A. Find a curve with a positive derivative through the point...

a. Find a curve with a positive derivative through the point $(1,1)$ whose length integral (Equation 3$)$ is
$$L=\int_{1}^{4} \sqrt{1+\frac{1}{4 x}} d x$$
b. How many such curves are there? Give reasons for your answer.

Key Concepts

Arc Length of a Curve

The arc length of a curve defined by y = f(x) is given by the integral ??(1 + (f?'(x))²) dx over an interval. This formula is fundamental in differential calculus and calculus of variations, allowing one to measure the distance along a curve. It forms the basis for relating the properties of the derivative of the function (slope) to the geometric concept of length.

Setting Up a Differential Equation from an Arc Length Condition

When a problem specifies an arc length integral with a particular integrand, one can equate it to the general form ?(1 + (f?'(x))²). By doing so, a relationship is established between f?'(x) and x. This results in a differential equation that the function must satisfy. In this case, the comparison yields (f?'(x))² = 1/(4x), linking the derivative directly to the independent variable.

Solving an Initial Value Problem

Once the differential equation is established from the arc length condition, it can be solved by integration. An initial condition, such as specifying a point through which the curve passes, is needed to determine the constant of integration. This process, characteristic of initial value problems in differential equations, ensures the precise determination of a function that meets both the differential equation and the given initial condition.

Uniqueness of the Solution

The uniqueness of a solution to a differential equation subject to an initial condition is guaranteed under conditions such as Lipschitz continuity of the function defining the derivative. In this context, having restricted the derivative to be positive selects the appropriate branch of the solution. Therefore, there is a unique curve that not only satisfies the differential equation derived from the arc length integral but also passes through the specified point.

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