Question

Find the equation of the secant line for the given interval f(x)=x³ for the interval [3,6}

Answer 1

To find the equation of the secant line for the function f(x) = x³ over the interval [3, 6], calculate the function values at the endpoints, find the slope, and write the point-slope form of the line to obtain the equation y = 63x - 162.

Step-by-step explanation:

The question asks us to find the equation of the secant line for the function f(x) = x³ over the interval [3, 6]. The secant line is the line that intersects the curve at two points, corresponding to the endpoints of the interval.

Steps to Find the Secant Line:

1. Calculate the function values at the endpoints of the interval: f(3) = 3³ = 27 and f(6) = 6³ = 216.
2. Find the slope (m) of the secant line using the slope formula m = (f(6) - f(3)) / (6 - 3). Thus, m = (216 - 27) / (6 - 3) = 189 / 3 = 63.
3. Use point-slope form to write the equation of the secant line. We can use the point (3, 27): y - 27 = 63(x - 3). Simplifying, we get the equation of the secant line: y = 63x - 162.

The equation of the secant line for the function f(x) = x³ over the interval [3, 6] is y = 63x - 162.