Question

If f(x) = 5x² - x³, find f'(1) and use it to find an equation of the tangent line to the curve y = 5x² - x³ at the point (1, 4).

Answer 1

To find f'(1), differentiate f(x) = 5x² - x³ using the power rule. Then use the point-slope form of a line to find the equation of the tangent line at (1,4), given the slope f'(1) = 7.

Step-by-step explanation:

To find f'(1), we need to differentiate the function f(x) = 5x² - x³. Applying the power rule of differentiation, we get f'(x) = 10x - 3x². Evaluating f'(1) gives us f'(1) = 10(1) - 3(1)² = 10 - 3 = 7.

Now, to find the equation of the tangent line, we use the point-slope form of a line: y - y₁ = m(x - x₁), where (x₁, y₁) represents the point of tangency and m represents the slope. Plugging in the values (x₁, y₁) = (1, 4) and m = f'(1) = 7, we have y - 4 = 7(x - 1). Simplifying, we get the equation of the tangent line as y = 7x - 3.