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.

Practice

1. Which method in numerical differentiation uses points on both sides of the target point to estimate the derivative?

easy

A. Forward difference method

B. Euler's method

C. Backward difference method

D. Central difference method

Solution

1. Step 1: Understand difference methods

   Forward difference uses the point and the next point; backward difference uses the point and the previous point.

2. Step 2: Identify method using both sides

   Central difference uses points before and after the target point for better accuracy.

3. Final Answer:

   Central difference method -> Option D

4. Quick Check:

   Both sides points = Central difference [OK]

Hint: Central difference uses points on both sides [OK]

Common Mistakes:

• Confusing forward with central difference
• Thinking Euler's method is for differentiation
• Assuming backward difference uses both sides

2. Which of the following MATLAB code snippets correctly computes the forward difference approximation of the derivative of function f at point x with step size h?

easy

A. df = (f(x+h) - f(x)) / h;

B. df = (f(x) - f(x-h)) / h;

C. df = (f(x+h) - f(x-h)) / (2*h);

D. df = (f(x) + f(x+h)) / h;

Solution

1. Step 1: Recall forward difference formula

   Forward difference formula is (f(x+h) - f(x)) / h.

2. Step 2: Match code to formula

   df = (f(x+h) - f(x)) / h; matches this formula exactly.

3. Final Answer:

   df = (f(x+h) - f(x)) / h; -> Option A

4. Quick Check:

   Forward difference = (f(x+h)-f(x))/h [OK]

Hint: Forward difference subtracts current from next point [OK]

Common Mistakes:

• Using backward difference formula instead
• Dividing by 2*h instead of h
• Adding function values instead of subtracting

3. Consider the MATLAB code:

f = @(x) x^2;
x = 2;
h = 0.01;
df = (f(x+h) - f(x-h)) / (2*h);

What is the value of df approximately?

medium

A. 5.00

B. 2.00

C. 4.00

D. 3.99

Solution

1. Step 1: Calculate f(x+h) and f(x-h)

   f(2+0.01) = (2.01)^2 = 4.0401; f(2-0.01) = (1.99)^2 = 3.9601.

2. Step 2: Compute central difference

   df = (4.0401 - 3.9601) / (2*0.01) = 0.08 / 0.02 = 4.00.

3. Step 3: MATLAB precision

   In MATLAB, df is computed and displayed as 4.0000.

4. Final Answer:

   4.00 -> Option C

5. Quick Check:

   Central difference ≈ 4 [OK]

Hint: Central difference approximates slope near x [OK]

Common Mistakes:

• Using forward difference formula instead
• Forgetting to divide by 2*h
• Incorrect function evaluation

4. The following MATLAB code attempts to compute the derivative using forward difference but gives incorrect results:

f = @(x) sin(x);
x = pi/4;
h = 0.001;
df = (f(x) - f(x+h)) / h;

What is the error in this code?

medium

A. Step size h is too large

B. The subtraction order is reversed; should be f(x+h) - f(x)

C. Function handle syntax is incorrect

D. Variable x should be a vector

Solution

1. Step 1: Recall forward difference formula

   Forward difference is (f(x+h) - f(x)) / h, not (f(x) - f(x+h)) / h.

2. Step 2: Identify error in code

   The code subtracts f(x+h) from f(x), reversing the order and causing incorrect sign.

3. Final Answer:

   The subtraction order is reversed; should be f(x+h) - f(x) -> Option B

4. Quick Check:

   Forward difference = (f(x+h)-f(x))/h [OK]

Hint: Forward difference subtracts future minus current value [OK]

Common Mistakes:

• Reversing subtraction order
• Using too large step size without reason
• Misunderstanding function handle syntax

5. You want to estimate the derivative of f(x) = exp(-x^2) at x = 1 using central difference in MATLAB. Which code snippet correctly implements this with step size h = 0.0001?

hard

A. f = @(x) exp(-x^2);
x = 1;
h = 0.0001;
df = (f(x+h) - f(x-h)) / (2*h);

B. f = @(x) exp(-x^2);
x = 1;
h = 0.0001;
df = (f(x) - f(x-h)) / h;

C. f = @(x) exp(-x^2);
x = 1;
h = 0.0001;
df = (f(x+h) + f(x-h)) / (2*h);

D. f = @(x) exp(-x^2);
x = 1;
h = 0.0001;
df = (f(x+h) - f(x)) / h;

Solution

1. Step 1: Recall central difference formula

   Central difference formula is (f(x+h) - f(x-h)) / (2*h) for better accuracy.

2. Step 2: Match code to formula

   f = @(x) exp(-x^2); x = 1; h = 0.0001; df = (f(x+h) - f(x-h)) / (2*h); matches the central difference formula exactly with correct function and step size.

3. Final Answer:

   f = @(x) exp(-x^2); x = 1; h = 0.0001; df = (f(x+h) - f(x-h)) / (2*h); -> Option A

4. Quick Check:

   Central difference = (f(x+h)-f(x-h))/(2*h) [OK]

Hint: Central difference uses symmetric points around x [OK]

Common Mistakes:

• Using forward or backward difference instead
• Adding function values instead of subtracting
• Dividing by h instead of 2*h