Problem

(a) Find the vertical and horizontal asymptotes. …

04:26

Problem 41 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 $f .$
$f(x)=e^{-x^{2}}$

Answer

A. horizontal asymptote: $y=0, \quad$ no vertical asymptotes
B. increasing: $(-\infty, 0) \quad$ decreasing: $(0, \infty)$
C. $(0,1)$
D. inflection $x=\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}}$
concave downward on $\left(-\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)$
concave upward $\left(-\infty,-\frac{1}{\sqrt{2}}\right) U\left(\frac{1}{\sqrt{2}}, \infty\right)$
E.SEE GRAPH

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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? Um, we first right out of the charity affair for fire convenience. So pat A for the synthetic behavior on since the domain off F is from minus infinity to infinity. So there's no vertical sympathetic, but there is a horizontal sympathetic. We see that because if Lim x goes to infinity, if we have the results zero and leave the limit exposed to minus infinity off. If the resources of zero, that means we have for his own toes. Maturity at why Coast zero for part B for the increasing anti Christian of always set Ex Prime Mickelson zero we have X equals zero is our solution. So this is a critical point in the throne, minus infinity to zero from zero to infinity. Um, we have a promise Positive on the first interval. So the function is increasing. We have f prime. It's negative. On the second around, the function is decreasing. That means that we have a local necks X X equals zero with Manlio F very cost one for policy. The concave it is. We stayed a second regenerative Toby zero. So we have X equals two plus minus root off 1/2 So we have the race of intervals from minus infinity to minus with all can have and the throw minus root off one have to wrote off one have in the front wrote off going have to infinity on the first interval If double prime is positive so they can keep up on the second trouble If the premise negative so to come Keep down on the last centavo. If Tebow promise positive against its can keep up That means we can sketch of the graph off. If so, there is a local maximum here and we have some. We have to inflection points and the graph is increasing. First increasing con cave up and the increasing conclave down, not decreasing in conclave down and the crazy in concrete up. So we have to re inflection points. We have ah vertical sympathetic horizontal synthetic and this is a sketch of the graph off it

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