Question

Find an equation of the tangent line to the graph of \(f\) at the point \((2, f(2)) .\) Use a graphing utility to check your result by graphing the original function and the tangent line in the same viewing window. $$ f(x)=3(9 x-4)^{4} $$

Short answer

The equation of the tangent line for the given function \(f(x) = 3(9x - 4)^4\) at point (2,f(2)) is \(y = 54000x - 79232\)

Step-by-step solution

Calculate the derivative

First, calculate the derivative of \(f(x)\). The derivative of \(f(x) = 3(9x - 4)^4\) is found through chain rule. First, find the derivative of the outer function \(f(x) = 3u^4\) where \(u = 9x - 4\). Then, find the derivative of the inner function \(u=9x-4\). Let's perform these actions:\n\nThe derivative of the outer function, \(f'(u) = 12u^3\). Next, the derivative of the inner function, namely \(u'=9\). Now apply the chain rule. The chain rule states that \(f'(x) = f'(u) * u'\). Applying this gives \(f'(x) = 12(9x - 4)^3 * 9\).

Evaluate the derivative at \(x=2\)

Now that we have the derivative function, we need to evaluate it at the given point \(x = 2\). This will give us the slope of the tangent line at \(x = 2\). Using the derivative function: \(f'(x) = 12(9x - 4)^3 * 9\), replace \(x\) with 2, resulting in: \(f'(2) = 12(18 - 4)^3 * 9\). Simplifying this gives the slope as \(f'(2) = 54000\).

Find the y-coordinate of the tangent point

Before we can find the equation of the line, we need the y-coordinate of the point of tangency (2, f(2)). Use the original function to calculate this: \(f(2) = 3(18 - 4)^4 = 285768\).

Use point-slope form to find the equation of the line

Now we can use the point-slope form: \(y - y_1 = m(x - x_1)\) to find the equation of the tangent line. Substituting our known values gives: \(y - 285768 = 54000(x - 2)\). After clearing this out, we get \(y = 54000x - 79232\).