Errors and Approximations

Errors in Using Derivative Rules:

  1. Algebraic Mistakes:

    • Errors can occur due to mistakes in applying derivative rules like the power rule, product rule, quotient rule, or chain rule.
    • Example: Incorrectly differentiating a function such as f(x)=x4+3x22 leading to f(x)=4x3+6x instead of the correct derivative f(x)=4x3+6x.
  2. Chain Rule Errors:

    • Mistakes in applying the chain rule, especially in complex composite functions, may result in errors in derivative calculations.
    • Example: Incorrectly applying the chain rule while differentiating a function like f(x)=sin(3x2), leading to an incorrect derivative.

Approximations Using Derivative Rules:

  1. Linear Approximation:

    • Applying linear approximations by using derivative rules to simplify functions around specific points for easier calculations.
    • Example: Approximating f(x)=x+1 around x=3 using the linear approximation L(x)=f(3)+f(3)(x3).
    • Example: Approximating f(x)=x at x=4 using the tangent line equation: L(x)=f(a)+f(a)(xa) where a=4 and f(x)=12x yields L(x)=2+14(x4).
  2. Truncation and Rounding:
    • Simplifying complex expressions by truncating higher-order terms or rounding terms involving derivatives for computational ease.
    • Example: Approximating f(x)=e3x+ln(2x) by neglecting terms involving higher-order derivatives beyond a certain point.

Quadratic Approximation (Second Derivative):

    • Using the second derivative to construct a quadratic approximation around a point for better accuracy than linear approximations.
    • Example: Approximating f(x)=ln(x) near x=1 using a quadratic approximation: Q(x)=f(a)+f(a)(xa)+f(a)2!(xa)2 where a=1 yields Q(x)=(x1)12(x1)2.

Example:

Let's consider the function f(x)=e2x+cos(3x). We want to find an approximation of the derivative of f(x) at x=0.1 using the linear approximation.

  1. Derivative Calculation:

    • Calculate the derivative of f(x) using the rules of differentiation: f(x)=2e2x3sin(3x).
  2. Linear Approximation:

    • Evaluate f(0.1) using the derivative obtained earlier: f(0.1)=2e2(0.1)3sin(3(0.1)).
    • This gives an exact value for f(0.1).
  3. Using Approximations:

    • To approximate f(0.1) without using exact calculations, one could employ linear approximation if exact calculations are challenging due to complexity or computational limitations.