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Basic Algebra

Constructing a polynomial

1 (a) Consider the function values... Consider a polynomial P(x) of least degree (the osculating polynomial) through the points xi=x0+i*h, i.e. polynomial that satisfies P(x0)=f0, P(x1)=f1, P'(x1)=f'(x1), P(x2)=f2 (b) Prove that df(x1)/dx = dP(x1)/dx for any smooth function f(x) (c) Construct the polynomial P(x) for the

Discharge over a spillway

The formula Q = 3.32LH^3/2 is used to find the maximum water discharge over a spillway before the water overflows the main part of the dam. Q is the discharge in cubic feet per second, L is the length of the spillway, and H is the depth of the spillway below the level of the dam. Find H when Q is 3000 cu. ft. per second and L =

Quadratic Equation Application Word Problems

A graphic artist is designing a poster that consists of a rectangular print with a uniform border. The print is to be twice as tall as it is wide, and the border is 3 inches wide. If the area of the poster is to be 680 square inches, find the demensions of the print. The foot of an extension ladder is 9 feet from the base

Ratios, LCD and Simplification

1.)The ratio of strawberry ice cream lovers to vanilla is 9 to 5. If there are 144 more strawberry ice cream lovers than vanilla, how many of each are there? 2.) Find the LCD and convert each rational expression into an equivalent rational expression with the LCD as the denominator: 4/15xy^2, 3x/6y 3.) Simplify and writ

How to solve for the variable "X"

How to solve for the variable "X" The following problems have tripped me up. I can't seem to figure out how to solve them. I've tried, several times, and had my algebra teacher explain them, but I still can't get the answers. 3x - 11 = 6 2x - 7 = 3 + x 3 + 3x = 9 + 2x 3x - 3

Ideals : Prime but not Maximal

Does the ring Z (integer) contain any ideals that are prime but not maximal? If so, give an example. If not, give a proof.

Extension Field : Kronecker's Theorem

For your information, Theorem 29.3 (Kronecker's Theorem): Let F be a field and let f(x) be a non-constant polynomial in F[x]. Then there exists an extention field E of F and an alpha in E such that f(alpha) = 0 Theorem 29.18: Let E be a simple extension F(alpha) of a field F, and let alpha be algebraic over F. Let

Gaussian Quadrature

1. Consider approximating integrals of the form I ( f ) = ∫ √x f(x)dx in which f(x) has several continuous derivatives on [0, 1] a. Find a formula ∫ √x f(x)dx ≈ w1 f(x1) ≡ I1( f ) which is exact if f(x) is any linear polynomial. b. To find a formula

Variables in an equation were reversed

If the variables in an equation were reversed, what would happen to the graph of the equation? For example, how would the graph of y = x² relate to the graph of x = y²? When might such a reversal of variables be useful in the real world? How does x=y relate to y=x? How about 2x = y and 2y=x? What about x^2 + y^2 = 49

Quadratic Equations : Maximum Value Word Problem

From this we can see that we have function that whose graph is a parabola and we want the values of w that make the function positive. The parabola opens downward, the vertex is at Let me show you an actual example of inequalities. Several years ago I was the chairman of the banquet committee for our ski club. Since one of ou

Physics and Algebra

1. Jeff rode his bike 45 miles to get to the park, and his co-worker Chris rode his bike 70 miles to meet Jeff at the park. Chris averaged 5 mph more than Jeff, and his trip took one-half hour longer than Jeff. How fast was each traveling? 2. Mike found a good deal on some golf clubs at an online auction site. The clubs were


Solve: 1/5 (x-15) + 1/9 (x+9)= x+7

Fourier Sine Series Solution - Wave Equation, Interval and Boundary Conditions

Please see attachment for complete questions (for the below "..." indicates equation to be found in attachment). Thanks! (a) Write down the Fourier (sine) series solution u(x,t) of the wave equation ... on the interval ... satisfying the boundary conditions ... and the initial conditions ... (b) Use the identity ... to sh

Ratio Problem

On a farm the ratio of sheep to cattle to goats is 8:3:1. How many cattle does the farmer own if there are 720 head of livestock, sheep, cattle and goats, altogether?