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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)=\ln (1-\ln x)$

A. $\mathrm{V} \cdot \mathrm{A} : x=0, x=e, \quad \mathrm{No} \mathrm{H.A.}$B. Decreasing: $0 < x < e$C. No Maxima or MinimaD. The function is concave up in $(0,1) .$The function is concave down in $(1, e)$Inflection point $-x=1$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

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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 question, we first realized that little main more five is from zero to me. So for the sympathetic, um, we first look at the vertical sympathetic if X approaches to zero phone, right. We can see that if goes to call the 10 pavement infinity. So, um, X equals zero. Yes. Vertical sympathetic. Also, if x approaches to e from the left, the function caused toe negative Infinity. That means X equals two e It's Ah vertical synthetic. Come. This X will not approach to infinity. So there's no horizontal. Oh, let's see Matteo take now for the for the increasing and decreasing Terrible we take the first of the narrative. That means we have one over x Thomas 1/1, minus loan off X on the interval from zero to e. This is always connective them use the function is decreasing. Overlook integrity. The second of dirty vehicles to remind us loan off X over X square one minus loan affects, uh, square. So it's said crying FBI bickering because it zero. So we have X equals to one. So from 0 to 1, have that the promise positive so the functions can keep out from one to if that look promise. Negative. So country is concave down? No, we are ready to graph. Uh, so X equals twice the inflection point so we can schedule a graph off the function. Since the Tomei's from zero to eat, we only focus on this interval. So the function is monotone Nicolle decreasing as we discussed before. And he has vertigo symptoms. So we have the graph like this. Uh, so we have a heretical I seem to tick another vertical synthetic which happens to be the why exists in the way have an inflection point at X equals 20 which happens to be the X intercept s o. This is the graph off the country.

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