Linear & Quadratic Approximations

Approximations allow you to find functions for values that you cannot (or don't want to, possibly) find directly.

Remember that:

Linearization (i.e. finding the tangent to a curve at a specific point, $a$) is itself a form of Polynomial Approximation.
Linear Approximation: $\quad y = f(a) + f'(a)(x-a)$
Higher-degree (e.g. quadratic) polynomials tend to make more accurate approximations.
Quadratic Approximation: $\quad y = f(a) + f'(a)(x-a) + \frac{f''(a)}{2}(x-a)^2$

Example

Approximate the function $f(x) = -4x^4-3x^3+4x^2$ at $x=-1$

The linearization of this function would be: $$y = f(a) + f'(a)(x-a)$$

The quadratic approximation of this function would be: $$y = f(a) + f'(a)(x-a) + \frac{f''(a)}{2}(x-a)^2$$



In [14]:

    
import sympy as sp
from matplotlib import pyplot as plt

%matplotlib inline

configure_matplotlib()

f = lambda x: -4*(x**4)-3*(x**3)+4*(x**2)

# First derivative of f
f1 = lambda x: -16*(x**3)-9*(x**2)+8*x

# Second derivative of f
f2 = lambda x: -48*(x**2)-18*x+8

# L(x) is a linearization of f(x) at x = -1
a = -1
L = lambda x: f(a)+f1(a)*(x-a)

# Q(x) is a quadratic approximation of f(x) at x = -1
Q = lambda x: f(a)+f1(a)*(x-a)+f2(a)/2*(x-a)**2

sp.mpmath.plot([f, L, Q], xlim=[-2,0], ylim=[-1,5], points=500)

As you can see from the plot above, the quadratic approximation is more accurate for $x$ near $a$.



In [15]:

    
def configure_matplotlib():
    # Customize figure size
    plt.rcParams['figure.figsize'] = 25, 15
    #plt.rcParams['lines.linewidth'] = 1
    #plt.rcParams['lines.color'] = 'g'
    #plt.rcParams['font.family'] = 'monospace'
    plt.rcParams['font.size'] = '16.0'
    plt.rcParams['font.monospace'] = 'Anonymous Pro, serif'
    plt.rcParams['text.hinting'] = 'either'



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