Numerical differentiation in MATLAB - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is numerical differentiation?

Numerical differentiation is a way to find the slope or rate of change of a function using numbers instead of formulas. It estimates the derivative by using values of the function at certain points.

beginner

What is the forward difference formula for numerical differentiation?

The forward difference formula estimates the derivative at a point x by using the function values at x and a point slightly ahead, x + h: f'(x) ≈ (f(x + h) - f(x)) / h, where h is a small step size.

intermediate

Why do we choose a small step size h in numerical differentiation?

A small step size h helps get a better estimate of the derivative because it measures the change over a tiny interval. But if h is too small, rounding errors can happen, so we balance between accuracy and errors.

intermediate

What is the central difference formula and why is it better than forward difference?

The central difference formula uses points before and after x: f'(x) ≈ (f(x + h) - f(x - h)) / (2h). It is usually more accurate than forward difference because it balances the slope from both sides.

beginner

Show a simple MATLAB code snippet to compute the numerical derivative of sin(x) at x = pi/4 using central difference.

h = 1e-5; x = pi/4; derivative = (sin(x + h) - sin(x - h)) / (2 * h); disp(derivative);

Which formula uses points on both sides of x to estimate the derivative?

A Central difference

B Forward difference

C Backward difference

D Euler's method

Central difference uses f(x + h) and f(x - h), points on both sides of x, to estimate the derivative.

In numerical differentiation, what happens if the step size h is too large?

A No effect on accuracy

B The estimate becomes more accurate

C The estimate becomes less accurate

D The function value becomes zero

A large h measures change over a big interval, which can miss details and reduce accuracy.

What MATLAB command displays the value of a variable on the screen?

A print()

B echo()

C show()

D disp()

disp() is used in MATLAB to display values or messages.

Which of these is NOT a common numerical differentiation method?

A Forward difference

B Integration by parts

C Backward difference

D Central difference

Integration by parts is a calculus technique, not a numerical differentiation method.

Why might very small h cause problems in numerical differentiation?

A Rounding errors increase

B Function values become negative

C Derivative becomes zero

D Step size is ignored

Very small h can cause rounding errors due to limited computer precision.

Explain how to estimate the derivative of a function at a point using numerical differentiation.

Think about how you find slope using points close to the target point.

Describe the difference between forward difference and central difference methods.

Consider which points around x each method uses.