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Basic Algebra

Differentiation : Radius of Convergence for Power Series

Consider the differnetial equation y'(x) + xy(x) = 0 with y(0) = 0 Look for a solution of this problem of the form y(x) = A + B + Ce^-x + De^-1/2x^2 Use the fact that y must satisfy the equation and the initial conditions to identify the constants A,B,C and D. By setting u = -x^2/2 in the power series for f(u) = exp{u},

Sides of a Cube Problem

Eight small cubes are put together to form one large cube. All six sides of the larger cube are painted, the paint is allowed to dry, and then the cube is taken apart. a) How many of the small cubes will have paint on just one side? On two sides, On three sides? On no sides? b) Complete the following table, assuming in turn

Algebra : Graphing Functions

Please see the attached file for the fully formatted problems. Can you please assist me with the problems listed below? P. 237 1. a) #6, b) #18 2. a) #22, b) #24 3. a) #32, b) #46 P.253 4. a) #1, b) #2, c) #3. d_ #4 5. a) #10, b) #12 6. a) # 28, b) #30 7. P.260, Matched Problem 1 P.271 8. # 2 - 22 (Eve

Linear Congruences: Example Problems

Find the solutions of (1) 153x is congruent to 6 (mod 12) (2) x + 1 is congruent to 3 (mod 7) (3) 8x is congruent to 6 (mod 422) (4) 363x is congruent to 345 (mod 624).

Proving that an equality is false.

I have two sets of 64 numbers (1.1 to 7.4). Both number sets are created using the same equation for values of i from 0 to 63. m = 1.1 + ( i * 0.1 ) n = 1.1 + ( i * 0.1 ) I am trying to understand if the following equality is false in all cases except when the terms in each expression are equal (e.g. m^-12 = n^-12 and

Logarithms and Exponents Rule

Logarithms and exponents. See attached file for full problem description. Q1: Express in log form 82=64 Q2: express in exponential form log4256=4 Q3: find unknown log327=x Q4: express as sum diff or multiple logarithm a. log314 b. log2 5/10 c. log5 (n3) Q5: express as log of single quantity a. long23 + log22

Algebra - Graphing a Function

Sketch the graph of each function.  Be sure to label three points on the graph. If f(x) = integral (2x), find: (a) f(1.2)  (b) f(1.6)  (c) f(-1.8)

Several proofs by induction

Show by induction that: a) Sum(n^2/[(2n-1)*(2n+1)],n=1..) = n(n+1)/[2*(2n+1)] b) Sum( r/(r-1)!, r=1..n) = 1-1/(n+1)! c) n^3+3n^2-10n is divisible by 3 d) 4^(2n+1) + 3^(n+2) is divisible by 13 See attached file for full problem description.

Induction

Prove by induction where n is a positive integer. (The questions are attached).

Algebra problem and rectangles

If the perimeter of a rectangle is 10 inches, and one side is one inch longer than the other, how long are the sides? Can you show me the steps to take to work out this and similar problems.

Investigation (Diagonals in a Rectangle)

Diagonals in a Rectangle. In the case of a 2 X 2 rectangle, or a 3 X 5 rectangle, we can simply count. However, can we make a decision about a 100 X 167 or a 3600 X 288 rectangle? In general, given an N X K rectangle, how many grid squares are crossed by its diagonal?

Algebra - Circle center, radius, equation

The endpoints of the diameter of a circle are P=(-3,2) and Q=(5,-6) Find: (i) the center of the circle (ii) The radius of the circle (iii) the equation of the circle.

Fibonacci Numbers for Mathematical Inductions

Application of Mathematical Induction It is an application of Mathematical Induction in proving the relations of Fibonacci Numbers. To prove: F 2n+1 - Fn Fn+2 = (-1) n for n > or, = 3.

Mixing Problem : Exponenetial Functions

A 5000 gallon aquarium is maintained with a pumping system that circulates 100 gallons of water per minute through the tank. To treat a certain fish malady, a soluble antibiotic is introduced into the inflow system. Assume that the inflow concentration of medicine is 10te-t/50 oz/gal, where t is measured in minutes. The well-

Simplifying Algebraic Expressions

I will use the ^ sign for the squared sign. Write the following algebraic expression in its simplest form: x^ + 2x + 3x^ + 2 + 4x + 7

Time, Rate, and Distance

I know the formula Rate x Time = Distance. However, I have a question where I am asked to find the time and not the distance? Can you give me some basic information about how to calculate the time using this formula or is there a different formula that I should be using? Thanks.

Distance, Rate and Time Problem

Karen can row a boat 10 kilometers per hour in still water. In a river where the current is 5 kilometers per hour, it takes her 4 hours longer to row a given distance upstream than to travel the same distance downstream. Find how long it takes her to row upstream, how long to row downstream, and how many kilometers she rows?

Exponential and Logarithmic Functions : Level of Sound

Determine the level of sound (in decibels) for the given sound intensity. (a) I=10^-3.5 watt per m^2 (jet 4 miles from takeoff) (b) I=10^-3 watt per m^2 (diesel truck at 25 feet) (c) I=10^-1.5 watt per m^2 (auto horn at 3 feet)

Exponential and Logarithmic Functions: Example Problems

Sales and Advertising: The sales S (in thousands of units) of a product after x hundred dollars is spent on advertising is: S = 10(1-e^kx). When $500 is spent on advertising, 2500 units are sold. (a) Complete the model by solving for K. (b) Estimate the number of units that will be sold if advertising expenditures are raised

Irreducible Polynomial over a Field

Please see the attached file for the fully formatted problems. 5. Find an irreducible polynomial f(x) over the field Z3 with Z3[x]/(f(x)) = F243. Note that 243 = 3^5 . Please explain your reasoning and solution in as much detail as possible. Thank You.