make A the subject of the formula sn=ar(rn-1)/r-1
(g) Prove that Rn - {0} and sn-1 X R are homeomorphic spaces. (Hint: Consider the function f: sn-1 X R → RN- {0} given by f(x,t) = 2'x.)
cos(()dr - (r sin(O) - e)de = 0, r(0) = 1 (make r the subject of the formula)
9. Given make the subject of the formula
(Derivation of perpetuity formula) Using the geometric series formula: if −1 < r < 1, then a+ar+ar2 +ar3 +···= a ,1−r derive the PV of perpetuity formula.
Let f : Rn × Rn → R be the inner product function: f(r,y)-(2,3) 1. Using the definition of multivariable derivative, calculate D fab and the Jacobian matrix f'(a, b) 2. If f, g : Rn → R are differentiable and h : R → R is defined by h(t)-(f(t), g(t)), show that 3. If f : R → Rn is differentiable and Ilf(t)ll = 1 for all t, show that(f,(t)T,f(t))-0
Given the formula for interest compounded nn times per year A=P(1+rn)ntA=P(1+rn)nt, solve each of the following and round your answer to the nearest penny. solve (a) and (b) Given the formula for interest compounded n times per year A = P(1 + )nt , solve each of the following and round your answer to the nearest penny. (a) P-190, 000, r-9%, n-26, t-48 A $ 1.4180 Toggle Clear Help 1.4180 Submit Answer Incorrect. Tries 5/99 Previous Tries (b) P 160,...
Problem 5. A subset A c Rn is an affine subspace of Rn if there exists a vector b є R', and a underlying vector subspace W of Rn such that (a) Describe all the affine subspaces of R2 which are not vector subspaces of R2. (b) Consider A є Rnxn, b є Rn and the system of linear equations Ax-b Prove that (i) if Ar= b is consistent, then its solution set is an affine subspace of Rn with...
2. Which of the following sets are convex? (a) A slab, i.e., a set of the forn {rE Rn l α-ar-β} (b) A rectangle, i.e., a set of the forin {2. E Rn | Qi-Z'i is sometimes called a hyperrectangle when n > 2. ,n). A rectangle A, i = 1, (d) The set of points closer to a given point than a given set, i.e., where SCR (e) The set of points closer to one set than another, i.e.,...
: R → Rn be a Ci path which solves the 1. Let F : Rn → R be a C1 function, and let differential equation, E'(t)--VF(C(t)), te R. (a) Show that f(t) F((t)) is a non-increasing function of t (ie, f'(t) 30 Vt.) (b) For any t for which F(E(t)) * 0, decreasing in t (ie, f'(t) <0.) show that č is a smooth path, and f(t) is strictly
Problem 5. A subset A C R', is an afǐпє subspace of Rn if there exists a vector b underlying vector subspace W of R" such that Rn and an (a) Describe all the affine subspaces of IR2 which are not vector subspaces of R2 (b) Consider A e R"Xn, beR" and the system of linear equations Ar- b. Prove that: (i) if A-b is consistent, then its solution set is an affine subspace of R" with underlying (ii) if...