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

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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. The quadratic approximation is one way to approximate a curve. In addition to modeling functions, approximations are used to:

The general form of quadratic approximations 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

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

Step 1: Find the derivative of the function. 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!

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