Table of Contents
Approximation of Differential Equation
Differential equations are mathematical equations that describe the change in a certain quantity, or function, over time. In order to approximate a differential equation, a numerical method is used to approximate the solution to the equation. This involves breaking the differential equation down into a series of smaller, simpler equations, which can then be solved using a computer. The approximation of a differential equation is used to calculate the approximate solution to the equation, which can be used to predict the behavior of the quantity or function over time.
Approximation Example
In the approximation example, we will approximate the value of pi. Pi is the ratio of a circle’s circumference to its diameter and is approximately 3.14.
We will use the following equation to approximate pi:
pi ≈ (C/d)^2
In this equation, C is the circumference of the circle and d is the diameter of the circle.
We can use the value of C and d from a circle with a diameter of 10 cm and a circumference of 31.4 cm.
pi ≈ (31.4 cm/10 cm)^2
pi ≈ 3.14
Riemann Sum Example
A Riemann sum is a way to estimate the value of a function over an interval by adding up the function values at evenly spaced points within the interval.
The example below illustrates how to find a Riemann sum for a function over the interval [0, 3].
The function being estimated is f(x) = x^3.
The points at which the function is evaluated are:
0, 1, 2, and 3.
The width of the interval is 1.
The sum of the function values at the points evaluated is:
0 + 1 + 4 + 9 = 14.
Thus, the Riemann sum for the function over the interval [0, 3] is 14.
Problems for Practice
Problem 1
A plumber charges $25 for a service call plus $50 per hour of service. Write an equation in slope-intercept form for the cost, C, after h hours of service.
C = 25 + 50h
