Use Stoke's Theorem to evaluate $\int F \cdot d r$ where C is oriented counterclockwise as viewed from above. $F(x, y, z)=e^{-x} i+e^{x} j+e^{z} k$ ,C is the boundary of the part of the plane 4x + y + 4z = 4 in the first octant.

Consider the set C(R) = {continuous functions of real numbers} and consider its subsets. Show that C(R) is a vector space a. Show that the set of differentiable functions is a subspace. b. Show that the sets O of odd functions and E of even functions are subspaces. c. Determine if the set U of functions whose definite integral over (0,1) is a subspace or not.

Q−1: [6+6 marks] a) ​​​​​​​Given the functions 〖f(x)=x〗^2-4x+4 and g(x)=1/(√x-1) i) Find the domain of f(x) and g(x). ii) Compute (g o f)(x). b) Find the points of intersection of the curves y=5x^2-3x-2 and y=2x-1

Mr. Hollywood has been trading for some years as a retail trader. Given below are the ledger balances as at 30 June 20X7, the end of his most recent financial year. $ Capital 83,887 Sales 259,870 Trade creditors 19,840 Returns out 13,407 Provision for doubtful debts 512 Discount allowed 2,306 Discount received 1,750 Purchases 135,680 Returns inwards 5,624 Carriage outwards 4,562 Drawings 18,440 Carriage inwards 11,830 Rent, rates and insurance 25,973 Heating and lighting 11,010 Postage, stationery &telephone 2,410 Advertising 5,980 Salaries & wages 38,521 Bad debt 2,008 Cash in hand 534 Cash at bank 4,440 Stock as at 1 May 20X6 15,654 Trade debtors 24,500 Fixtures & Fittings- at cost 120,740 Provision for depreciation on fixtures& fittings-30 April 20X7 63,020 Depreciation 12,074 You are given the following additional information as at 30 April 20X7. (a) Stock at the close of business was valued at $ 17,750 (b) Insurance has been prepaid by $ 1,120 © Heating and lighting is accrued by $ 1,360 (d) Rates have been prepaid by $ 5,435 (e) The provision for doubtful debts is to be adjusted so that it is 3% of trade debtors Required: Prepare Mr. Hollywood’s statement of profit or loss for the year ended 30 April 20X8 and a statement of financial position as at that date

(4 points) If \[ A=\left[\begin{array}{cc} -5 & 6 \\ 1 & -5 \end{array}\right] \] then \[ A^{-1}=\left[\begin{array}{l|c} \hline-5 / 19 & \# \\ \hline-1 / 19 & \# \\ \hline \end{array}\right. \] Given $\vec{b}=\left[\begin{array}{l}4 \\ 3\end{array}\right]$, solve $A \vec{x}=\vec{b}$ (4 points) If \[ A=\left[\begin{array}{ccc} -2 & -2 & 9 \\ -5 & -4 & 22 \\ 1 & 1 & -5 \end{array}\right] \] then Given $\vec{b}=\left[\begin{array}{c}-4 \\ 2 \\ -2\end{array}\right]$, solve $A \vec{x}=\vec{b}$

Please spend extra effort to write up this problem’s solution as an exposition that can be read and understood by a beginning algebra student. That student knows function notation and standard properties of polynomials (as taught in a high school algebra course). (a) Find all polynomials f that satisfy the equation: f(x + 2) = f(x) + 2 for every real number x. (b) Find all polynomials g that satisfy the equation: g(2x) = 2g(x) for every real number x. (c) The problems above are of the following type: Given functions H and J, find all polynomials Q that satisfy the equation: J(Q(x)) = Q(H(x)) for every x in S where S is a subset of real numbers. In parts (a) and (b), we have J = H and S is all real numbers, but other scenarios are also interesting. For example, the choice J(x) = 1/(x − 1) and H(x) = 1/(x + 1), generates the Find all polynomials Q that satisfy the equation: 1/(Q(x) − 1) = Q(1/(x + 1)) for every real number x such that those denominators are nonzero. Is this one straightforward to solve? (d) Make your own choice for J and H, formulate the problem, and find a solution. Choose J and H to be non-trivial, but still simple enough to allow you to make good progress toward a solution

