# Approximating solutions using Euler's method (BC only)

> AP Calculus BC · CED Differential Equations
> Source: https://www.owlsprep.com/study/ap-calculus-bc-u7-approximating-solutions-using-euler-s/

This subtopic covers Euler’s recursive formula, step size calculation, approximating solutions to first-order initial value problems, identifying over/under-approximation, and step size-error relationships, all tested exclusively on AP Calculus BC.

**Prerequisites:** [First-order initial value problems](https://www.owlsprep.com/study/ap-calculus-bc-u7-introduction-to-differential-equations/); Point-slope form of a line; [Derivative tests for concavity](https://www.owlsprep.com/study/ap-calculus-ab-u5-analyzing-functions/)

## Learning objectives

- Use Euler's method to approximate solutions to first-order initial value problems
- Calculate correct step size for a given target x-value
- Identify whether an approximation is over or under the exact solution
- Describe how global error scales with step size for Euler's method

## What is Euler's Method?

Euler's method is a numerical iterative technique for approximating values of the unknown solution $y(x)$ to a first-order initial value problem (IVP) of the form $\frac{dy}{dx} = f(x,y)$, $y(x_0) = y_0$. Unlike analytical methods such as separation of variables that give an exact closed-form solution, Euler's method works even when an exact solution cannot be found easily.

It is tested exclusively on the AP Calculus BC exam, making up approximately 2-3% of the total exam score, and appears in both multiple-choice and free-response questions, often paired with other differential equation topics like slope fields or population growth. The core intuition uses tangent line approximation at each known point to step to the next x-value iteratively.

## The Euler's Method Recursive Formula

The core of Euler's method is breaking the interval from the initial $x$-value $x_0$ to the target $x$-value $x_t$ into $n$ equal steps, each of size $h$ (called step size). The formula for step size is:

$$h = \frac{x_t - x_0}{n}$$

Starting from the known initial point $(x_0, y_0)$, we use the slope from the differential equation at the start of the step to approximate the y-value at the end of the step. The general recursive formulas are:

$$x_k = x_0 + k \cdot h$$

$$y_k = y_{k-1} + h \cdot f(x_{k-1}, y_{k-1})$$

**Worked example:** Given the initial value problem $\frac{dy}{dx} = 2x - y$, with $y(0) = 1$, use Euler's method with step size $h=0.5$ to approximate $y(1.0)$.

1. Identify parameters: Initial $x_0 = 0$, $y_0 = 1$, step size $h=0.5$, target $x=1.0$, so we need 2 total steps.
2. First step to $x_1 = 0 + 0.5 = 0.5$: Calculate the slope at $(x_0, y_0)$:

   $$f(0,1) = 2(0) - 1 = -1$$
3. Then calculate the approximation for $y_1$:

   $$y_1 = 1 + (0.5)(-1) = 0.5$$
4. Second step to $x_2 = 0.5 + 0.5 = 1.0$ (our target): Calculate the slope at $(x_1, y_1)$:

   $$f(0.5, 0.5) = 2(0.5) - 0.5 = 0.5$$
5. Calculate the final approximation:

   $$y_2 = 0.5 + (0.5)(0.5) = 0.75$$
6. Final result: $y(1.0) \approx 0.75$

> **Exam tip:** Always write down the number of steps you need before starting calculations. It is extremely common to do one fewer step than required, especially when step size is a fraction and target x is a whole number.

## Identifying Over- and Under-Approximation

After calculating an approximation, AP questions often ask if it is higher or lower than the exact value. This depends on the concavity of the solution over the entire interval, since each step uses the tangent line at the start of the step to approximate the whole interval:

- If $y'' > 0$ (concave up), tangent lines lie below the solution curve, so the approximation is an **under-approximation**
- If $y'' < 0$ (concave down), tangent lines lie above the solution curve, so the approximation is an **over-approximation**

To find the sign of $y''$, differentiate the original differential equation implicitly, then substitute $y' = f(x,y)$ to get $y''$ in terms of $x$ and $y$ before checking its sign.

**Worked example:** For the IVP $\frac{dy}{dx} = 2x - y$, $y(0) = 1$, the approximation for $y(1.0)$ is 0.75. Is this an over- or under-approximation of the exact value?

1. Differentiate the differential equation to find $y''$:

   $$y' = 2x - y \implies y'' = 2 - y'$$
2. Substitute $y' = 2x - y$ into the expression for $y''$:

   $$y'' = 2 - (2x - y) = 2 - 2x + y$$
3. Check the sign of $y''$ over $[0, 1]$: $y(x)$ is positive on this interval, so $y''$ is always positive: at $x=0$, $y'' = 3 > 0$; at $x=1$, $y'' = y > 0$.
4. Since $y'' > 0$ everywhere on $[0,1]$, the solution is concave up, so 0.75 is an under-approximation, and the exact value of $y(1.0)$ is greater than 0.75.

> **Exam tip:** You can only conclude over/under approximation if $y''$ has constant sign over the entire interval. If concavity changes, you cannot make a general claim about the final approximation.

## Step Size and Error Behavior

> **warning**
>
> Do not confuse Euler's first-order error with second-order error for higher-order methods (like Runge-Kutta) that are not tested on AP BC. Only Euler's linear proportionality to step size is tested.

