### All Calculus 2 Resources

## Example Questions

### Example Question #1 : Euler's Method And L'hopital's Rule

Suppose we have the following differential equation with the initial condition:

Use Euler's method to approximate , using a step size of .

**Possible Answers:**

**Correct answer:**

We start at x = 0 and move to x=2, with a step size of 1. Essentially, we approximate the next step by using the formula:

.

So applying Euler's method, we evaluate using derivative:

And two step sizes, at x = 1 and x = 2.

And thus the evaluation of p at x = 2, using Euler's method, gives us p(2) = 2.

### Example Question #1 : Euler's Method

Approximate by using Euler's method on the differential equation

with initial condition (which has the solution ) and time step .

**Possible Answers:**

**Correct answer:**

Using Euler's method with means that we use two iterations to get the approximation. The general iterative formula is

where each is

is an approximation of , and , for this differential equation. So we have

So our approximation of is

### Example Question #3 : Euler's Method

Use Euler's method to find the solution to the differential equation at with the initial condition and step size .

**Possible Answers:**

**Correct answer:**

Euler's method uses iterative equations to find a numerical solution to a differential equation. The following equations

are solved starting at the initial condition and ending at the desired value. is the solution to the differential equation.

In this problem,

Starting at the initial point

We continue using Euler's method until . The results of Euler's method are in the table below.

Note: Solving this differential equation analytically gives a different answer, . Future problems will explain this discrepancy.

### Example Question #4 : Euler's Method

Use Euler's method to find the solution to the differential equation at with the initial condition and step size .

**Possible Answers:**

**Correct answer:**

Euler's method uses iterative equations to find a numerical solution to a differential equation. The following equations

are solved starting at the initial condition and ending at the desired value. is the solution to the differential equation.

In this problem,

Starting at the initial point

We continue using Euler's method until . The results of Euler's method are in the table below.

Note: Solving this differential equation analytically gives a different answer, . As the step size gets larger, Euler's method gives a more accurate answer.

### Example Question #5 : Euler's Method

Use Euler's method to find the solution to the differential equation at with the initial condition and step size .

**Possible Answers:**

**Correct answer:**

Euler's method uses iterative equations to find a numerical solution to a differential equation. The following equations

are solved starting at the initial condition and ending at the desired value. is the solution to the differential equation.

In this problem,

Starting at the initial point

We continue using Euler's method until . The results of Euler's method are in the table below.

### Example Question #6 : Euler's Method

**Possible Answers:**

**Correct answer:**

In this problem,

Starting at the initial point

We continue using Euler's method until . The results of Euler's method are in the table below.

Note: Due to the simplicity of the differential equation, Euler's method finds the exact solution, even with a large step size, Using a smaller step size is unnecessary and more time consuming.

### Example Question #7 : Euler's Method

**Possible Answers:**

**Correct answer:**

In this problem,

Starting at the initial point

We continue using Euler's method until . The results of Euler's method are in the table below.

### Example Question #8 : Euler's Method

**Possible Answers:**

**Correct answer:**

In this problem,

Starting at the initial point

We continue using Euler's method until . The results of Euler's method are in the table below.

### Example Question #1 : New Concepts

**Possible Answers:**

**Correct answer:**

In this problem,

Starting at the initial point

We continue using Euler's method until . The results of Euler's method are in the table below.

### Example Question #10 : Euler's Method

**Possible Answers:**

**Correct answer:**

In this problem,

Starting at the initial point

We continue using Euler's method until . The results of Euler's method are in the table below.

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