Numerical Differentiation & Integration: Rules, Errors & Applications

TAKEAWAYS:

• Numerical differentiation approximates derivatives using finite differences (forward, backward, central) with error bounds $O(h^2)$ or $O(h)$.
• Integration rules (Trapezoidal, Simpson’s) convert continuous functions into discrete sums, with Simpson’s 1/3 rule offering $O(h^5)$ accuracy for smooth functions.
• Error analysis distinguishes truncation, rounding, and propagation errors, critical for validating computational results.
• Real-world applications include calculating loan interest (integration), optimizing traffic routes (differentiation), and modeling stock prices (ODEs).
• Trapezoidal rule is simple but less accurate ($O(h^3)$) than Simpson’s rules for smooth functions.
• Always check function smoothness and step size ($h$) when choosing a method—smaller $h$ reduces error but increases computation time.

Core Concepts: Why Numerical Methods?

Numerical methods solve mathematical problems approximately when exact solutions are impossible or impractical. For differentiation and integration:

• Differentiation: Finds the rate of change of a function (e.g., velocity from position, slope of a curve).
• Integration: Computes areas under curves (e.g., total distance traveled, accumulated interest).

Why not exact methods?

• Some functions (e.g., $f(x)=e^{-x^2}$) have no closed-form antiderivative.
• Real-world data is often discrete (e.g., sensor readings), not continuous.

1. Numerical Differentiation: Finite Difference Methods

Key Idea

Approximate the derivative $f^{\prime }(x)$ using function values at nearby points. The error depends on the step size $h$ and the method used.

Three Methods

Visual: Error vs. Step Size Key Insight: Central difference is twice as accurate for the same $h$ because its error term is $O(h^2)$.

Worked Example: Loan Interest Rate Calculation (Real-World Tie-In)

Problem: A bank offers a loan with monthly interest calculated as $I(t)=1000\cdot e^{0.05t}$, where $t$ is time in years. Estimate the interest rate of change at $t=1$ year using:

1. Forward difference ($h=0.1$),
2. Central difference ($h=0.1$).

Solution:

1. Forward Difference: $$I^{\prime }(1)\approx \frac{I(1.1)-I(1)}{0.1}=\frac{1000(e^{0.055}-e^{0.05})}{0.1}\approx 52.76$$
2. Central Difference: $$I^{\prime }(1)\approx \frac{I(1.1)-I(0.9)}{0.2}=\frac{1000(e^{0.055}-e^{0.045})}{0.2}\approx 52.52$$ (Exact value: $I^{\prime }(1)=1000\cdot 0.05\cdot e^{0.05}\approx 52.51$.)

Visual: Interest Growth Curve Exam Tip: Always compare methods! Central difference is preferred unless data is limited.

2. Numerical Integration: Rules for Area Under Curves

Key Idea

Approximate ${\int }_a^{b}f(x)dx$ by dividing $[a,b]$ into subintervals and summing areas of simple shapes (rectangles, trapezoids, parabolas).

Three Rules

Visual: Rule Comparison

Worked Example: Daraz Order Processing Time (Real-World Tie-In)

Problem: Daraz’s order processing time $T(t)$ (in minutes) follows $T(t)=10+5t-0.2t^2$ for $0\le t\le 5$ hours. Estimate the total processing time for 5 hours using:

1. Trapezoidal rule ($n=5$),
2. Simpson’s 1/3 rule ($n=5$).

Solution:

1. Trapezoidal Rule:

   • Divide $[0,5]$ into 5 subintervals: $h=1$.
   • Values: $T(0)=10$, $T(1)=14.8$, $T(2)=18.8$, $T(3)=22$, $T(4)=24.4$, $T(5)=26$.
   • Sum: $$\frac{1}{2}\left[10+2(14.8+18.8+22+24.4)+26\right]=98.4$$

2. Simpson’s 1/3 Rule:

   • Same $h=1$, but weights: $1,4,2,4,2,4,1$.
   • Sum: $$\frac{1}{3}\left[10+4(14.8+22+26)+2(18.8+24.4)\right]=98.33$$
   • Exact integral: ${\int }_0^5(10+5t-0.2t^2)dt=98.33$.

