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(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)=\sqrt{x^{2}+1}-x$

A. There is No vertical asymptote.There is one horizontal asymptote, that is $y=0$B. The function is decreasing for all $x \in(-\infty, \infty)$C. No local maxima or minima.D. No inflection point, concave up for all $x$E. SEE GRAPH

Calculus 1 / AB

Chapter 4

APPLICATIONS OF DIFFERENTIATION

Section 3

Derivatives and the Shapes of Graphs

Derivatives

Differentiation

Applications of the Derivative

Oregon State University

Harvey Mudd College

Baylor University

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

44:57

In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.

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(a) Find the vertical and …

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for this program. Um, first part, this about the house methodic. So for the for today, the horizontal symptomatic. We need to evaluate this to limits. So we send X two positive infinity off. If, um, so the result off this will be zero. Because, um, this is because when we evaluates this kind of limits we meet you converts this function to a fraction toe a kind the difference of the square formula. So we have, um, one the white divided If I something are really, really large. So the result is there and the on only on a hand, we also need to evaluates, um, when x approaches to negative infinitely. What's it for? FEC's? So, in this case, um, the result will be just knew finished. So that means very so. 100 Santos in the totemic. Which is why, because zero as x goes toe positive. Infinity. As for the verticals, sympathetic, there's neng because, um, this function is defined it on the whole, um, interval from minus infinity toe quality of the affinity. This is product. And for probably we need to find, um, the purity of first said your activities this, um So it's just about the chin role so we can see. Um, the denominator is always positive because it's a squared off, some quality of thing and the for the numerator. It's also positive because eggs is always sorry. You should be negative because X is always less than Rudolph X Square Pass 12 cities. We just square boats that we have X square less than next passport, which is always true. Okay, so there's no quit your point. That means ah, if prime is always negative. So if is one anatomically decreasing on minus infinity joke infinity. And for the inflection point and the concave Biti's we need to take the second curative. So we have one over one process X square to, ah, 3/2 which is also positive. Full cool, thanks in the interval. Next. Injecting infinity to apart it here for you? Definitely. So if is, um Com cave up for it and the least known infection point. Based on this information, we can sketch the graph off this function, which is a mountain Nikolay decrease in function with with the horizontal sympathetic XY Y causes zero. So it looks like this if I want to be decorated. We can finally y intercept. So the Y intercept way Just project X equals zero wherever one year. Okay, so this is the graph. The sketch off graph off this function.

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