Questions: The arc length function for a curve y=f(x), where f is an increasing function, is s(x)=∫ from 0 to x √(7t+10) dt. (a) If f has y-intercept 4, find an equation for f.

Solution Steps

To find the equation for \( f(x) \) given the arc length function \( s(x) \), we need to use the relationship between the arc length and the function itself. The arc length function is given by \( s(x) = \int_{0}^{x} \sqrt{7t + 10} \, dt \). We can differentiate \( s(x) \) to find the integrand, which will help us determine \( f(x) \).

1. Differentiate the arc length function \( s(x) \) with respect to \( x \) to find \( \sqrt{7x + 10} \).
2. Use the fact that \( \frac{ds}{dx} = \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \) to find \( \frac{dy}{dx} \).
3. Integrate \( \frac{dy}{dx} \) to find \( f(x) \).
4. Use the given \( y \)-intercept to determine the constant of integration.

Given the arc length function: \[ s(x) = \int_{0}^{x} \sqrt{7t + 10} \, dt \] we differentiate \( s(x) \) with respect to \( x \): \[ \frac{ds}{dx} = \sqrt{7x + 10} \]

Using the relationship: \[ \frac{ds}{dx} = \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \] we solve for \( \frac{dy}{dx} \): \[ \sqrt{7x + 10} = \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \] \[ 7x + 10 = 1 + \left( \frac{dy}{dx} \right)^2 \] \[ \left( \frac{dy}{dx} \right)^2 = 7x + 9 \] \[ \frac{dy}{dx} = \sqrt{7x + 9} \]

Integrate \( \frac{dy}{dx} \) to find \( f(x) \): \[ y = \int \sqrt{7x + 9} \, dx \] \[ y = \frac{2}{21} (7x + 9)^{\frac{3}{2}} \]

Given the \( y \)-intercept is 4, we add this constant to the function: \[ f(x) = \frac{2}{21} (7x + 9)^{\frac{3}{2}} + 4 \]

Final Answer