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:
Rank the following compounds in order of increasing heat of hydrogenation. A B D E OB<C<E<A<D OD A<E<C<B OB«E<C<A<D O E<B<C<A<D O DA<C<<
Suppose f, g are two functions mapping positive real numbers to positive real numbers and f = O(g). Prove why each statement is true or false. (a) log2 f = O(log2 g) (b) √f = O(f) (c) fk + 100fk−1 = O(gk), for k ≥ 1
Suppose that A is diagonalizable and all eigenvalues of A are
positive real numbers. Prove that det (A) > 0.
(1 point) Suppose that A is diagonalizable and all eigenvalues of A are positive real numbers. Prove that det(A) > 0. Proof: , where the diagonal entries of the diagonal matrix D are Because A is diagonalizable, there is an invertible matrix P such that eigenvalues 11, 12,...,n of A. Since = det(A), and 11 > 0,..., n > 0,...
1.8.6 (a) If a, b, c are positive real numbers, and a < b + c, show that C 1c 1a
1.8.6 (a) If a, b, c are positive real numbers, and a
1. (a) Determine the smallest subring S (with identity) of the real numbers R that contains 3/5 (give a simple description of the elements of S and prove that S is a subring) (b) Is S an integral domain? (c) Find all units of S. (d) What is the characteristic of S? (e) Find the field of fractions of S (f) Find the smallest ideal I of R that contains 3/5 (of corse, justify all your answers).
(on this page, A, B, C, D are all positive integers and A/B <C/D.) We saw in the previous assignment that CA CB - AD 1 DB=BD ? BD (The numerator must be an integer, and since the two fractions are unequal, it can't be 0.) In other words, "the closest two unequal rational numbers and can be is BD" (9.1) A sort of average of two fractions: . Show that <A+O- We gave an intuitive explanation of this in...
For the equation 2x2 + 2y2 - 12x + 8y - 24 = 0, do the following. (a) Find the center (h,k) and radius r of the circle. (b) Graph the circle. (c) Find the intercepts, if any. (a) The center is (Simplify your answer. Type an ordered pair.) The radius is ra (Simplify your answer.) (b) Use the graphing tool to graph the circle. Click to enlarge graph X (c) Find the intercepts, if any. Select the correct choice...
Let α and β be real numbers with 0 < α < βく2m and let h : [α, β] → R>o be a continuous function that is always positive. Define Rh,a to be the region of the (x,y)-plane bounded by the following curves specified in polar coordinates: r-h(0), r-2h(0), θ α, and θ:# β. 3. (a) Show that (b) (c) depends only on β-α, not on the function h. Evaluate the above integral in the case where α = π/4...
You are given a sequence of positive real numbers a[1..n]. You can now add ‘+’ and ’×’ signs between these numbers, and your goal is to generate an expression that has the largest value. As an example, if a = {2, 3, 0.5, 2}, then you should output the expression 2 × 3 + 0.5 + 2 = 8.5. This is larger than any other expression (e.g. 2 × 3 × 0.5 × 2 = 6, 2 + 3 +...
Suppose that f(x) is a continuous function over all real numbers, f'(- 10) = 0, and f''( - 10) = 24 Which of the following is true? (Hint: 2nd derivative test) Which of the following is true? (Hint: 2nd derivative test) O A. f(x) has a relative minimum at x = - 10 W O B. f(x) is decreasing when x = - 10 O C. f(x) is increasing when x = - 10 O D. f(x) has a relative...