Graph e^(-x^2)

Problem

e(−x2)

Solution

1. Identify the function as the Gaussian function, which is symmetric about the y-axis because ƒ(x)=ƒ*(−x)

2. Determine the horizontal asymptote by taking the limit as x approaches ∞ or −∞ which results in y=0

3. Find the y-intercept by evaluating the function at x=0 giving ƒ(0)=e0=1

4. Analyze the first derivative to find extrema:

d(e(−x2))/d(x)=−2*x*e(−x2)

5. Locate the maximum point at (0,1) since the derivative is zero at x=0 and changes from positive to negative.

6. Analyze the second derivative to find inflection points:

d2(e(−x2))/(d(x)2)=(4*x2−2)*e(−x2)

7. Calculate the inflection points where the second derivative is zero, occurring at x=±1/√(,2)

8. Sketch the "bell-shaped" curve starting from the asymptote y=0 rising to the peak at (0,1) and falling back toward the asymptote.

Final Answer

The graph of the function is a bell-shaped curve symmetric about the y-axis with a maximum at (0,1) and horizontal asymptote y=0

ƒ(x)=e(−x2)

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