2. Give an example of an integral that is impossible to solve without using Taylor series. Approximate this integral over the range 0 to 1.

Integrals that are impossible to solve are usually ones that do not have a simple antiderivative. One example is the integral of ln(1+x) dx from 0 to 1. ln does not have a simple antiderivative so we have to use the Taylor series to solve.

3. Give an example of the use of a Taylor Series in finance

From researching, Finance uses the Taylor Series. The Taylor series is the main way to manage risk in financial markets. It is also used to approximate the movement in value options and bonds. The simple way is to say that the change in price or the change in yield will change the value of the option/bond, but the Taylor series shows that this is not this change is not linear. Finance normally uses the first two derivatives to calculate bond/option pricing. This helps them determine the price of the securities and evaluating the risk.

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