Lec 13 | MIT 18.01 Single Variable Calculus, Fall 2007

The script begins with a reminder that the content is provided under a Creative Commons license and encourages donations to MIT OpenCourseWare to support free educational resources. The professor then introduces the topic of related rates, referencing a previous problem involving geometry and a right triangle. The scenario involves a road, a police monitoring station, and a vehicle moving in one direction. The distances between the vehicle, the road, and the police station are given as part of a right triangle with sides 3, 4, and 5, and the rate of change of the distance to the police is discussed in the context of speeding.

The professor continues the lecture by discussing implicit differentiation as a method to solve related rates problems. Using the given scenario, the professor sets up the problem with the relationship x^2 + 30^2 = D^2 and the rate of change dD/dt = -80 feet per second. The process involves differentiating the equation with respect to time, without substituting constants prematurely, and solving for dx/dt to determine if the vehicle is speeding. The conclusion is that the vehicle is indeed speeding, as the rate of approach towards the police is faster than the speed limit.

The script moves on to a second example involving a conical tank being filled with water. The tank has a radius of 4 feet and a depth of 10 feet, and water is being added at a rate of 2 cubic feet per minute. The goal is to determine how fast the water level is rising when it reaches a depth of 5 feet. The professor explains the importance of setting up diagrams and variables, using similar triangles to establish a relationship between the radius of the water's surface (r) and its height (h), and then setting up equations for the volume of the water and its rate of change.

The professor demonstrates how to use implicit differentiation to solve for the rate at which the water level is rising (dh/dt). Starting with the volume of a cone formula, V = (1/3) * π * r^2 * h, and substituting r as a function of h, the professor shows the process of differentiating the volume with respect to time to find dv/dt. The chain rule is applied to solve for dh/dt when h equals 5 feet, concluding with the formula dh/dt = 1/(2 * π) feet per second.

The script presents a physical application of related rates by discussing a scenario with a ring and a flexible string. The professor explores where the ring will settle if the ends of the string are at different heights. Through a demonstration with volunteers, the professor shows that the ring settles at a point that can be determined mathematically, relating it to the concept of potential energy and the lowest energy state.

The professor introduces a problem related to the construction of a suspension bridge, using a string and a ring to illustrate the concept. The goal is to find where the ring settles when the string is fixed at two points with different heights. The problem is set up by labeling points on the diagram and identifying the constraint that the total length of the string is constant. The professor explains that the problem involves minimizing the y-coordinate of the ring's position, subject to this constraint.

The script delves into the analytical aspect of the suspension bridge problem by describing the constraint curve as an ellipse. The professor uses the Pythagorean theorem to express the lengths of the string segments and sets up an equation where the sum of the square roots of these lengths is constant. Implicit differentiation is used to find the relationship between x and y, leading to an equilibrium condition that reveals the symmetry in the problem.

The professor interprets the equilibrium condition derived from the previous step, explaining that it represents the angles alpha and beta being equal, indicating a hidden symmetry in the problem. This symmetry implies that the tension on the two lines of the string is the same, leading to a balanced distribution of load, which is a principle used in the construction of suspension bridges.

The script concludes with an introduction to Newton's method, a numerical technique for solving equations. The professor illustrates the method using the example of finding the square root of 5, by setting up a function f(x) = x^2 - 5 and aiming to solve f(x) = 0. The method involves making an initial guess, using the tangent line at that point as an approximation, and iteratively refining the guess. The professor outlines the formula for Newton's method and demonstrates its effectiveness through several iterations, showing rapid convergence to the accurate value.

The professor continues the discussion on Newton's method by providing the formula for the nth term and applying it iteratively to improve the guess for the square root of 5. The script shows the calculations for the first few iterations and notes the rapid improvement in accuracy, highlighting the power and efficiency of Newton's method for finding numerical solutions to equations.