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Example s(t), f(x,y) For example, the command syms A(x) 3 2 matrix currently errors Differentiation functions, such as jacobian and laplacian, currently do not accept symbolic matrix variables as input To evaluate differentiation with respect toTwo vectors x, y in R n are orthogonal or perpendicular if x y = 0 Notation x ⊥ y means x y = 0 Since 0 x = 0 for any vector x, the zero vector is orthogonal to every vector in R n We motivate the above definition using the law of cosines in R 2In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomialAccording to the theorem, it is possible to expand the polynomial (x y) n into a sum involving terms of the form ax b y c, where the exponents b and c are nonnegative integers with b c = n, and the coefficient a of each term is a specific positive

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Is xy=3 a function-Y^2 = x^32x^2 WolframAlpha Natural Language Math Input Extended Keyboard ExamplesLog b (x / y) = log b (x) log b (y) For example log 10 (3 / 7) = log 10 (3) log 10 (7) Logarithm power rule The logarithm of x raised to the power of y is y times the logarithm of x log b (x y) = y ∙ log b (x) For example log 10 (2 8) = 8∙ log 10 (2) Derivative of natural logarithm The derivative of the natural logarithm function

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Establish the domain by creating vectors for x and y (using linspace, etc);3(1) − 9 ≠ 0 Hence, R is neither reflexive, nor symmetric, nor transitive (ii) R = {(x, y) y = x 5 and x < 4} = {(1, 6), (2, 7), (3, 8)} It is seen that (113 Problem Using the Euler equation nd the extremals for the following functional Z b a x y(x)2 3( @x y(x))dx Hint elementary Solution We denote auxiliary function f

Residuals are = Y−X Want to minimize sum of squared residuals X 2 i = 1 2 0 n 2 6 6 6 4 1 2 n 3 7 7 7 5 = We want to minimize 0 =(Y−X )0(Y−X ), where the \prime" ()0 denotes the transpose of the matrix (exchange the rows and columns) We take the derivative with respect to the vector This is like a quadratic function think \(YX n with values y 0;y 1;Expectations Expectations (See also Hays, Appendix B;

Section 145 (3/23/08) Directional derivatives and gradient vectors Overview The partial derivatives fx(x0,y0) and fy(x0,y0) are the rates of change of z = f(x,y) at (x0,y0) in the positive x and ydirectionsRates of change in other directions are given by directional A very simple example x y = 5, and x y = 1 Adding the first equation to the second equation eliminates y and leaves you with 2x = 6, so that x = 3 Using that value for x in either of the original equations lets you see that y = 2{(x,y) x y ≥ 1}, which is the region above the line y = 1 − x See figure above, right To compute the probability, we double integrate the joint density over this subset of the support set P(X Y ≥ 1) = Z 1 0 Z 2 1−x (x2 xy 3)dydx = 65 72 (c) We compute the marginal pdfs fX(x) = Z ∞ −∞ f(x,y)dy = ˆR 2 0 (x 2 xy 3)dy

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We will be using the formula of the exact length of the curve to solve this The Exact Length of the Curve x = 1/3 √y (y − 3), 1 ≤ y ≤ 9 is 32/3 unitsC A Bouman Digital Image Processing 3 Projection Property of Chromaticity Coordinates • Fact Straight lines in (X,Y,Z)space project to straightY nLet S(x) bet the spline Let M j = S00(x j);

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 mathatan2 (y, x) ¶ Return atan(y / x), in radians The result is between pi and pi The vector in the plane from the origin to point (x, y) makes this angle with the positive X axis The point of atan2() is that the signs of both inputs are known to it,Figure 2 Draughtsman's spline 3 Equations of cubic spline Let data be given at x 0;x 1;Y 1 2 x = ln y − 1 3 1 y 1 x = ln 2 y − 1 1 x = (ln(y 1) − ln(y − 1)) 2 There are many equivalent correct answers to this question The best answer is the one that is

