Please answer the attached question. It involves calc 3. Thanks

Let F be the solid sphere 0 < x* + y? + z? $ 1 of radius 1 centered on the origin and let /, be the portion of F that lies in the first octant. Assume that f(x, y, z) is a continuous function that issymmetric with respect to reflections through the coordinate planes. That is:f(-x, y, z) = f(x, y, z), f(x, -y, z) = f(x, y, z), f(x, y, -z) = f(x, y, z).If1, "(x, Y, 2) dv = 317, find the value of((x, Y. 2) av.

 

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Let G be a group of order 42.

Prove both (a) and (b).

Prove both (a) and (b).

Prove both (a) and (b).

7 . Let G be a group of order 42 .( a ) Prove that G has a normal subgroup of order 7 .`( b ) Prove that G is a semidirect product of a normal subgroup of order 21 and 2.2 .

 

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Let G be a connected bipartite graph (i.e the set of vertices is split as X S Y so that the edges are connecting some pairs xi ∈ X with yj ∈ Y ). Show that if each vertex has multyplicity 3 then there is a complete matching( i.e the number of vertices in X is the same as in Y and there is matching of all vertices in X with vertices in Y )

 

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This is a statistics question. We need to use RStudio to get the answer of this question. Therefore, this is actually an coding problem in R

Let * follow an exponential distribution with rate parameter &* = 2. Let I" follow a Poisson distributionwith rate parameter^` = $.We write sall I ; for the true standard deviation of I and mall"; for the true median of I".Let s _ he the sample standard deviation of I which is an estimate of sillly. Also let my he the samplemedian which is an Estimate of mill’ ] .Suppose we take samples of size `= = 101 from* and take samples of size my = ]. Consider the statisticWhat is the ( sampling ; distribution of* ! We could ask a statistician who specializes in theary . Instead oflising mathematics . simulate* Good times and stare the results . Plot a histogram of the observed values ofAL . Comment on the shape of the histogram and empirical distribution of* . Before running your code , Set.the same seed used for the previous exercise . For full credit , do not use a For Long.

 

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10. 11. 12. Let f(x) = x3 + x — 1 and 30:) = x3 — x2 + 2x + 1. Find the area A of thebounded region between the graphs offand g. (10 pts) Consider the graph of the curve with parametric equations x(t) = —3 sin(2t),y(t) = 4cos(2t), for t 2 0. a. Sketch the graph of the curve, and indicate the direction of increasing t. (Lookingahead to part c might be helpful.) (5 pts) b. Find the slope of the line tangent to the graph of this curve when t = E. No approximations. Find 2—: n by using the formula in the box of section 2.2, p. 169.t:—6(5 NS)2 2c. Show that x? + {—6 = 1. A curve with this equation is called an (fill in theblank). (5 pts) d. By using implicit differentiation on the equation in part c), find 2—: in terms ofxand y. Calculate x (E) and y (E), and reconcile your answer to part b). (5 pts) Consider the function f(x) = i lnx’a. Determine the domain off. (5 pts)b. Find the vertical asymptote(s), if any, of this function. Use limits to justify thatyou have in fact found the asymptote(s). (5 pts) c. Find the horizontal asymptote(s), if any, of this function. Use limits to justify thatyou have in fact found the asymptotes. (5 pts)

 

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