Question

Find the equation of the line tangent to the graph of f at (3, 141), where f is given by f(x) = 8x³ - 9x² + 6. Use y as the dependent variable when you write your equation.

Answer 1

To find the equation of the tangent line to the graph of f at (3, 141), we first need to find the derivative of f(x) = 8x³ - 9x² + 6. Then, we substitute x = 3 to find the slope of the tangent line. Finally, we use the point-slope form to write the equation of the line.

Step-by-step explanation:

To find the equation of the tangent line to the graph of f at (3, 141), we first need to find the derivative of f(x). The derivative represents the slope of the tangent line at any given point. Differentiating f(x) = 8x³ - 9x² + 6, we get f'(x) = 24x² - 18x.

Next, we substitute x = 3 into f'(x) to find the slope of the tangent line at (3, 141):

f'(3) = 24(3)² - 18(3) = 216 - 54 = 162

So, the slope of the tangent line is m = 162. Now we can use the point-slope form of a line, y - y₁ = m(x - x₁), where (x₁, y₁) is the point (3, 141) and m is the slope. Plugging in the values, we have:

y - 141 = 162(x - 3)

This is the equation of the line tangent to the graph of f at (3, 141) using y as the dependent variable.