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The family of bell-shaped curves$$y=\frac{1}{\sigma \sqrt{2 \pi}} e^{-(x-\mu)^{2} /\left(2 \sigma^{2}\right)}$$occurs in probability and statistics, where it is called the normal density function. The constant $\mu$ is called the mean and the positive constant $\sigma$ is called the standard deviation. For simplicity, let's scale the function so as to remove the factor 1$/(\sigma \sqrt{2 \pi})$ and let's analyze the special case where $\mu=0 .$ So we study the function$$f(x)=e^{-x^{2} /\left(2 \sigma^{2}\right)}$$(a) Find the asymptote, maximum value, and inflection points of $f .$(b) What role does $\sigma$ play in the shape of the curve?(c) Illustrate by graphing four members of this family on the same screen.

a. HA: $y=0, \quad$ local $\&$ abs. max: $f(0)=1$inflection points: $f( \pm \sigma)=\frac{1}{\sqrt{e}}$b. since we have IP at $x=\pm \sigma,$ the inflection points move away from the$y-$axis as $\sigma$ increases.

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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Okay, so for this problem, for us away, look at the, um a sympathetic behavior. So there's no Rocky come sympathetic because Because that old man off f is from minus infinity to infinity Eso four have so far resulted from total We evaluate Lim x goes to you mean to you because toe zero and the allergy meter x close to minus infinity yourself. Zero So we can crew. Let's for the horizontal synthetic We have y coastal zero. And for part two, we've take a dirty if off. So we have each of the x squared minus over minus a so Italy minus X squared over two sigma squared times minus X over to stick of my square. So it's set f crime to be zero. When we have one solution, X equals zero because it's sick. Um isa constant. So from minus infinity to zero if prime is politics so the function is increasing from zero to infinity. If prime is an active so far, she is decreasing. So if has, uh, no coal makes hman at X equals zero with value X zero you cost one not for the inflection point or another can cavities. We computer a second narrative So we have need to the minus x squared, divided by two Sigma Square times one over to a Sigma Square, times X squared over two Sigma squared minus one. So we said F double prime because zero so X equals two house of miners wrote off two Sigma own minus infinity to minus wrote off to a sigma F double promise for the tape. So functions can keep up from minus no tough to a sigma to root off two Sigma the function of the second of purity of his connective. So functions concrete. Come, um, and the own root off to a sick amount you divinity f type of promise positives Lafont into can keep up That means we have inflection point at X equals to cross of minus wrote off two Sigma. Based on this information we can of the function So the fuck the function is increasing And the conclave up that bikini when he hits Ah minus I wrote off two sigma It becomes come cave complex down. So when you hit, I'm a tough toast taking my here. So you become conclave up and d Kwesi and the local maximum attribute at X equals zero with value one and that we have to inflection point here. And we have ah, horizontal sympathetic. Um so horizontal, I think totally why he causes your which is the XX is so if we if we change if we traded a value off Sigma so we'll see another and then the curve looks like this. So if if sick amount goes to a smaller and smaller we have something like this so busy the curfew will be more concentrated close to X equals toe zero. So if synchronized, um, bigger and bigger So have something like this, so the current will be like more flat.

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