Solving f(x)=e^(x^2): A Step-by-Step Guide

Thread starter leftwing1018
Start date May 3, 2009

In summary, solving f(x)=e^(x^2) involves finding the value of x that makes the equation true, which can be helpful in solving problems related to exponential functions and finding maximum or minimum values. This can be done using algebraic manipulation or graphing techniques. The steps involved include writing the equation in the form e^(x^2) = a, taking logarithms, solving for x, and checking the solution. Special techniques such as using logarithm properties and substitution can also be used. It is possible for f(x)=e^(x^2) to have multiple solutions due to the one-to-one nature of the exponential function.

May 3, 2009

leftwing1018

Here's the problem:

If f(x)=e^(x^2), show that f^(2n)(0)=(2n)!/(n!).

Really I don't even know where to begin. Any help on where to start would be great.

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May 3, 2009

Homework Helper

Gold Member

Hi leftwing1018,

Welcome to PF. Here's a hint: write f(x) as an infinite series using the well-known series expansion of e^x, and then compare that with the Taylor expansion of a general function near x = 0.

1. What is the purpose of solving f(x)=e^(x^2)?

The purpose of solving f(x)=e^(x^2) is to find the value of x that makes the equation true. This can be helpful in solving problems related to exponential functions and can also be used to find the maximum or minimum values of a function.

2. How do I solve f(x)=e^(x^2)?

To solve f(x)=e^(x^2), you can use algebraic manipulation or graphing techniques. Algebraically, you can use logarithms to isolate the variable x. Graphically, you can use a graphing calculator or a computer program to plot the function and find the x-value where the function equals e^(x^2).

3. What are the steps involved in solving f(x)=e^(x^2)?

The steps involved in solving f(x)=e^(x^2) are as follows:

Write the equation in the form e^(x^2) = a, where a is a constant.
Take logarithms of both sides of the equation.
Solve for x using algebraic manipulation or a graphing calculator.
Check your solution by plugging it back into the original equation.

4. Are there any special techniques for solving f(x)=e^(x^2)?

Yes, there are some special techniques that can be used to solve f(x)=e^(x^2). These include using the properties of logarithms, using the inverse of the natural logarithm function, and using the substitution method. It is important to choose the most appropriate technique based on the specific problem at hand.

5. Can f(x)=e^(x^2) have multiple solutions?

Yes, f(x)=e^(x^2) can have multiple solutions. This is because the exponential function is a one-to-one function, meaning that each input (x-value) has only one output (y-value). However, it is possible for different input values to produce the same output value, resulting in multiple solutions for the equation.