By Edgar Asplund; Lutz Bungart

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Because the determinant of a matrix is a polynomial function of the elements of the matrix, the expression p(λ ) = det(M − λ I) is is a polynomial in λ , called the characteristic polynomial of M. The equation p(λ ) = 0 is the characteristic equation of M. 1. Each eigenvalue of M is a root of its characteristic equation. 1 2i 1 = = 2i ; −i 2 −i 0 −2 2 0 Characteristic equation ⊔ ⊓ But real polynomials can have complex roots, too. For example, our rotation matrix M3 has the characteristic polynomial p(λ ) = λ 2 +4 whose roots are λ = ±2i.

A. Sketch the curve in the (x, y)-plane given parametrically as x= 2t , 1 + t2 y= 1 − t2 . 1 + t2 In particular, label the points where t = −2, −1, 0, +1, +2. b. Each of the following limits exists; determine the location of each as a point in the (x, y)-plane: lim (x(t), y(t)) t→+∞ lim (x(t), y(t)) t→−∞ c. , independent of t) that is consistent with the sketch of the curve you made in part (a). What is the curve and how does α relate to it? 24. Determine the work done by the force field F in moving a particle along the oriented curve C, where: a.

The manufacturer can expect that the weights X will be dispersed around the central value (here, 5 ounces) in a certain predictable way. For many random variables like X , the dispersion follows what is called a normal distribution. If Xµ ,σ is a random variable that follows a normal distribution with mean µ (its central value) and standard deviation σ (its measure of dispersion), then the probability that the value of Xµ ,σ lies between a and b is equal to the fraction of the area under the entire graph of Random variables Normal distribution y = g µ ,σ (x) = e−(x−µ ) y 1 that lies between the vertical lines x = a and x = b.