The table gives the values of a function obtained from an experiment. Use them to estimate 9 f(x) dx 3 using three equal subintervals with right endpoints, left endpoints, and midpoints. x 3 4 5 6 7 8 9 f(x) −3.5 −2.3 −0.5 0.2 0.9 1.5 1.9 (a) Estimate 9 f(x) dx 3 using three equal subintervals with right endpoints. R3 = If the function is known to be an increasing function, can you say whether your estimate is less than or greater than the exact value of the integral? less thangreater than one cannot say (b) Estimate 9 f(x) dx 3 using three equal subintervals with left endpoints. L3 = If the function is known to be an increasing function, can you say whether your estimate is less than or greater than the exact value of the integral? less thangreater than one cannot say (c) Estimate 9 f(x) dx 3 using three equal subintervals with midpoints. M3 = If the function is known to be an increasing function, can you say whether your estimate is less than or greater than the exact value of the integral? less thangreater than one cannot say

The Cantor set, named after the German mathematician Georg Cantor (1845–1918), is constructed as follows. We start with the closed interval [0,1] and remove the open interval ( 1/3 , 2/3 ). That leaves the two intervals [0, 1/3 ] and [ 2/3 , 1] and we remove the open middle third of each. Four intervals remain and again we remove the open middle third of

Find the extremal function of the following:  a) Integral from a to b of [1/2 x^2 (y')^2 + y^2 + x^2y] dx                                                                      b) Integral from a to b [(y')^2 - y^2 + 2y sec x] dx

Find a particular solution to y'' - 6y' + 9y = (-3.5e^(3t))/(t^2+1).   Prefer via Laplace Transform please. I just need the particular solution, not the complimentary or the general solution. I'd like to understand it via Laplace Transform, but if it can't be done then I understand. 

(1 point) Find a possible formula for the trigonometric function graphed below. Use $x$ as the independent variable in your formula. \[ f(x)= \] Iti help (formulas)

Let $f$ be the function defined by $f(x)=\left\{\begin{array}{ll}\frac{1}{2}(x+2)^{2} & \text { for }-2 \leq x<0 \\ 2-2 \sin \sqrt{x} & \text { for } 0 \leq x \leq \frac{x^{2}}{4}\end{array}\right.$. The graph of $f$ is shown in the figure above. Let $\boldsymbol{R}$ be the region bounded by the graph of $\boldsymbol{f}$ and the $\boldsymbol{z}$-axis. (a) Find the area of $\boldsymbol{R}$ (b) Region $\boldsymbol{R}$ is the base of a solid. For this solid, each cross section perpendicular to the $\boldsymbol{m}$. axis is a square. Write, but do not evaluate, an expression involving one or more integrals that gives the volume of the solid. (c) The portion of the region $\boldsymbol{R}$ for $\mathbf{1} \leq \boldsymbol{v} \leq \boldsymbol{2}$ is revolved about the $\boldsymbol{\pi}$-axis to form a solid. Find the volume of the solid. Let f be the function defined by F(x)=(1/2)(x+2)^2 for [-2,0) and 2-2sin(sqrtx) for [0, (x^3)/4]. the graph of f is shown in the figure above. Let R be the regiok bounded by the graph of f and the x-axis.

Consider your student ID as eight decimal digits, $d_{1}, d_{2}, \ldots, d_{8}$, read leftto-right. For example, for the student ID 01234868 , \[ d_{1}=0, d_{2}=1, d_{3}=2, d_{4}=3, d_{5}=4, d_{6}=8, d_{7}=6, d_{8}=8 \text {. } \] Fill in the $d_{i}$ values using your student ID and select all true statements. $\left(d_{1}, d_{2}, \ldots, d_{8}\right)$ is a string of length 8 from the alphabet set $\mathrm{D}$. $\left[d_{1}, d_{2}, \ldots, d_{8}\right]$ is an 8-combination with repetition of elements in the set $\mathrm{D}$. $\left(d_{1}, d_{2}, \ldots, d_{8}\right)$ is a length 8 permutation of elements from the set $\mathrm{D}$. $\left(d_{1}, d_{2}, \ldots, d_{8}\right)$ is a length 8 sequence of elements from the set $\mathrm{D}$. $\left\{d_{1}, d_{2}, \ldots, d_{8}\right\}$ is an element of the power set of set $\mathrm{D}$. $\left\{d_{1}, d_{2}, \ldots, d_{8}\right\}$ is a length 8 combination of elements from the set $\mathrm{D}$.

