📚

All Subjects

>

♾️

AP Calc

>

💎

Unit 7

# 7.5 Approximating Solutions Using Euler’s Method Jacob Jeffries

### AP Calculus AB/BC♾️

Bookmarked 7.1k • 259 resources
See Units ## What is Euler's Method?

Euler’s method is a way to find the numerical values of functions based on a given differential equation and an initial condition. We can approximate a function as a set of line segments using Euler’s method. Before introducing this idea, it is necessary to understand two basic ideas. This information allows us to do an algorithmic process to approximate function values when given a differential equation and an initial condition.
To showcase this method, let’s consider the following differential equation with a consequent initial condition: Let’s say we want to approximate y(7). We will create a table that essentially creates that line-segment link in Fig. 7.1: Notice that we can fill in the rest of the table and continue the process to get closer and closer to x = 7. Also notice that the change in x is a constant value (which is typically called the step-size).
We use the differential equation to find the slope at the given point and use Eq. 41 to find the change in y: We can then find the new value of y by adding the change in y from the original y value: We can then fill in the rest of the table: Note that as the step size approaches zero, the approximation becomes more and more exact. As an exercise, find an approximate value for y(9). This means that x = 7 corresponds to y = 249, which is our approximate solution.

## Practice

Using Euler’s method, approximate the value of y(2) using a step size of 0.25 given the following: Then find the absolute error in the approximation by directly solving for y(2)  by using a calculator. Approximate y(2) again using a step size of 0.2 and compare the absolute error in this approximation to the original absolute error.   Thousands of students are studying with us for the AP Calculus AB/BC exam.
#####   Studying with Hours = the ultimate focus mode
Start a free study session
##### 🔍 Are you ready for college apps?
Take this quiz and find out!
Start Quiz
Browse Study Guides By Unit
📆Big Reviews: Finals & Exam Prep
✍️Free Response Questions (FRQ)
🧐Multiple Choice Questions (MCQ)
Unit 10: Infinite Sequences and Series (BC Only)
👑Unit 1: Limits & Continuity
🤓Unit 2: Differentiation: Definition & Fundamental Properties
🤙🏽Unit 3: Differentiation: Composite, Implicit & Inverse Functions
👀Unit 4: Contextual Applications of the Differentiation
Unit 5: Analytical Applications of Differentiation
🔥Unit 6: Integration and Accumulation of Change
💎Unit 7: Differential Equations
🐶Unit 8: Applications of Integration
🦖Unit 9: Parametric Equations, Polar Coordinates & Vector Valued Functions (BC Only)
#####  FREE AP calc Survival Pack + Cram Chart PDF
Thousands of students are studying with us for the AP Calculus AB/BC exam.  