Lecture 14: Calculators and Approximations

This paragraph introduces Zeno's Arrow Paradox to explain how calculators and computers perform operations involving infinity. It describes the paradox where an arrow must travel infinite steps to reach its target, but in reality, it reaches the target in a finite time. The resolution is found through the concept of infinite series, where the sum of the distances the arrow travels converges to one, symbolizing the completion of the journey. The paradox is used to illustrate the mathematical concept of limits and how infinite series can be used to solve problems that, at first glance, seem unsolvable due to infinite processes.

The second paragraph delves into the application of infinite series in calculus to compute values of mathematical constants, such as pi, which cannot be known exactly. It introduces the Leibniz series for pi/4 and demonstrates how adding more terms of the series brings the approximation closer to the actual value of pi. The paragraph also mentions another series that approximates pi^2/6, showing how calculators can use these series to provide increasingly accurate values of pi. The underlying theme is the power of infinite series in numerical computation and the concept of convergence in series.

This paragraph explains how calculators evaluate the sine function for any given angle using an infinite series. It presents the Taylor series expansion of the sine function, which is an infinite series that approximates the sine of any radian measurement x. The paragraph illustrates how by adding more terms of the series, the calculator can compute the sine of x to any desired degree of accuracy. It also explains the concept of factorials in the context of the series and provides examples, including the sine of 0 and pi/6, to demonstrate the accuracy of the approximation.

The fourth paragraph provides a graphical representation of how the infinite series approximation of the sine function works. It describes how the graph of the sine function is approximated by the sum of the first few terms of its series. The paragraph explains that as more terms are added, the polynomial approximation becomes indistinguishable from the actual sine function, especially near the origin. This visual demonstration reinforces the concept that polynomials can closely mimic the behavior of the sine function.

In this paragraph, the concept of derivatives is used to explain the behavior of functions and their relationship to the motion of a car. It discusses how the position, velocity, and acceleration of a car relate to the derivatives of its position function. The paragraph uses the example of a car with a maximum velocity of one mile per minute to illustrate how the car's position changes over time. It also introduces the idea that if all derivatives of a function are zero at a point, the function does not change at that point, which is a key insight into the nature of functions and their properties.

The sixth paragraph introduces the Newton-Raphson method, a powerful iterative technique for finding the roots of a function. The method involves making an initial guess for the root, drawing a tangent to the function at that point, and finding where the tangent intersects the x-axis. This new x-value becomes the next guess, and the process is repeated to refine the approximation. The paragraph provides a step-by-step explanation of the method and demonstrates its effectiveness using the function f(x) = x^2 - 5, showing how quickly the method converges to the actual root, the square root of 5.

The final paragraph concludes the lecture by summarizing how calculators and computers use infinite series and iterative methods to solve complex mathematical problems. It highlights the Newton-Raphson method as an example of how these tools can approximate solutions to equations. The paragraph also teases the next lecture, which will explore how calculus can be used for optimization, promising further insights into the power of these mathematical tools.