Read Theorem 5.14 (Remainders in Alternating Series) in the textbook $\mathrm{e}^{\text {T }}$. The alternating series $\sum_{n=1}^{\infty}(-1)^{n} \frac{1}{3^{n} n}$ converges. (You do not need to prove this, but you can if you want to.) Suppose we approximate the sum of the series by the partial sum $S_{k}$. What is the smallest number $k$ could be if we want the error to be at most 0.001 ?

Question 1 (a) Show that the Euler-Lagrange equation for the functional \[ S[y]=\int_{0}^{\frac{\pi}{2}}\left(y^{\prime 2}-y^{2}+4 y e^{-x}\right) d x, \quad y(0)=0, y(\pi / 2)=1, \] is \[ y^{\prime \prime}+y=2 e^{-x} . \] (b) Show that the stationary function is given by \[ y=e^{-x}-\cos x+\left(1-e^{-\pi / 2}\right) \sin x . \] Question 2: (a) Use the first integral of the Euler-Lagrange equation to show that the functional \[ S[y]=\int_{0}^{1} \frac{y^{4}}{y^{5}} d x, \quad y(0)=1, y(1)=\frac{1}{16}, \] has two real-valued stationary paths, given by \[ y(x)=(x+1)^{-4} \text { and } y(x)=(3 x-1)^{-4} . \] (b) Find the value of the functional on each stationary path.

23. (a) Compute the eigenvalues of each of the following matrices. (i) $\left[\begin{array}{ll}4 & 0 \\ 2 & 2\end{array}\right]$ (ii) $\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right]$ (iii) $\left[\begin{array}{ll}2 & 1 \\ 2 & 3\end{array}\right]$ (iv) $\left[\begin{array}{rr}4 & -1 \\ 1 & 2\end{array}\right]$ (v) $\left[\begin{array}{rrr}0 & 2 & 1 \\ 3 & -1 & -3 \\ -2 & 2 & 3\end{array}\right]$ (b) Determine an eigenvector associated with the largest eigenvalue, using the method in Example 4, for the matrices in part (a). Please show ALL work no matter how small the step is.

Emma and Tom are playing a game in turns. Each player stands on their own number line, which has spaces numbered 1 through n for some integer n ≥ 3. Emma goes first, and starts at position 1 on her number line, and Tom starts at position n on his. On each player's turn, that player must move to another number on their number line. The new number need not be adjacent to the old one. But no repetitions are allowed in the pair of positions: if Emma moves back to position 1, then Tom cannot move back to position n on his next move, because the ordered pair (1,n) has already occurred. The first player who cannot move loses.  1) For each n, who wins?  2)  Now suppose they are on the same number line, and cannot simultaneously occupy the same spot. (So if Emma is at 1 and Tom is at 3, Emma can move to 2, 4, 5, …, n but not 1 or 3.) Furthermore, a position is considered the same if the same spots are occupied (so N=1, T=3 is a repetition of N=3, T=1). For each n, who wins? 3) Same as (b), but the players are considered distinct (so N=1, T=3 isn’t a repetition of N=3, T=1).  

