Numerical differentiation in MATLAB - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of forward difference approximation

What is the output of this MATLAB code that approximates the derivative of f(x) = x^2 at x = 2 using the forward difference method with h = 0.01?

MATLAB

f = @(x) x.^2;
x = 2;
h = 0.01;
derivative = (f(x + h) - f(x)) / h;
disp(derivative);

A 4.01

B 4.00

C 3.99

D Error: Undefined function or variable 'f'

💡 Hint

Recall the formula for forward difference: (f(x+h) - f(x)) / h.

The exact derivative of x^2 at x=2 is 4. The forward difference with h=0.01 gives approximately (4.0401 - 4)/0.01 = 4.01.

❓ Predict Output

intermediate

Output of central difference approximation

What is the output of this MATLAB code that approximates the derivative of f(x) = sin(x) at x = pi/4 using the central difference method with h = 0.001?

MATLAB

f = @(x) sin(x);
x = pi/4;
h = 0.001;
derivative = (f(x + h) - f(x - h)) / (2*h);
disp(derivative);

A 0.7060

B 0.7070

C 0.7071

D Error: Index exceeds matrix dimensions.

💡 Hint

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

The exact derivative of sin(x) at pi/4 is cos(pi/4) ≈ 0.7071. The central difference with h=0.001 approximates this closely.

🔧 Debug

advanced

Identify the error in backward difference code

This MATLAB code attempts to compute the derivative of f(x) = exp(x) at x = 1 using the backward difference method with h = 0.01. What error will it produce when run?

MATLAB

f = @(x) exp(x);
x = 1;
h = 0.01;
derivative = (f(x) - f(x - h)) / h;

A Index exceeds matrix dimensions.

B No error; outputs approximately 2.7183

C SyntaxError: Missing semicolon

D Undefined function or variable 'h'

💡 Hint

Check if the code syntax is correct and variables are defined.

The code correctly computes the backward difference and outputs approximately exp(1) ≈ 2.7183. No error occurs.

❓ Predict Output

advanced

Output of second derivative approximation

What is the output of this MATLAB code that approximates the second derivative of f(x) = x^3 at x = 2 using the central difference formula with h = 0.001?

MATLAB

f = @(x) x.^3;
x = 2;
h = 0.001;
second_derivative = (f(x + h) - 2*f(x) + f(x - h)) / (h^2);
disp(second_derivative);

A 12.0000

B Error: Undefined function or variable 'second_derivative'

C 13.0000

D 11.9990

💡 Hint

Recall the second derivative central difference formula: (f(x+h) - 2f(x) + f(x-h)) / h^2.

The exact second derivative of x^3 is 6x. At x=2, it is 12. The central difference with h=0.001 gives approximately 12.0000.

🧠 Conceptual

expert

Error behavior in numerical differentiation

Which statement correctly describes the error behavior when using the central difference method to approximate the first derivative of a smooth function as the step size h approaches zero?

A The error increases exponentially as h decreases.

B The error is constant regardless of h.

C The error always decreases as h approaches zero without limit.

D The error decreases linearly with h and then increases due to round-off errors for very small h.

💡 Hint

Consider both truncation and round-off errors in numerical differentiation.

Central difference error has two parts: truncation error decreases with smaller h, but round-off error increases for very small h due to floating-point precision limits, causing total error to first decrease then increase.

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