Differentials & Euler's Method

Front

What is a differential?

Back

A differential represents an infinitesimally small change in a function's output relative to a small change in its input, denoted as dy = f'(x)dx.

Front

What is Euler's Method?

Back

Euler's Method is a numerical technique used to approximate solutions of ordinary differential equations (ODEs) using tangent line approximations.

Front

How do you calculate the differential of a function?

Back

To calculate the differential of a function f(x), use the formula: dy = f'(x)dx, where f'(x) is the derivative of f with respect to x.

Front

What is the formula for the area of a circle?

Back

The area A of a circle is given by the formula A = πr^2, where r is the radius.

Front

How do you approximate changes in area using differentials?

Back

To approximate the change in area (dA) when the radius changes (dr), use the formula: dA = 2πr dr.

Front

What is the derivative of f(x) = x^(1/3)?

Back

The derivative f'(x) = (1/3)x^(-2/3).

Front

What is the value of f(4) for the function f(x) = x^2 - 2x + 3?

Back

f(4) = 4^2 - 2(4) + 3 = 3.

Front

What is the relationship between dx and Δx?

Back

dx represents an infinitesimally small change in x, while Δx represents a finite change in x.

Front

How do you find Δy for a function given Δx?

Back

To find Δy, use the formula: Δy = f'(x)Δx, where f'(x) is the derivative at the point of interest.

Front

What is the significance of the point (4, f(4)) in calculus?

Back

The point (4, f(4)) is used as a reference point for calculating differentials and approximating changes in the function.