Question

The arc length function for a curve y = f(x), where f is an increasing function, is s(x) = x 5t 10 dt 0 . (a) if f has y-intercept 9, find an equation for f.

Answer 1

The equation for the increasing function f(x), with a y-intercept of 9 and a slope of 3, is y = 3x + 9.

Step-by-step explanation:

In this mathematics question, we are given that the curve y = f(x) is an increasing function with a y-intercept of 9. This information, along with the additional context provided by the related figures, allows us to deduce an equation for the function f(x). From the supporting details, we learn that for a line, the y-intercept is the point where the line meets the y-axis. FIGURE A1 indicates that the y-intercept is 9 and the slope of the line is 3. This means for every 1 unit increase in x, y increases by 3 units.

Thus, we can express the equation of the line as y = 3x + 9. Now concerning arc length, we understand from the additional information that arc length is directly proportional to the radius in a circular path, and inversely proportional to the angle of rotation. Although interesting, this fact is not directly applicable to determining the equation for f(x) based on the y-intercept and slope information provided.