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If x=0 , then y=0 so that y=0 is the yintercept If y=0 , then x 33x 2 =x 2 (x3)=0 so that x=0 and x=3 are the xintercepts There are no vertical or horizontal asymptotes since f is a polynomial See the adjoining detailed graph of f Click HERE to return to the list of problemsIb/rm @X y l o i dConnect with friends and the world around you on Facebook Create a Page for a celebrity, band or business

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 Example 17 Solve the pair of equations 2/𝑥 3/𝑦=13 5/𝑥−4/𝑦=−2 2/𝑥 3/𝑦=13 5/𝑥−4/𝑦=−2 So, our equations become 2u 3v = 13 5u – 4v = –2 Hence, our equations are 2u 3v = 13 (3) 5u – 4v = – 2 (4) From (3) 2u 3v = 13 2u = 131 Suppose the joint pmf of X and Y isgiven byp(1,1) = 05, p(1,2) = 01, p(2,1) = 01, p(2,2) = 03 Find the pmf of X given Y = 1 Solution pXY=1(1) = p(1,1)/pY (1) = 05/06 = 5/6 pXY=1(2) = p(2,1)/pY (1) = 01/06 = 1/6 2 If X and Y are independent Poisson RVs with respective means λ1 and λ2, find the conditional pmf of XPython(x, y) was concieved, developed and maintained by Pierre Raybaut since 08 with the above goals Gabi Davar joined the project as a maintainer since 11 Pierre moved to work on other projects since 13 leaving Gabi as the primary maintainer

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SOLUTION 1 Begin with x 3 y 3 = 4 Differentiate both sides of the equation, getting D ( x 3 y 3) = D ( 4 ) , D ( x 3) D ( y 3) = D ( 4 ) , (Remember to use the chain rule on D ( y 3) ) 3x 2 3y 2 y' = 0 , so that (Now solve for y' ) 3y 2 y' = 3x 2, and Click HERE to return to the list of problems SOLUTION 2 Begin with (xy) 2 = x y 1 Differentiate both sides Given A linear equation #color(red)(y=f(x)=3x2# Note that the parent function is #color(blue)(y=f(x)=x# #color(green)("Step 1"# Consider the parent function and create a data table followed by a graph to understand the behavior of a linear graph #color(red)(y=f(x)=3x2# compares with the parent function #color(blue)(y=f(x)=x# Graph of the parent functionSection 35 Minterms, Maxterms, Canonical Form & Standard Form Page 2 of 5 A maxterm, denoted as Mi, where 0 ≤ i < 2n, is a sum (OR) of the n variables (literals) in which each variable is complemented if the

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Plot the surface The main commands are mesh(x,y,z) and surf(z,y,z)Calculate z for the surface, using componentwise computations;Y=x^3 WolframAlpha Area of a circle?

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Example 3 Graph using intercepts 2x−3y=12 2 x − 3 y = 12 Solution Step 1 Find the x and y intercepts Step 2 Plot the intercepts and draw the line through them Use a straightedge to create a nice straight line Add an arrow on either end to indicate that the line continues indefinitely in either direction2 days ago 5 Data Structures — Python 397 documentation 5 Data Structures ¶ This chapter describes some things you've learned about already in more detail, and adds some new things as well 51 More on Lists ¶ The list data type has some Explanation graph { (ysqrt (9x^2))=0 6, 6, 2, 4} The Volume of Revolution about Ox is given by V = ∫ x=b x=a πy2 dx So for for this problem, Noting that 9 − x2 = 0 ⇒ x = ± 3, and that by symmetry we can double the volume for the region x ∈ 0,3 V = 2∫ 3 0 π(√9 −x2)2 dx = 2π ∫ 3 0 (9 − x2) dx = 2π 9x − x3 33 0

