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

let f(x,y)=x^2y(2-x+y^2)-4x^2(1+x+y)^7+x^3y^2(1-3x-y)^8 find the coefficient of x^5y^3

The Eiffel Tower is 325 metres tall. Assuming the earth is perfectly spherical, how far can a person with perfect eye-sight can see from the top of the Eiffel Tower? Again, a good sketch would come in very handy for this problem. Since the precise definition of the word ”far” is somewhat unclear, an approximate answer should suffice. If you make any assumption, then explain why they are reasonable assumption.

Tom has taken out a loan for college. He started paying off the loan with a first payment of $200. Each month he pays, he wants to pay back 1.2 times as the amount he paid the month before. Explain to Tom how to represent his first 30 payments in sigma notation. Then explain how to find the sum of his first 30 payments, using complete sentences. Explain why this series is convergent or divergent.

Calculate the length of the curve r (t) = (3cost, 5t, 3sint) and calculate what is indicated below
a) Unit tangent vector T=
b) Main Normal Vector N =
c) Binormal vector B =
d) Function curvature k =
e) Torsion function t =
f) the tangential and normal acceleration components at = and aN =

Problem 3. An isometry between inner-product spaces V and W is a linear
operator L in B (V ,W) that preserves norms and inner-products. If x, y in V
and if L is an isometry, then we have <L(x),L(y)>_W = <x, y>_V .
Suppose that V and W are both real, n-dimensional inner-product spaces.
Thus the scalar field for both is R and both of them have a basis consisting of
n elements. Show that V and W are isometric by demonstrating an isometry
between them.
Hint: take both bases, and cite some linear algebra result that says that
you can orthonormalize them. Prove (or cite someone to convince me) that you
can define a linear function by specifying its action on a basis. Finally, define
your isometry by deciding what it should do on an orthonormal basis for V , and
prove that it preserves inner-products (and thus norms).

Problem 1. Show that the cross product defined on R^3 by [x1 x2 x3] X [y1 y2 y3]
= [(x2y3 − x3y2), (x3y1 − x1y3), (x1y2 − x2y1)] makes R^3 into an algebra.
We already know that R^3 forms a vector space, so all that needs to be shown is that
the X operator is bilinear.
Afterwards, show that the cross product is neither commutative nor associative.
A counterexample suffices here. If you want, you can write a program that
checks the commutative and associative laws for x, y, z in R^3, and then simply
generate random integer vectors x, y, z in Z^3 a subset of R^3 until those laws fail.

5. The cost of heart procedures for a county-run hospital is $125,000, with a fixed cost of $50,000 and a variable cost of $75,000. The hospital completed 250 procedures last year. [A] Assuming no changes in the number of procedures the hospital will perform next year, calculate the cost of surgery for next year, if (a) a 2% increase in variable and 1.5% increase in fixed costs, and (b) it wants to maintain 3.5% safety margin. [B] What would be the cost if the number of procedures (1) will increase by 2%, and (2) will decrease by 4%, with and without safety margin, assuming no changes in cost conditions next year?

7.

Prove that the folowing is a valid argument, explaining each rule of inference used to arrive at the conclusion.

1. “I take the bus or I walk. If I walk I get tired. I do not get tired. Therefore I take the bus.”

2. “All lions are fierce. Some lions do not drink coffee. Therefore, some fierce creatures to not drink coffee."

If technology is so wonderful, why do i need to understand the basics of matrix operations? Why cant i just an app on a smart phone?

4(a). Find the equation y = ax + b of the line passing through (2,3) and (5,8). (Your answer should be an equation of the form y = ax + b, for some constants a and b.)

(b) Find the equation y = ax^2 + bx + c of the parabola passing through the points (-2, -6), (1,6), and (3,4). (Your answer should be an equation of the form y = ax^2 + bx + c, for some constants a, b, and c.)

Hint: Substituting the x- and y-coordinates of a point into the equation y = ax^2 + bx + c will produce a linear equation in a, b, and c. Do this for the three given points to get the three linear equations. Then solve the system of linear equations.

Consider the solid S with the following properties: • The base of S is the triangle T with vertices (0, 0),(1, 0), and (0, 1). • When S is sliced perpendicularly to the x-axis, it has square cross sections.

(a) (4 points) Sketch the base T and determine the equations of its three edges.

(b) (3 points) Set up an integral to compute the volume of S.

(c) (3 points) Evaluate the integral from part (b).

Show that the integers have infinite index in the additive group of rational numbers.