2. For two real numbers a and b with a > b>0 find the valuels) of the constant C for which the following inte...
theorem1 let an and bn be squences of real numbers theorem 2 let an and bn and cn be squences of real numbers if an<bn<cn theorem 3 let an be squences of real numbers if an=L and L defined at all an,f(an)=f(L) theorem 4 f(x) defined for all x>n0 then limit f(x)=L and limit an =L theorem 5 follwing six squences converage to be limit limit lnn\n =0 ,limit (1+x/n)n=ex .... Based on Theorems 1 to 5 in Section 10.1...
3. Let a, b, and c be real numbers, with c +0. Show that the equation ax2 + bx + c = 0 (a) has two (different) real solutions if 62 > 4ac, (b) has one real solution if 62 = 4ac, and (c) has two complex conjugate solutions if 62 < 4ac.
Fix two real numbers a > 0) and b < 0. The logarithmic spiral is the plane curve ol→R², determine the signed curvature and remark that is never zero. o(t) = (aebt cost, aebt sint).
Suppose that 0 < a < b < c < d are Real numbers (they are all positive and from smallest to greatest in alphabetical order). Order the following fractions from SMALLEST to LARGEST: Suppose that 0 <a<b<c<d are Real numbers (they are all positive and from smallest to greatest in alphabetical order). Order the following fractions from SMALLEST to LARGEST: daba cbbc Choose the correct ordering from options below: O A. da a b C'b'c'b OB. a a d...
Let a, b, and c be three strictly positive real numbers. Two sub-intervals of the interval (0, a + b + c) are chosen at random. One of sub-intervals of length a, and the other of length b. Find the probability that the sub-intervals do not overlap (that is, that their intersection is empty).
Given the nxn matrices A,B,C of real numbers, which satisfy the Condition: A+B+C+λΑΒ=0 Α+Β+C+λBC=0 A+B+C+λCA=0 for some λ≠0 ∈ R (α) Prove that I+λΑ,Ι+λΒ,Ι+λC are invertible and AB=BC=CA. (b) Prove that A=B=C
1. Find scalars a,b,c that are real numbers such that at least one of a,b,c is non zero: 2. Find a nonzero vector v in R^4 orthogonal to: at] +b]+cE]=R] -3
5. (a) Show that the following improper real integral is absolutely convergent cos 2x dr I (1+?}% " (b) If CR is the semicircle of radius R in the upper half plane with centre at z = 0, show carefully that e2iz lim R00JCR (1+ z2)2 d% = 0 (c) Use residue calculus to evaluate the real integral I of part (a) 5. (a) Show that the following improper real integral is absolutely convergent cos 2x dr I (1+?}% "...
Please explain in full detail! For two fixed positive numbers a, b, 0 < a < b (for example, a = 1, b = 2), let a = V ab and b = - - Define aj+1 = a;b; and b;+1 = - T for any positive interger j. a+b (1) Prove that the sequence {aj} is convergent. (2) Prove that the sequence {b;} is convergent. (3) Prove that lim a;= lim bj. 1700
b where v7 = (2, -1), wł = (-1,2), b7 = 2. Find real numbers c and d so that cv + dw (1,0).