Visual: Order Processing Curve

3. Error Analysis in Numerical Methods

Types of Errors

Visual: Error Propagation

flowchart TD
    A["Initial Error"] --> B["Step 1\n(Addition)"]
    B --> C["Step 2\n(Multiplication)"]
    C --> D["Step 3\n(Division)"]
    D --> E["Final Error\n(Grows with steps)"]

Worked Example: Ncell Data Usage Calculation

Problem: Ncell’s daily data usage $D(t)$ (in GB) follows $D(t)=2+0.5t-0.01t^2$ for $0\le t\le 24$ hours. Estimate the total data used in a day using:

1. Trapezoidal rule ($n=4$),
2. Simpson’s 1/3 rule ($n=4$), and compare to the exact integral.

Solution:

1. Exact Integral: $${\int }_0^{24}(2+0.5t-0.01t^2)dt=2t+0.25t^2-\frac{0.01}{3}{t}^3{|}_0^{24}=48+14.4-57.6=4.8\text{ GB}$$

2. Trapezoidal Rule ($h=6$):

   • Values: $D(0)=2$, $D(6)=4.8$, $D(12)=6.8$, $D(18)=8.2$, $D(24)=9$.
   • Sum: $$\frac{6}{2}\left[2+2(4.8+6.8+8.2)+9\right]=4.9\text{ GB}$$
   • Error: $|4.9-4.8|=0.1$ GB.

3. Simpson’s 1/3 Rule ($h=6$):

   • Sum: $$\frac{6}{3}\left[2+4(4.8+8.2)+2(6.8)+9\right]=4.8\text{ GB}$$
   • Error: $0$ (exact for this case).

Visual: Data Usage Curve

## In the Real World

1. eSewa/Khalti Payments:

   • Idea Used: Numerical integration calculates total transaction fees over time for dynamic pricing models.
   • How: If fee $F(t)=1+0.02t$ (in NPR), the total fee for a month is ${\int }_0^{30}F(t)dt$, approximated using Simpson’s rule for efficiency.

2. Pathao Driver Earnings:

   • Idea Used: Numerical differentiation estimates instantaneous demand growth (e.g., $D^{\prime }(t)$) to optimize driver dispatch.
   • How: If demand $D(t)=50+2t-0.1t^2$ (rides/hour), $D^{\prime }(t)$ is approximated using central differences to predict peak times.

3. NTC Electricity Billing:

   • Idea Used: Trapezoidal rule approximates energy consumption for time-of-use tariffs.
   • How: For a household with usage $U(t)=10+0.5\sin (t)$ (kWh), monthly consumption is ${\int }_0^{720}U(t)dt$ (720 hours), computed via trapezoidal rule for billing.

## Exam Tip: How to Score Full Marks

1. For Differentiation:

   • Always state the method (forward/backward/central) and step size $h$.
   • Show the formula and substitute values clearly.
   • Compare with exact value if possible (e.g., $f(x)=x^2$ has $f^{\prime }(x)=2x$).

2. For Integration:

   • Label subintervals $x_0,x_1,\dots ,x_n$ and compute $h=\frac{b-a}{n}$.
   • For Simpson’s 1/3, alternate weights (4, 2, 4, 2, ...) and include endpoints.
   • Verify with exact integral if the function is simple (e.g., polynomials).

3. Error Analysis:

   • Distinguish truncation (method error) vs. rounding (precision error).
   • For integration, state the error order (e.g., "Trapezoidal rule has $O(h^2)$ error").

4. Real-World Applications:

   • Tie examples to Nepal: Use Ncell, NTC, or Daraz in problems.
   • Show units: If integrating time, answer should be in "hours" or "minutes," not just numbers.

5. Algorithm Questions:

   • Write pseudo-code with steps like:

     1. Input: f(x), a, b, n
     2. h = (b - a)/n
     3. sum = f(a) + f(b)
     4. For i = 1 to n-1:
            sum += 2*f(a + i*h)  [Trapezoidal]
            OR sum += 4*f(a + i*h) if i odd, 2*f(...) if i even  [Simpson’s]
     5. Return (h/2)*sum  [Trapezoidal] or (h/3)*sum  [Simpson’s]
     

Final Visual: Method Selection Guide

mindmap
  root((Choose Method))
    Differentiation
      Forward: Limited data
      Backward: Past data only
      Central: Best accuracy
    Integration
      Trapezoidal: Quick estimate
      Simpson’s 1/3: Smooth functions
      Simpson’s 3/8: Rare cases
    Error
      Small h: Higher accuracy
      Large h: Faster but less precise