a. Find a curve with a positive derivative through the point (1,1) whose length integral (Equati

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

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.

Transcript

00:01 Section 6 .3, problem 25, we're dealing here with arc length formula.

00:06 So they tell us to find a curve that has a positive derivative passes through the point 1 -1, and has an arc length from 1 to 4 given by 1 plus 1 over 4x dx, and then how many such curves would exist.

00:21 So we know from this section that our formula for the arc length is given by some integral, so the arc length from a to v is to the arc length is, the square root of 1 plus f prime of x d x and what we were given here i was given an arc length for my curve 1 to 4 so now i'm given the arc length is from 1 to 4 of what was it square root of 1 plus 1 over 4x d x so what that means is if i look at this and sorry about there here's a typo this f prime of x squared so what it means is if i compare the the arc length formula with the arc link of the problem that i'm dealing with i know then that f prime of x squared is going to be equal to one over four x for all x values between one to four so that means if i take the square root of both side so the square root of f prime of x absolute value is going to be equal to if i take the square root of the right side i get one over to the square root of x now they tell us initially that this function has a positive derivative if it has a positive derivative that means that f prime is greater than zero the absolute value of a positive number is positive i can get rid of the absolute value signed.

01:59 So it means that f prime of x is equal 1 over 2 root x.

02:08 And now if i were to integrate both sides of this equation, that tells me that f of x is going to be the square root of x plus a constant of integration.

02:24 So here's where you might be tempted to say, oh, there are infinite many solutions that can make this happen...

Please give Ace some feedback