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Quadratic Approximation in Calculus: How to Use it, Step by Step

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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 known as Taylor polynomials. The basic idea is that you want to approximate a function with a line. However, a straight line normally won’t do, because most functions are curves.

General Form of Quadratic Approximation

The general form of a quadratic approximation is:
Q(f) = f(a) + f'(a)x + f”(a)2*x2.
If it looks complicated, don’t worry—you don’t have to solve the equation; all you have to do is insert a few terms and then graph it.

Quadratic Approximation: Example

Example problem: Find the quadratic approximation for f(x) = ex + x2 near x = 0

Step 1: Find the derivative of the function. You’re solving for the second part of the general form (f'(a)x). For this example, use the chain rule. Click here for an explanation of the chain rule.
f’ ex + x2 = ex + x2 * (1 + 2x) = (2x + 1) ex + x2

Step 2: Find the second derivative of the function. In other words, find the derivative of the derivative you calculated in Step 1. This particular function requires use of the product rule.
f”(x) = 2ex + x2 + (2x + 1) ex + x2 * (2x + 1)
Combining terms: f”(x) = (4x2 + 4x + 3) ex + x2

Step 3: Find values at x = 0 for the function, and the first and second derivatives you calculated in Steps 1 and 2:

  • f(0) = e0 + 02 = 1
  • f'(0)= (2(1) + 1)e1 + 1) = 3e2
  • f”(0)= (3) * e0) = 3

Step 4: Take the three values you calculated in Step 3 and insert them into the general formula for Q(f):
Q(f) = f(a) + f'(a)x + f”(a)⁄2*x2
= 1 + x + + 3x22

The quadratic approximation (green) for the original function (red).

The quadratic approximation (green) for the original function (red).

That’s it!

Use of Quadratic Approximation

In addition to modeling functions, approximations are used to:

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

Stephanie Glen. "Quadratic Approximation in Calculus: How to Use it, Step by Step" From Calculus for the rest of us!

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