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\begin{equation}\begin{array}{l}{\text { Evaluate numerically } \frac{1}{3 \sqrt{2 \pi}} \int_{14.5}^{\infty} e^{-(z-10)^{2} / 18} d z}. \end{array}\end{equation}

And this problem. We're finding the linear approximation of the given function at the given point. So first will find all partial derivatives with respect to W X, Y and Z. So far, Phil were through with respect to W with David Negative three w square times x squared. Oh, partial with respect to X would be equal to negative y z times dying Uh, X, y and Z minus two w cute times x The partial with respect to why would be equal to negative xy times dying of X. Why and no partial with respect Z would be equal to negative X y time Sign Why? So at our first point, we will leave either with the function and the partial derivative. So to know the new 140 gives this cool sign of zero minus who? Cubed times negative ones squared and what we get one minus eight. This negative seven. It's, um w ah to negative Born born here A. What we get here is negative three thanks to square times negative born squared and that gave negative 12 in value in the partial derivative of X gives us, uh, there are minus two times too cute. Terms negative born and that looks out to be positive. 16. Well, partial respect to why, at a given point, Uh, that works out to be zero. And so does the partial with respect to see at a given point, given coordinates that's also equal to zero. So they will find that I will learn your approximation is equal to negative seven minus 12 times w minus two birth 16 times X plus one There were for distributes. We have negative. 12 W I was 24 six 16 which would simplify too negative. 12 w the 16 x close 33. There is all in your proclamation on the first point. The first set of coordinates. Well, part B, we repeat the process for this 0.2101 The first evaluate all function at these coordinates in what we have. This cool sign of zero minus through huge times once where that gives negative seven devaluating The partial with respect to W gives this negative three times two square times one squared that gives us negative. 12 are impartial with respect to X. When we evaluate gives zero minor two times two Cute. I'm so worn, and that gives us negative 16. When we evaluate for departure with respect to y and see, they both end up being zero. Here we find that the linear approximation is equal to negative seven minus 12 times W minus two minus 16 times X minus one Rex Distributing We have negative 12 w mind Most 24 My It's been x 16. So are linear approximation is equal to negative tone W minus 16 plus 33.

All right. So this question here, if evaluate this integral numerically, it's really just a matter of finding a calculator, uh, capable of doing this. And so just for the sake of it, um, we can rewrite this into something a little more familiar looking. So 1/3 times Ah, this function which is not f of ze since f of Z would be from negative infinity to Z instead. Instead, we have, um, one. Okay, so one minus f z first of all is gonna be, ah, normal effort, See? But it's gonna be one over to pie. What's one over route to pine times. Times the integral from Z to infinity of feed to the mass X squared over two d x. And so that looks close. Here we have 1/3 times one over written by So from 14.5 to infinity of E to the minus. And so instead of X here Ah, we have, uh, z Z minus 10 squared over two. You're not over to over 18 over it. Tz So how can we express this in terms of the standard if of Z function? Well, it's 1/3 times something that's almost f of Z Ah, er one minus F Uzi. Ah, but it's It's not quite since we have this Z minus 10 term in this, uh, this e over 18 terms, so we're best off just plugging this into a calculator.

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