Let $\left\{x_{n}\right\}_{n}$ and $\left\{y_{n}\right\}_{n}$ be positive sequences. We say that $\left\{x_{n}\right\}_{n}$ is asymptotically similar to $\left\{y_{n}\right\}_{n}$ if $\lim _{n \rightarrow \infty} \frac{x_{n}}{y_{n}}$ exists and is non-zero. Prove each of the following true statements. IF $\left\{x_{n}\right\}_{n}$ is asymptotically similar to $\left\{y_{n}\right\}_{n}$ THEN there exists $C>0$ such that for all $n$ large enough \[ C y_{n}<x_{n}<2 C y_{n} \] Define the sequence $\left\{a_{n}\right\}_{n=0}^{\infty}$ by \[ \begin{aligned} a_{0} & =0, \\ \forall n \in \mathbb{N}^{+}, \quad a_{n} & =\ln (n !)-\left(n+\frac{1}{2}\right) \ln n+n . \end{aligned} \] (a) Show that IF $\left\{a_{n}\right\}_{n}$ converges THEN $n$ ! is asymptotically similar to $n^{n+\frac{1}{2}} e^{-n}$.

(5) [Marked] Consider the downward-oriented (in the negative y-direction) quintic curve, \[ C=\left\{(x, y) \in \mathbb{R} \times(-1,1) \mid x=y^{5}+y+2\right\}, \] and consider the vector field $\mathbf{F}$ on $\mathbb{R}^{2}$ given by \[ \mathbf{F}(x, y)=(x-y, x+y)_{(x, y)} . \] (a) Give an injective parametrisation $\gamma$ of $\mathrm{C}$ such that the image of $\gamma$ differs from $\mathrm{C}$ by only a finite number of points. Which orientation of $\mathrm{C}$ does $\gamma$ generate? (b) Compute the (curve) integral of $\mathbf{F}$ over the curve $\mathbf{C}$.

The demand and supply functions for pans are q=710−5,5p and p=3,5q+57,5, respectively, with the price in rand and q the quantity in boxes. Find the point(s) where the demand and supply function intersect. a. (p;q)=(126;14,556) b. (p;q)=(14,556;126) c. (p;q)=(126;126) d. (p;q)=(14,556;14,556)

(1 point) Let $f(x)$ be a function that is defined and has a continuous derivative on the interval $(2, \infty)$. Assume also that \[ \begin{array}{c} f(3)=-5 \\ |f(x)|<x^{6}+1 \end{array} \] and \[ \int_{3}^{\infty} f(x) e^{-x / 5} d x=-5 \] Determine the value of \[ \int_{3}^{\infty} f^{\prime}(x) e^{-x / 5} d x \] point) Evaluate the following improper integral. If the integral is divergent, enter "divergent" as answer. \[ \int_{5}^{-5} \frac{1}{|x|^{2 / 3}} d x \]

Let, $\mathrm{x}$ has a pdf of $f(x)=\left\{\begin{array}{c}x+1,-1<x<0 \\ 1-X, 0<x<1 \\ 0, \text { Otherwise }\end{array}\right.$ Convert to $\mathrm{cdf}$ all x are the same and lower case, disregard the upper case X and treat as x

