Problem

Suppose the derivative of a function $f$ is $f^{\…

02:19

Problem 44 Hard Difficulty

(a) Find the vertical and horizontal asymptotes.
(b) Find the intervals of increase or decrease.
(c) Find the local maximum and minimum values.
(d) Find the intervals of concavity and the inflection points.
(e) Use the information from parts ( d ) to sketch the graph of $f .$
$f(x)=e^{\arctan x}$

Answer

A. No Vertical Asymptotes.
Horizontal Asymptotes: $y=e^{\frac{\pi}{2}}$ and $y=e^{-\frac{\pi}{2}}$
B. $f(x)$ is increasing for all $x \in(-\infty, \infty)$
C. No local maxima or minima.
D. Concave UP on $\left(-\infty, \frac{1}{2}\right)$
Concave DOWN on $\left(\frac{1}{2}, \infty\right)$
E.SEE GRAPH

Related Courses

Calculus 1 / AB

Essential Calculus Early Transcendentals

Chapter 4

APPLICATIONS OF DIFFERENTIATION

Section 3

Derivatives and the Shapes of Graphs

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Video Transcript

for this program. First, we need to figure out the s Antarctic. Um so because little man often this function a yes for O minus infinity to infinity. So there is no vertical A sympathetic, but for the for is not so sympathetic. We need to evaluate the following two limits. X goes to infinity and X goes to negative infinity off so I can see that act engine goes toe pie half when x goes to infinity name Is this the mediators to eat the pie have and the act engine goes to negative. I have as x goes to connective infant. So the second image because to you to the ah minus pi half that means we have 213 with politics. Um so the first nice white close to you to the pie half has X goes to positive infinity the second ways. Why cause to e to the minus by half as X goes to minus the affinity. As for the increasing and decreasing cover with tech literate tive And there we see that this the first of the narrative it's known active because both components are asked tricky, positive them use if you see Increasing. Oh, on minus Infinity to infinity. No way! I want to find a can carry t So we take the steak on the narrative than we have you to be. Car engine Hicks. Um Times one minus two x over one plus x square in Madison Square here. So he said the second of the directive because of your we have x equals to 1/2. So from from negative infinity to 1/2 second, the narrative is, um what is positive? The muse. Ah, if his concrete fuck only the interval one have to infinity a safe on the purity of these Negative. So you can keep down now we are ready to schedule graph off f. So, um first, let's put to hurry down toe. I've seen Tom Dick here. Um, so there's another inflection point at 1/2. So the function the great for function Well, looks like this. So first this increasing Montagny clean crazy bats can keep up. Um, when you passed this one, have it becomes conclave thumb. So there is the inflection point of this. At this point, X equals Do I have? And then there are two horizontal sympathetic. So this point is e to the minus. I have another point is to the Italy pie half

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