Refer a friend and earn $50 when they subscribe to an annual planRefer Now

Get the answer to your homework problem.

Try Numerade Free for 30 Days

Like

Report

Use the guidelines of this section to sketch the curve.$$y=x e^{-x}$$

$\left(2, \frac{2}{e^{2}}\right)$

Calculus 1 / AB

Chapter 4

APPLICATIONS OF DIFFERENTIATION

Section 4

Curve Sketching

Derivatives

Differentiation

Applications of the Derivative

Campbell University

Harvey Mudd College

Baylor University

Boston College

Lectures

03:09

In mathematics, precalculu…

31:55

In mathematics, a function…

07:53

Use the guidelines of this…

04:46

05:43

03:27

16:02

15:02

01:39

15:37

05:05

10:32

so we're given the equation. F of X is equal to e equals Teoh X times e to the minus X and weaken. Just rewrite this as X over e to the X sketch this We can easily find out the domain of this function, and you might have an instinct to find to set the denominator equal to zero e to the X is equal to zero and solve for X. But if the X will never equal zero for all X, that's why the domain is actually negative. Infinity to infinity or all real numbers. So X is old real numbers. The we can figure out the intercepts so we can solve for the X intercept by setting it equal to zero. And we could essentially just look at the numerator when we set it equal to a number. So we have X is equal to zero. So if I plug in zero into this equation roundup just getting zero back, and if we plug and that will be our only intercept isn't we tried to find the y intercept, it will have to plug in zero, which we already 10. If we did so, this will be our only intercept. Now we can focus on the symmetry of dysfunction so we can do the even test. So is our function the same when we replace X with minus X? Well, we have minus X over E to the minus FEC's and it's not the same thing as the original function. What about the odd test? So it is not even our test is. Do we get the same results from the even tests? If we just negate the entire equation? And if we negate the entire equation, it just becomes negative. X over 82. The X and these two are not the same thing. So it is not even. It is not odd, so there is no symmetry. What about Assam toads? Well, there are no vertical asking totes because our denominator doesn't really equal zero. So what about horizontal asking toads? So I'll just abbreviated as h A. And you can think of it as sort of take the limit as X tends to infinity infinitely for our equation. X over eighth e x even think about it as what if we just constantly increase the value of X like, let's say we have five over E to the fifth. Well, the numerator is going to increase, but the denominator will explode vastly increased much faster than the new than the numerator, because each of the fifth is vastly larger than the number five, and we can continue this trend. So, like if we have X is equal to 20 then we have 20 over E to the 20 which is fastly larger so it's can equal to zero, since the denominator will always be will be very larger, then be numerator. So what about what if we decrease the numbers significantly so very large negative numbers? So let's say if we want. So let's take the limit as extends to negative infinity for our equation. What is this equal to? Well, let's just say negative five. Let's say exited with a negative five. So it will be negative five over you to the minus five. So Ethan, minus five is going to be well, less than one. So that means negative five over eat. The negative five will be a very via large negative value. In fact, this is actually equal to, well, approximately negative. 700 and 740. And if we do this for negative six, then it is. And if we keep decreasing the number from negative five to negative six, then that is around negative. 2000 420. So you can see that is getting much, much, uh, less. It's decreasing significantly, so this limit will be negative. Infinity. So we know the end behavior of this function as well as that. There is a horizontal asking tote on for the essentially the right side of the function. Let's focus instead of the original equation. But let's try to take the derivative of our equation. So we're given this equation. So it is a quotient so we can use the quotient to rule and just a reminder. But the question to rule appear so it's F g f prime G minus F g prime over g swear. So G square is e to the X squared. And we could solve this as each of the two X but you might want to keep it like this. So f prime is just ex prime, which is just one. So that is just e to the X minus f then g prime in that just the same thing as e to the X. We noticed that there's an E to the exit each term in the numerator, so we could, in fact, to that out. So that's one minus X over E to the X squared. If we can cancel out the E to the X in the square, and then this is equal to one minus X over E to the X, and then we can set this equals zero to find our critical point. And then we can just focus on the numerator portion. So it's one minus X equals zero, and then it just X is equal to one. So this is our critical point, and with this we can figure out a few intervals. So the interval of negative infinity toe one and then from one to infinity so we can get a test value from each interval. So the test value we could use, let's say zero and two. And what is the sign when we plug this value back into the first derivative? Well, we have zero, so it's one minus zero. That's just 1/1. So that is positive. About two. It's one minus two. That's a negative and then you know that each the X is always gonna be positive. So it's negative over positive, which is negative. So that means it is increasing on this interval and it is decreasing. On this interval, we could plug back in one to the original function to figure out the extreme of it to evaluate any extreme. And it looks like it is increasing and then it is decreasing, so it looks like it's going to be a maximum. So we have a local Max at one comma. When we plug one back in its one over E to the one, I'll just leave that as e to the minus one. Then we have our intervals and I'll just clean this up for enough in a second. After we were done with the first derivative, we can figure out the second derivative. Do you figure out points of inflection, end concave ity And again, this is a quotient. So we're going to use the quotient rule. So it is Eat the X squared and you might guessed it. We were going to leave it like this, so f prime is just negative. One times e x minus beaches. I'll keep this in parentheses, one minus X, and either the ex prime is just a to the X, then weaken. Distribute at this E to the X first, so it's minus E to the X minus a to the X minus x e to the X over e to the X squared. Then we can distribute out this negative. No distribute out here, so it's negative and plus, so we have minus E to the X twice. So we have X E to the X minus two easily x over E to the X squared, and then we could factor out and e to the X. So it's X minus two over each of the X squared. You can cancel those out and then we are left with X minus two over E to the X and then to find the point of inflection, we can just set it equal to zero. You can only really care about the numerator and then set that equal to zero. Solve for X, which were allowed with X, is equal to two. And then to find concave ity, we can have to have two intervals negative infinity to to to to infinity for test values. We can have zero in three. So our interval are a test value and our sign. So when we plug zero back into our equation here zero minus two, that's negative. Over a positive that negative and then three minus two That will be positive over a positive, which is positive. So this will be con cave downwards and this will be con cave upwards. So that means this point and we have to figure out the Y coordinate of that point is a point of inflection of in reflection. So oh, I will organize this. So, for the part of inflection we have to do is put it back into the equation, which is so we have to over e squared and we can just leave it like fat so to over East squared, or is to e to the minus two. And then we can finally graph our equation of X over e to the X so I can just draw that in some blue. So the intercepts, the only intercept is at 00 the local maximum one comment e to the, uh, minus one. So there's one. I will say it right there is easily minus one and then we know it will have. It will start at Assam tote when it is very increasing and it is coming from this way because you know, for the end, behavior sort of looks like more linear. So it's increasing for infinity to this point. Touches it and then it is decreasing and approaching zero but will never reach zero. Oops. And it will never reach zero. And this is our function f of x legal two x e to the minus x.

