Calculus How To

Quadratic Approximation: Step by Step Example

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The quadratic approximation is one way to approximate a curve. Quadratic polynomial approximations are specific examples of a useful class of quadratic approximations called Taylor polynomials.

The basic idea is that you want to approximate a function with a parabola.

General Form of Quadratic Approximation

The general form of a quadratic approximation is:


If it looks complicated, don’t worry: you don’t have to solve the equation; all you have to do is plug in some terms.

Let’s say you were trying to approximate a function at x = 1. The formula is basically saying to find three values at point x = 1 and add them up:

Quadratic Approximation: Example

Example problem: Find the quadratic approximation for f(x) = xe-2x near x = 1

Step 1: Find the first derivative of the function. Use the product rule for this function (with x and e-2x) and then the chain rule (for e-2x):

f′(x) = e-2x – 2xe-2x = e-2x(1 – 2x).

Step 2: Find the second derivative of the function. In other words, find the derivative of the derivative you calculated in Step 1.

f′′(x) = -2e-2x – 2(e-2x – 2xe-2x)
= -4e-2x + 4xe-2x
= -4e-2x(1 – x).

Step 3: Find values at x = 1 for the function, and the first and second derivatives you calculated in Steps 1 and 2:

  1. f(1) = (1)e-2(1) = e-2
  2. f′(1) = e-2(1)(1 – 2(1)) = -e-2
  3. f′′(1) = -4(1)-2(1)(1 – (1)) = 0.

Step 4: Take the three values you calculated in Step 3 and insert them into the general formula . Note that the second derivative (from Step 3) was zero, so we can ignore the third part of the formula:
f(x) ≈ f(1) + f′(1)(x – 1)
≈ e-2 – e – 2(x – 1).


That’s it!

quadratic approximation graph

The quadratic approximation is a good fit near x = 1 (Graphed with Desmos).

Use of Quadratic Approximation

In addition to modeling functions, approximations are used to:

Next: How to use Taylor polynomials to approximate a function.

References

Desmos Graphing Calculator.

CITE THIS AS:
Stephanie Glen. "Quadratic Approximation: Step by Step Example" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/quadratic-approximation/
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