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λ2 = −3 ↔ v2 = −1,1T x'=x4y, y'=2x−y −5 0 5 −5 0 5 x y Time Plots for 'thick' trajectory −05 0 05 1 −30 − −10 0 10 30 t x and y x y Nodal Source Ex A = 3 1 1 3 λ1 = 4 ↔ v1 = 1,1T λ2 = 2 ↔ v2 = −1,1T x'=3xy, y'=x3y −5 0 5 −5 0 5 x y Time Plots for 'thick' trajectory −25 −2X,Y = meshgrid(x,y) returns 2D grid coordinates based on the coordinates contained in vectors x and y X is a matrix where each row is a copy of x, and Y is a matrix where each column is a copy of yThe grid represented by the coordinates X and Y has length(y) rows and length(x) columnsCreate a "grid" in the xyplane for the domain using the command meshgrid;

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X and Y, ie corr(X,Y) = 1 ⇐⇒ Y = aX b for some constants a and b The correlation is 0 if X and Y are independent, but a correlation of 0 does not imply that X and Y are independent 33 Conditional Expectation and Conditional Variance Throughout this section, we will assume for simplicity that X and Y are discrete random variablesI(x, y) = 2y 3 − x 2 y 3y x 3 2x C And the general solution is of the form I(x, y) = C and so (remembering that the previous two "C"s are different constants that can be rolled into one by using C=C 1 C 2) we get 2y 3 − x 2 y 3y x 3 2x = C Solved!Harnett, ch 3) A The expected value of a random variable is the arithmetic mean of that variable,

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Easy as pi (e) Unlock StepbyStep Natural Language Math InputAnswer (1 of 3) Solve for x x^2 y y x^3 = 0 Eliminate the quadratic term by substituting z = y/3 x y y (z y/3)^2 (z y/3)^3 = 0 Expand out terms of the left hand side z^3 (y^2 z)/3 (2 y^3)/27 y = 0 Change coordinates by substituting z = u λ/u, where λ is a constanX 2 ( − 2 x − 8) y y 2 − 8 x = 0 All equations of the form ax^ {2}bxc=0 can be solved using the quadratic formula \frac {b±\sqrt {b^ {2}4ac}} {2a} The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction

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Given random variables,, , that are defined on a probability space, the joint probability distribution for ,, is a probability distribution that gives the probability that each of ,, falls in any particular range or discrete set of values specified for that variable In the case of only two random variables, this is called a bivariate distribution, but the concept generalizes to anyGraph y=x Use the slopeintercept form to find the slope and yintercept Tap for more steps The slopeintercept form is , where is the slope and is the yintercept Find the values of and using the form The slope of the line is the value of , and the yintercept is the value of Slope In this section we will discuss implicit differentiation Not every function can be explicitly written in terms of the independent variable, eg y = f(x) and yet we will still need to know what f'(x) is Implicit differentiation will allow us to find the derivative in these cases Knowing implicit differentiation will allow us to do one of the more important applications of derivatives

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H j = x j x j 1 be 'moment' at jth point Then between x Article Summary X To find the vertex of a quadratic equation, start by identifying the values of a, b, and c Then, use the vertex formula to figure out the xvalue of the vertex To do this, plug in the relevant values to find x, then substitute the values for a and b to get the xvalueX(x) 0 010 1 030 2 0 3 030 4 010 y f Y (y) 0 014 1 016 2 018 3 025 4 027 Exercise 3 Give two pairs of random variables with different joint mass functions but the same marginal mass functions The definition of expectation in the case of a finite sample space S is a straightforward generalization of

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So x 3 y 2 is NOT homogeneous And notice that x and y have different powers x 3 vs y 2 For polynomial functions that is often a good test But not all functions are polynomials How about this one Example the function x cos(y/x) Start with f(x, y) = x cos(y/x)(x y) 5 = x 5 5x 4 y 10x 3 y 2 10x 2 y 3 5xy 4 y 5 We can make several observations In each expansion, there are n 1 terms In each expansion, x and y have symmetric roles The powers of x decrease by 1 in successive terms, whereas the powers of y increase by 1Desmos offers bestinclass calculators, digital math activities, and curriculum to help every student love math and love learning math

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