View More Answers From This Book

Find Another Textbook

In mathematics, precalculus is the study of functions (as opposed to calculu…

In mathematics, a function (or map) f from a set X to a set Y is a rule whic…

Use the guidelines of this section to sketch the curve.

$ y = e^{2x}…

$ y = (1 - x…

$ y = e^x/x^…

$ y = (1 + e…

$ y = \frac{…

$ y = 1/(1 +…

$ y = e^{\ar…

Use the guidelines of this section to sketch the curve.$$y=x \ln x$$

01:08

Let $$f(x)=c x+\ln (\cos x) .$ For what value of $c$ is$f^{\prime}(\pi /…

01:10

The graph of a function $f$ is given. Estimate $\int_{0}^{10} f(x) d x$u…

00:25

Evaluate the integral by making the given substitution.$$\int e^{-x} d x…

00:44

$7-10=$ Sketch the graph of a function $f$ that is continuouson $[1,5]$ …

01:16

Evaluate the limit and justify each step by indicating the appropriate Limit…

11:39

The frame for a kite is to be made from six pieces of wood.The four exte…

01:25

Let $$f(x)=\log _{a}\left(3 x^{2}-2\right) .$ For what value of $a$ is $f^{\…

07:03

A cone-shaped drinking cup is made from a circular piece of paper of radius …

01:58

$51-60=$ Use logarithmic differentiation or an alternativemethod to find…

00:33

Use Formula 14 to graph the given functions on acommon screen. How are t…

92% of Numerade students report better grades.

Try Numerade Free for 30 Days. You can cancel at any time.

Annual

0.00/mo 0.00/mo

Billed annually at 0.00/yr after free trial

Monthly

0.00/mo

Billed monthly at 0.00/mo after free trial

Earn better grades with our study tools:

Textbooks

Video lessons matched directly to the problems in your textbooks.

Ask a Question

Can't find a question? Ask our 30,000+ educators for help.

Courses

Watch full-length courses, covering key principles and concepts.

AI Tutor

Receive weekly guidance from the world’s first A.I. Tutor, Ace.

30 day free trial, then pay 0.00/month

30 day free trial, then pay 0.00/year

You can cancel anytime

OR PAY WITH

Your subscription has started!

The number 2 is also the smallest & first prime number (since every other even number is divisible by two).

If you write pi (to the first two decimal places of 3.14) backwards, in big, block letters it actually reads "PIE".

Receive weekly guidance from the world's first A.I. Tutor, Ace.

Mount Everest weighs an estimated 357 trillion pounds

Snapshot a problem with the Numerade app, and we'll give you the video solution.

A cheetah can run up to 76 miles per hour, and can go from 0 to 60 miles per hour in less than three seconds.

Back in a jiffy? You'd better be fast! A "jiffy" is an actual length of time, equal to about 1/100th of a second.