Euler's method is a first-order numerical method, meaning total global error is approximately proportional to step size $h$, or $E = k \cdot h$ where $k$ is constant for a given IVP and target. Smaller step size always produces a more accurate approximation, and on the AP exam you will never need more than 3-4 steps for any question.

**Worked example:** An Euler approximation of $y(2)$ with step size $h=0.4$ has a total error of 0.12. Assuming error follows the expected proportionality for Euler's method, what is the expected error when using step size $h=0.1$ to approximate $y(2)$?

1. Recall that for Euler's method, global error $E = k \cdot h$, where $k$ is constant for this IVP and target.
2. Solve for $k$ using the known error:

   $$0.12 = k(0.4) \implies k = \frac{0.12}{0.4} = 0.3$$
3. Calculate the expected error for $h=0.1$:

   $$E = k \cdot h = 0.3(0.1) = 0.03$$
4. The expected error for step size 0.1 is 0.03.

## Common pitfalls

- **Wrong:** Stopping one step early because you count the starting point as step 1, e.g. stopping at $x=0.5$ when targeting $y(1)$ with $h=0.5$ starting at $x_0=0$.
  - Why it fails: Students forget the initial point is step 0, not step 1, so they miscount the number of steps needed.
  - Correct: Before starting any calculation, explicitly calculate $n = \frac{x_{\text{target}} - x_0}{h}$ to get the required number of steps, then cross off each step as you complete it.
- **Wrong:** Using $f(x_k, y_k)$ instead of $f(x_{k-1}, y_{k-1})$ when calculating $y_k$.
  - Why it fails: Students mix up the recursive order; $y_k$ is unknown when calculating the step.
  - Correct: Always remember you use the slope at the *start* of the step (the known point you already have) to calculate the y-value at the end of the step.
- **Wrong:** Claiming a concave up solution gives an over-approximation, or a concave down solution gives an under-approximation.
  - Why it fails: Students mix up the position of the tangent line relative to the solution curve.
  - Correct: Draw a quick sketch: concave up curves bend upward above their tangent lines, so the approximation is low; concave down curves bend below their tangent lines, so the approximation is high.
- **Wrong:** Stating that cutting the step size in half cuts the error by one-quarter for Euler's method.
  - Why it fails: Students confuse Euler's first-order error with second-order error for untested higher-order methods.
  - Correct: Remember Euler's method is first-order: error scales linearly with step size, so halving $h$ halves the error.
- **Wrong:** Calculating step size as $h = \frac{n}{x_{\text{target}} - x_0}$ instead of $h = \frac{x_{\text{target}} - x_0}{n}$.
  - Why it fails: Students mix up the formula when given a fixed number of steps $n$.
  - Correct: Step size is total distance to the target divided by number of steps, so the difference in x goes in the numerator.
- **Wrong:** Leaving $y'$ in the expression for $y''$ when checking concavity, so you cannot determine the sign of $y''$.
  - Why it fails: Students stop after implicit differentiation and forget to substitute the original differential equation.
  - Correct: After finding $y'' = ... + ...y'$, always substitute $y' = f(x,y)$ to get $y''$ in terms of $x$ and $y$ before checking its sign.

## Cheatsheet

| Category | Formula | Notes |
| --- | --- | --- |
| Step Size (n steps from $x_0$ to $x_t$) | $h = \frac{x_t - x_0}{n}$ | Calculate this first; equal step size is assumed for all AP questions |
| Recursive Euler Step | $y_k = y_{k-1} + h \cdot f(x_{k-1}, y_{k-1})$ | Use slope at the start (known point) of the step |
| Second Derivative for Concavity | $y'' = f_x + f_y \cdot f(x,y)$ | Substitute $y' = f(x,y)$ before checking sign |
| Over-Approximation Condition | $y'' < 0$ on entire interval | Concave down: tangent line lies above curve |
| Under-Approximation Condition | $y'' > 0$ on entire interval | Concave up: tangent line lies below curve |
| Euler Global Error | $E \propto h$ | Error scales linearly with step size |
| Euler's Method Input | $\frac{dy}{dx} = f(x,y), \; y(x_0) = y_0$ | Only for first-order initial value problems |

## What's next

Mastering Euler's method is a critical prerequisite for the remaining topics in Unit 7 (Differential Equations), specifically applied modeling with differential equations, where numerical approximations are often required when exact solutions are too complex to derive. This topic builds on your prior knowledge of tangent line approximations, extending that local linear idea to an iterative process that can reach any desired x-value. It also introduces core numerical analysis concepts widely used in engineering, biology, and physics for solving real-world differential equations that lack closed-form exact solutions. Without mastering the recursive step and error behavior of Euler's method, you will lose easy points on both MCQ and FRQ questions that explicitly test this BC-only topic.

- [General solutions via separation of variables](https://www.owlsprep.com/study/ap-calculus-bc-u7-general-solutions-via-separation-of/)
- [Particular solutions with initial conditions](https://www.owlsprep.com/study/ap-calculus-bc-u7-particular-solutions-with-initial-conditions/)
- [Exponential models with differential equations](https://www.owlsprep.com/study/ap-calculus-bc-u7-exponential-models-with-differential-equations/)

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