Q3 a) to h) 3. Hen we consider warious solids of recolution whose aris is the diagonal line $y=x$. The goal is to adopt the "pile of thin disks" idea from Question 2(b) to this new situation. For each real constant $m \neq 1$, the curve $y=m x+(1-m) x^{2}$ is a parabola that croses the line $y=x$ at the points $(0,0)$ and $(1,1)$. Let $R(m)$ denote the finite region between the parabola and the line. Then let $S(m)$ denote the solid generated by rotating $R(m)$ around the line $y=x$. We are interested in the volume of the solid $S(m)$; call this volume $V(m)$. (a) Using the same set of axes, draw the line $y=x$ and several of the parabolas $y=m x+(1-m) x^{2}$. Your sketch should show enough parabolas to communicate all the different types of posible shapes. (b) Suppose $m=-6$. With reference to a suitable sketch, explain why the "plle of thin disks" idea cannot be used directly to find the volume $V(-6)$. Then determine the largest closed interval $\left[m_{0}, m_{1}\right]$ of $m$-values for which this idea can be used directly. (c) Suppose $m=0$. Let $r(x)$ denote the radius of the disk of revolution whose centre has coordinates $(x, x)$. Find a formula for $r(x)$, valiel for each number $x$ in $[0,1]$. Check that the values for $r(0)$ and $r(1)$ are compatible with the geometry of the situat ion. (d) Repeat part (c), but use $m=2$. (e) Extend your reasoning in parts (c)-(d) to produce a formula for $r(x)$ that involves $m$, and holds for each $m$ (except $m=1$ ) in the interval $\left[m_{0}, m_{1}\right]$ found in part (b). Note: Your formula should reproduce your earlier findings when $m$ is replaced with either 0 or 2 . (f) Suppose $m \neq 1$ lies in the interval $\left[m_{0}, m_{1}\right]$ found above. Set up, but do not evaluate, a definite integral whose exact value equals the volume $V(m)$, in terms of the function $r(x)$ found earlier. Hint: The answer is not quite $\int_{0}^{1} \pi r(x)^{2} d x$. (g) Find the exact volume $V\left(m_{0}\right)$, where $m_{0}$ is the smallest number in the interval found in part (b). (h) Find the exact volume $V(0)$. 3. Herr we consider various solids of revolufion whose axis is the diagonal line $y=x$. The goal is to adnpt the "pile of thin disks" idea from Question $2(b)$ to this new situation. For each real constant $m \neq 1$, the curve $y=m x+(1-m) x^{2}$ is a parabola that croeses the line $y=x$ at the points $(0,0)$ and $(1,1)$. Let $R(m)$ denote the finite region between the parabola and the line. Then let $S(m)$ denote the solid generated by rotating $R(m)$ around the line $y=x$. We are interested in the volume of the solid $S(m)$ : call this volume $V(m)$. (a) Using the same set of axes, draw the line $y=x$ and several of the parabolas $y=m x+(1-m) x^{2}$. Your shetch should show enough parabolas to communicate all the different types of possible shapes. (b) Suppose $m=-6$. With reference to a suitable sketch, explain why the "pile of thin disks" idea cannot be used directly to find the volume $V(-6)$. Then determine the largest cloeed interval $\left[m_{0}, m_{1}\right]$ of $m$-values for which this idea can be used directly. (c) Suppose $m=0$. Let $r(x)$ denote the radius of the disk of revolution whose centre has coordinates $(x, x)$. Find a formula for $r(x)$, valid for each number $x$ in $[0,1]$. Check that the values for $r(0)$ and $r(1)$ are compatible with the geometry of the situation. (d) Repeat part (c), but use $m=2$. (e) Extend your reasoning in parts (c) -(d) to produce a formula for $r(x)$ that involves $m$, and holds for each $m$ (except $m=1$ ) in the interval $\left[m_{0}, m_{1}\right]$ found in part (b). Note: Your formula should reproduce your earlier findings when $m$ is replaced with either 0 or 2. (f) Suppose $m \neq 1$ lies in the interval $\left[m_{0}, m_{1}\right]$ found above. Set up, but do not evaluate, a definite integral whose exact value equals the volume $V(m)$, in terms of the function $r(x)$ found earlier. Hint: The answer is not quite $\int_{0}^{1} \pi r(x)^{2} d x$. (g) Find the exact volume $V\left(m_{0}\right)$, where $m_{0}$ is the smallest number in the interval found in part (b). (h) Find the exact volume $V(0)$.

2. Explicitly compute the following commutators: $\left[L_{x}, X\right],\left[L_{x}, Y\right],\left[L_{x}, Z\right],\left[L_{x}, r^{2}\right]$, $\left[L_{x}, P_{x}\right],\left[L_{x}, P_{y}\right],\left[L_{x}, P_{z}\right],\left[L_{x}, \mathbf{P}^{2}\right]$. Here $r^{2}=X^{2}+Y^{2}+Z^{2}$.

