UNIT 2 - PART 3 NOTES

2.7 Evaluation of the Derivatives of a Polynomial

Definition of Derivatives of a Polynomial

• "Derivatives of a polynomial" refers to the process of finding the rate of change of a polynomial function.

Polynomials

• Expressions consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents.

Derivatives

• Measures how the function's output changes as its input changes.

• Represents the slope of the tangent line to the function's graph at any given point.

Applications of Derivatives of Polynomials

1. Optimization: Used to find local maxima and minima by identifying where the derivative is zero.

2. Root Finding: Many algorithms, like Newton-Raphson, use derivatives to approximate roots.

3. Taylor Series: Approximates functions using an infinite sum of terms, requiring derivative evaluation.

4. Curve Fitting: Helps fit polynomial curves to data.

5. Computer Graphics: Ensures smooth curves and surfaces.

Methods for Evaluating Derivatives

1. Symbolic Differentiation

2. Horner's Method (with Modification)

3. Algorithmic Differentiation (Automatic Differentiation)

   • Breaks down complex functions into elementary operations and applies the chain rule repeatedly.

   • Evaluates both the polynomial value and its derivative at a specific point.

Key Differentiation Rules

1. Power Rule: ddx(xn)=nxn−1\frac{d}{dx} (x^n) = nx^{n-1}

2. Constant Rule: ddx(c)=0\frac{d}{dx} (c) = 0 where c is a constant.

3. Sum/Difference Rule: The derivative of a sum or difference of terms is the sum or difference of their derivatives.

4. Constant Multiple Rule: ddx(cf(x))=c⋅ddx(f(x))\frac{d}{dx} (c f(x)) = c \cdot \frac{d}{dx} (f(x))

Examples

1. Example 1:

   • ddx(x3)=3x2\frac{d}{dx} (x^3) = 3x^2

2. Example 2:

   • ddx(x)=1\frac{d}{dx} (x) = 1

3. Constant Multiple Rule Example:

   • Function: y=5x2y = 5x^2

   • Derivative: ddx(5x2)=10x\frac{d}{dx} (5x^2) = 10x

4. Sum Rule Example:

   • y=x3+4x2y = x^3 + 4x^2

   • Derivative: ddx(x3)+ddx(4x2)=3x2+8x\frac{d}{dx} (x^3) + \frac{d}{dx} (4x^2) = 3x^2 + 8x

5. Combined Example:

   • f(x)=5x4−3x2+2x−7f(x) = 5x^4 - 3x^2 + 2x - 7

   • f′(x)=20x3−6x+2f'(x) = 20x^3 - 6x + 2

Algorithmic Differentiation Process (Forward Mode)

Example Polynomial: f(x)=x2+3x+2f(x) = x^2 + 3x + 2

• Evaluate at x=4x = 4.

• Break Down into Elementary Operations:

  • v1=x⋅xv_1 = x \cdot x

  • v2=3⋅xv_2 = 3 \cdot x

  • v3=v1+v2v_3 = v_1 + v_2

  • v4=v3+2v_4 = v_3 + 2

• Forward Pass (Value and Derivative):

  • f(4)=30f(4) = 30

  • f′(4)=11f'(4) = 11

2.8 Evaluation of Polynomials with Complex Argument

Polynomials

• Expressions involving variables and coefficients, combined using addition, subtraction, and non-negative integer exponents.

Complex Numbers

• Extend the real number system by including the imaginary unit ii, where i2=−1i^2 = -1.

• A complex number is written as a+bia + bi, where 'a' and 'b' are real numbers.

Polynomials with Complex Arguments

• The variable (argument) can be a complex number.

• The coefficients can also be complex numbers.

Example of a Polynomial with a Complex Argument

• Polynomial: P(z)=z2+(2+i)z−3iP(z) = z^2 + (2 + i)z - 3i

• Evaluate P(z)P(z) when z=1+iz = 1 + i

  • P(1+i)=(1+i)2+(2+i)(1+i)−3iP(1 + i) = (1 + i)^2 + (2 + i)(1 + i) - 3i

  • Expand and simplify:

    • (1+i)2=2i(1 + i)^2 = 2i

    • (2+i)(1+i)=1+3i(2 + i)(1 + i) = 1 + 3i

    • P(1+i)=1+2i+3i−3i=1+2iP(1 + i) = 1 + 2i + 3i - 3i = 1 + 2i

END OF UNIT 2