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Math 10350 – Example Set 11A
- Find the vertical asymptotes of the following functions.
(a) y = x − 3 x^2 − 9
(b) y = x^2 + 9 x^2 − 9
- Draw a graph with horizontal asymptotes y = 1 and y = −4.
Horizontal Asymptote. If lim x→∞ f (x) = A (finite number) or lim x→−∞ f (x) = A. Then y = f (x) has horizontal asymptote y = A.
- Find the limits (^) x→−∞lim
1 − ex 1 + ex^ and (^) xlim→∞
1 − ex 1 + ex^
. Give graphical interpretations of your answers.
- Sketch the graph of g(x) = xe−x 2 by completing the steps below. a. Find all x-intercepts and y-intercept of the graph of g(x) whenever possible.
b. Find coordinates of all critical points, vertical asymptotes, and places where g(x) are undefined.
„ g′(x) = (1 − 2 x^2 )e−x^2
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c. Determine where g(x) is increasing and where it is decreasing.
d. Determine the concavity and coordinates of inflection points of g(x).
„ g′′(x) = (4x^3 − 6 x)e−x^2
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e. Find all asymptotes and limit at infinity whenever applicable. Check for any symmetry.
f. Sketch the graph below labeling all important features. Your picture should be large and clear.
e. Find all asymptotes and limit at infinity whenever applicable. Check for any symmetry.
f. Sketch the graph below labeling all important features. Your picture should be large and clear.
Math 10350 – Example Set 11C
1a. Find the absolute (global) maximum and minimum of f (x) = xe−x^ on the interval [0. 5 , 2].
1b. Using the steps below, find the global maximum and minimum of f (x) = xe−x^ on [0. 5 , ∞).
Step 1: Find all critical points in the domain of f (x) and the values of f (x) there. Classify them using first derivative test.
Step 2: Find the values of f (x) at the end-points (if any) of its domain.
Step 3: If end-point not included, or ±∞, find all limits of f (x) towards end of interval.
Step 4: Give a schematic sketch (ignore concavity) of the graph of f (x) clearly indicating where the global maximum and minimum are. State the global maximum and minimum of f (x) on [0. 5 , ∞) if any.
