Question

For y = f(x) = 9x³, x = 4, and Δx = 0.08, find:

a) Δy for the given x and Δx values,
b) dy = f'(x) dx,
c) dy for the given x and Δx values.

Answer 1

The question deals with finding the change in y (∆y) for a cubic function, its derivative (dy), and the approximate change in y based on the derivative for given x and ∆x values.

Step-by-step explanation:

The student asks for calculations related to the function y = 9x³ when x = 4 and a change in x (∆x) equals 0.08.

1. To find ∆y, we calculate the difference in y values for x and x + ∆x: ∆y = f(4 + 0.08) - f(4).
2. The derivative of the function [dy = f'(x) dx] can be found by differentiating f(x) = 9x³, which gives us f'(x) = 27x². By substituting x = 4, we can calculate dy.
3. To find dy for the given x and ∆x values, we multiply 27x² by ∆x at x = 4.