PLEASE DO THIS ASAP THANKS WILL LEAVE AMAZING REVIEW! 8. A function = (t) represents the position of an object based on time t in seconds. Determine what the following represent in the context of the problem. [2 marks each] a) $z(5)$ b) $\frac{s(22)-s(5)}{17}$ c) $\frac{s(4)-s(-2)}{6}$

DISCRETE MATH Question 1: Let A be the set of all statement forms in the three variables p, q, and r, and let R be the relation defined on A as follows: For all S and T in A, S R T ⇔ S and T have the same truth table. What are the distinct equivalence classes of R? Thus, there are ________ distinct equivalence classes. Question 2: There are as many equivalence classes as there are which of the following? (Select all that apply.) a. distinct real numbers b. distinct horizontal lines in the plane c. distinct integers d. distinct lines in the plane whose coordinates equal each other e. distinct vertical lines in the plane

Suppose that α, β are disjoint cycles in the symmetric group Sm for m ≥ 9.   a) Let α by a cycle of length 3 and β a cycle of length 9. What is the order of αβ. Is the permutation αβ even or odd? b/  Show that for every positive integer n we have (αβ)raised to the power n = α raised to the power n times  β raised to the power n

1. (10 points) Given the return matrix \[ R=\left(\begin{array}{cc} 2 & 0 \\ -1 & 0.1 \end{array}\right) \] and the probabilities $p_{1}=0.8, p_{2}=0.2$, compute without using any software the maximum expected return that we can get while the ADR is zero.

Q3 a) to h) 3. Here we consider various solids of revolution whose aris is the diagonal line $y=x$. The goal is to adapt the "pile of thin disks" idea from Question $2(b)$ to this new situation. For each real constant $m \neq 1$, the curve $y=m x+(1-m) x^{2}$ is a parabola that crosses the line $y=x$ at the points $(0,0)$ and $(1,1)$. Let $R(m)$ denote the finite region between the parabola and the line. Then let $S(m)$ denote the solid generated by rotating $R(m)$ around the line $y=x$. We are interested in the volume of the solid $S(m)$ : call this volume $V(m)$. (a) Using the same set of axes, draw the line $y=x$ and several of the parabolas $y=m x+(1-m) x^{2}$. Your sketch should show enongh parabolas to communicate all the different types of possible shapes. (b) Suppose $m=-6$. With reference to a suitable sketch, explain why the "pile of thin disks" idea cannot be used directly to find the volume $V(-6)$. Then determine the largest closed interval $\left[m_{0}, m_{1}\right]$ of $m$-values for which this idea can be used directly: (c) Suppose $m=0$. Let $r(x)$ denote the radius of the disk of revolution whose centre has coordinates $(x, x)$. Find a formula for $r(x)$, valid for each number $x$ in $[0,1]$. Check that the values for $r(0)$ and $r(1)$ are compatible with the geometry of the situation. (d) Repeat part (c), but use $m=2$. (e) Extend your reasoning in parts (c)-(d) to produce a formula for $r(x)$ that involves $m$, and holds for each $m$ (except $m=1$ ) in the interval $\left[m_{0}, m_{1}\right]$ found in part (b). Note: Your formula should reproduce your earlier findings when $m$ is replaced with either 0 or 2. (f) Suppose $m \neq 1$ lies in the interval $\left[m_{0}, m_{1}\right]$ found above. Set up, but do not evaluate, a definite integral whose exact value equals the volume $V(m)$, in terms of the function $r(x)$ found earlier. Hint: The answer is not quite $\int_{0}^{1} \pi r(x)^{2} d x$. (g) Find the exact volume $V\left(m_{0}\right)$, where $m_{0}$ is the smallest number in the interval found in part (b). (h) Find the exact volume $V(0)$.

A hiker is standing beside a stream on the side of a mountain, examining her map of the region. The height of land in kilometres) at any point (x, y) is given by 20 h(x, y) = 3 + x2 + 2y2' where x and y (also in kilometres) denote the coordinates of the point on the hiker's map. The hiker is at the point (3,2). (a) What is the direction of flow of the stream at (3, 2) on the hiker's map? How fast is the stream descending at her location? y hop, 7 (b) Find the equation of the path of the stream on the hiker's map. (c) At what angle to the path of the stream (on the map) should the hiker set out if she wishes to climb at a 15° inclination to the horizontal? (d) Make a sketch of the hiker's map, showing some curves of constant elevation, and showing the stream