1)
We have the point (x1, y1) = (2, (1/2^3 - 2)^2), and we found the slope (m) to be 285/64.
Using the point-slope form, we can write the equation of the tangent line:
y - (1/2^3 - 2)^2 = (285/64)(x - 2)
Simplifying this equation will give you the equation of the tangent line.
To check if it looks correct, you can graph both the original function and the tangent line using a graphing calculator or software.