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Calculus/graph sketching

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Expert: - 1/20/2011


Question
Let f be the function defined by f(x) = x.e^-x?
a. Determine the y-intercept
b. Determine the horizontal and vertical asymptote
c. Use the sign pattern for f'(x) to determine
(1) the intervals over which f rises and where it falls
(2) the local extrema
d. Use the sign pattern for f''(x) to determine
1. where te graph of f is concave up and where it is concave down;
2. the inflection points if any

Answer

Graph
Questioner: Peter
Country: South Africa
Category: Calculus
Private: No
Subject: help
Question: Let f be the function defined by f(x) = x.e^-x?
a. Determine the y-intercept
b. Determine the horizontal and vertical asymptote
c. Use the sign pattern for f'(x) to determine
(1) the intervals over which f rises and where it falls
(2) the local extrema
d. Use the sign pattern for f''(x) to determine
1. where te graph of f is concave up and where it is concave down;
2. the inflection points if any
....................................................

Exactly what part of this is giving you trouble?

f(x) = x(e^x - 1)  might be a start.  Then you will note that:

1) x = 0 -->  y = 0.
2) it is never undefined.
3) when x --> infinity, it looks like x e^x.

Then you will find  f'(x), using the product rule:

f' = e^x - 1 + xe^x

and set it = 0:

e^x(1 + x) - 1

I think you can go on from there.  If you have trouble, let me know where you get stuck.

Paul Klarreich

Expertise

All topics in first-year calculus including infinite series, max-min and related rate problems. Also trigonometry and complex numbers, theory of equations, exponential and logarithmic functions. I can also try (but not guarantee) to answer questions on Analysis -- sequences, limits, continuity.

Experience

I taught all mathematics subjects from elementary algebra to differential equations at a two-year college in New York City for 25 years.

Education/Credentials
(See above.)

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