Show that for any square-free integer n > 1, √ n is an irrational number

Topology question:

Show that a function f : ℝ → ℝ is continuous in the ε − δ definition of continuity if and only if, for every x ∈ ℝ and every open set U containing f(x), there exists a neighborhood V of x such that f(V) ⊂ U.

Use a direct proof to prove that 6 divides (n^3)-n whenever n is a non-negative integer.

Permutations and combinations.

a.)A typesetter has before him 26 trays, one for each letter of the alphabet. Each tray contains

10 copies of the same letter. In how many ways can he form a three letter word that

requires at most two different letter?

b.)Determine the number of ways of forming words which use exactly two different letters.

Solve h(x) =l -2x l

Solve y = 2 (x + 4)2 - 3

Please explain for each of the above provide examples · write down five (5) points on the graph of each equation · For each of your equation, does the graph of each equation have any intercepts? · State the domain for each of your equations. Write them in interval notation. · State the range for each of your equations. Write them in interval notation. · State whether each of the equations is a function or not giving your reasons for the answer. · Select one of your graphs and assume it has been shifted three units upward and four units to the left. · Incorporate the following four math vocabulary words into your discussion. Use bold font to emphasize the words in your writing. Do not write definitions; use sentences describing the thought behind your math.

o Function

o Relation

o Vertical Line test

o Transformation

P.0.2 Show that (a) the diagonal entries of a Hermitian matrix are real; (b) the diagonal entries of a skew-Hermitian matrix purely imaginary; c) the diagonal entries of a skew-symmetric matrix are zero.

P.0.5 Let A ∈ Mn be invertible. Use mathematical induction to prove that (A-1)k = (Ak)-1 for all integers k.

P.0.25 Let A ∈ Mn be idempotent. Show that A is invertible if and only if A = I

P.0.26 Let A,B ∈ Mn be idempotent. Show that tr((A-B)3) = tr(A-B).

Find the area of △ABC. Where:A=(3,2), B=(2,4), C=(0,2)

Suppose that a×b=〈−1,1,−1〉〉 and a⋅b=−4 Assume that θ is the angle between aand b. Find:

tanθ=

θ=

Find a nonzero vector orthogonal to both a=〈5,1,5〉, and b=〈2,−4,−1〉

Find a nonzero vector orthogonal to the plane through the points: A=(0,2,2), B=(−3,−2,2), C=(−3,2,3).

Find a nonzero vector orthogonal to the plane through the points: A=(0,1,−1), B=(0,6,−5), C=(4,−3,−4)..

Integral

Let f:[a,b]→R and g:[a,b]→R be two bounded functions. Suppose f≤g on [a,b]. Use the information to prove thatL(f)≤L(g)andU(f)≤U(g).

Information:

g : [0, 1] —> R be defined by if x=0, g(x)=1; if x=m/n (m and n are positive integer with no common factor), g(x)=1/n; if x doesn't belong to rational number, g(x)=0

g is discontinuous at every rational number in[0,1].

g is Riemann integrable on [0,1] based on the fact that Suppose h:[a,b]→R is continuous everywhere except at a countable number of points in [a,b]. Then h is Riemann integrable on[a,b].

f : [0,1]→R defined by (f(x) =0 if x = 0) and (f(x)=1 if 0 < x≤1)

f is integrable on [0,1]

**Please show all work and explain** Very confused

We can express insertion sort as a recursive procedure as follows. In order to sort A[1..n], we recursively sort A[1..n-1] and then insert A[n] into the sorted array A[1..n-1]. Write a recurrence for the running time of this recursive version of insertion sort.

Show that if S is bounded above and below, then there exists a number N > 0 for which - N < or equal to x < or equal to N if x is in S

Hi. I wanted to know how to prove this:

From the fact that the sum of two continuous functions is continuous, prove by induction that the sum of n continuous functions is continuous. I need to prove this by induction.

Thanks!

(1) Research online an alternative form of measuring angles (for example: radian or turn) and compare it with the degree measure. In your description include, if possible, its origin, applications, benefits or drawbacks.

(2) Provide an example of unit circle trigonometry being applied in real life.

Napoleon is contemplating four institutions of higher learning as options for a Master’s in Business Administration. Each university has strong and weak points and the demand for MBA graduates is uncertain. The availability of jobs, student loans, and financial support will have a significant impact on Napoleon’s ultimate decision. Vanderbilt and Seattle University have comparatively high tuition, which would necessitate Napoleon take out student loans resulting in possibly substantial student loan debt. In a tight market, degrees with that cachet might spell the difference between a hefty paycheck and a piddling unemployment check. Northeastern State University and Texas Tech University hold the advantage of comparatively low tuition but a more regional appeal in a tight job market. Napoleon gathers his advisory council of Jim and Pedro to assist with the decision. Together they forecast three possible scenarios for the job market and institutional success and predict annual cash flows associated with an MBA from each institution. All cash flows in the table are in thousands of dollars.

Suppose that the likelihood for each of scenarios 1 through 3 is 0.3, 0.4, and 0.3, respectively. What is the optimal decision under the EVM criterion?

Check if each of the following ODEs is an exact. If it is not an exact find an integrating factor to make it an exact. Then solve each of the following ODEs.

(e) dx + (x/y−siny)dy = 0

(f) ydx + (2x−ye^y)dy = 0

(g) (3x²y + 2xy + y³)dx + (x² + y²)dy = 0