Numerical differentiation in MATLAB - Interactive Code Practice

Practice - 5 Tasks

1fill in blank

easy

Complete the code to calculate the forward difference approximation of the derivative.

MATLAB

h = 0.01;
x = 1;
f = @(x) x^2;
derivative = (f(x + [1]) - f(x)) / [1];

Drag options to blanks, or click blank then click option'

Ax

B0.1

Ch

D0

Attempts:

3 left

💡 Hint

Common Mistakes

Using x instead of h as the step size.

Using zero as the step size, which causes division by zero.

The forward difference formula uses a small step size h to approximate the derivative as (f(x+h) - f(x))/h.

2fill in blank

medium

Complete the code to calculate the central difference approximation of the derivative.

MATLAB

h = 0.01;
x = 2;
f = @(x) sin(x);
derivative = (f(x + [1]) - f(x - [1])) / (2 * h);

Drag options to blanks, or click blank then click option'

A2

B0.1

Cx

Dh

Attempts:

3 left

💡 Hint

Common Mistakes

Using different values for the forward and backward steps.

Using x instead of h for the step size.

The central difference formula uses (f(x+h) - f(x-h)) divided by 2h to approximate the derivative.

3fill in blank

hard

Fix the error in the code to correctly compute the numerical derivative using backward difference.

MATLAB

h = 0.001;
x = 3;
f = @(x) exp(x);
derivative = (f(x) - f(x [1] h)) / h;

Drag options to blanks, or click blank then click option'

A-

B+

C*

D/

Attempts:

3 left

💡 Hint

Common Mistakes

Using plus instead of minus, which changes the formula.

Using multiplication or division operators incorrectly.

The backward difference formula subtracts the function value at (x - h) from f(x), so the operator must be '-'.

4fill in blank

hard

Fill both blanks to create a function that returns the central difference derivative approximation.

MATLAB

function d = central_diff(f, x, [1])
    d = (f(x + [2]) - f(x - h)) / (2 * h);
end

Drag options to blanks, or click blank then click option'

Ah

Bx

C0.01

Df

Attempts:

3 left

💡 Hint

Common Mistakes

Using x or a constant instead of the step size variable.

Using the function name f instead of the step size.

The function uses h as the step size, so both blanks should be h to correctly compute the central difference.

5fill in blank

hard

Fill all three blanks to create a function that computes the forward difference derivative with input validation.

MATLAB

function d = forward_diff([1], [2], [3])
    if h <= 0
        error('Step size must be positive');
    end
    d = (f(x + h) - f(x)) / h;
end

Drag options to blanks, or click blank then click option'

Af

Bx

Ch

Dd

Attempts:

3 left

💡 Hint

Common Mistakes

Mixing the order of inputs.

Using variable names not related to the function inputs.

The function inputs are the function handle f, the point x, and the step size h, in that order.

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