Calculus > Quadratic Approximation

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:

- Study asymptotic behavior,
- Evaluate definite integrals (i.e. integrals that have a defined starting and stopping point),
- Understanding the growth of functions,
- Solve differential equations.

The general form of quadratic approximations is Q(f) = f(a) + f'(a)x + ^{f”(a)}⁄_{2}*x^{2}. 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) = e^{x + 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’ e^{x + x2} = e^{x + x2} * (1 + 2x) = (2x + 1) e^{x + 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) = 2e^{x + x2} + (2x + 1) e^{x + x2} * (2x + 1)

*Combining terms*: f”(x) = (4x^{2} + 4x + 3) e^{x + 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) = e^{0 + 02} = 1

f'(0)= (2(1) + 1)e^{1 + 1}) = 3e^{2}

f”(0)= (3)*e^{0}) = 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*x^{2}

= 1 + x + + 3x^{2}⁄_{2}

*That’s it!*

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

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