Use the substitution x = et to transform the given Cauchy-Euler equation to a differential equation with constant coefficients. (Use yp for dy/dt and ypp for d2y/dt2.)
x2y'' − 3xy' + 13y = 2 + 3x
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1. For the equation x2+y2−2x−4y−11=0, do the following.
(a) Find the center (h,k) and radius r of the circle.
(b) Graph the circle.
(c) Find the intercepts, if any.
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The size of the raccoon population at a national park last year was 180. This year, the population is 189. If the population increases exponentially, find the population of raccoons in 4 years. Use P = P0ekt where P0 represent the initial population. Round to the nearest whole number.
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ln(cos6x) Calculate the first four terms with the maclaurin series?
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Solve the following system. If the system's equations are
dependent or if there is no solution, state this.
3x - 4y - z = 18
4x + 2y + z = 16
7x - 2y + 3z = 19
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a) Let f(x) = −x^4 − 4x^3 . (i) Find the intervals of increase/decrease of f. (ii) Find the local extrema of f (values and locations). (iii) Determine the intervals of concavity. (iv) Find the location of the inflection points of f. (v) Sketch the graph of f. (You can choose your own scale for the graph)
b) A farmer wants to fence in an area of 6 km2 in a rectangular field and then divide it in half with a fence parallel to one of the sides of the rectangle. Find the dimensions of the rectangular field that minimize the amount of fence used.
Express the area of the region bounded by the graph of f(x) = 2 + √ x and the x-axis in the interval [3, 8] as a limit of sums using right endpoints. Do not evaluate the limit.
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Let f(x) = −x^4 − 4x^3.
(i) Find the intervals of increase/decrease of f.
(ii) Find the local extrema of f (values and locations).
(iii) Determine the intervals of concavity.
(iv) Find the location of the inflection points of f.
(v) Sketch the graph of f. (You can choose your own scale for the graph)
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Separation of Variables:
Let v(t) be the velocity, in knots per hour, of a vessel at time
t, t = 0. After the second engine is turned on, its velocity
satisfies the differential equation ??/??=3?+22, with initial
condition v (0) = 2.
First use separation of variables to find an expression for v in
terms of t. Secondly, f the vessel cruising speed is 25 knots per
hour, how long will the vessel take to reach this speed?
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The number? H(t) of veterans from foreign wars who are homeless or at risk of becoming homeless can be approximated by the exponential? function,
H(t)=65(1.65)t where t is the number of years since 2000.
?a) In what year were there 14,000 veterans who were homeless or at risk of becoming? homeless?
?b) What is the doubling time of homelessness among? veterans?
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A room has volume 100m3, and initially contains normal air. Every minute, 2m3 of gas is pumped in, which is 1.9m3 air, and 0.1m3 xenon. Every minute, 2m3 of well-mixed gas is pumped out. What volume of xenon will be in the room after 1 hour?
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Factor the polynomial as a product of linear factors: x4-5x3+12x2-2x-20
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Separation of Variables:
The rate at which fuel is being used by a Wartsila-Sulzer RTA96-C Turbocharged Diesel Engine is proportional to the amount of fuel already consumed at time t. If there are 2 gallons consumed when t 0 and 3 gallons consumed when t 5, how many gallons will be used when t = 9?
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A group of retailers will buy 116 televisions from a wholesaler if the price is $300 and 156 if the price is $250. The wholesaler is willing to supply 104 if the price is $220 and 184 if the price is $310. Assuming that the resulting supply and demand functions are linear, find the equilibrium point for the market. (q, p) =
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(a) Show that B {[2, 3, 0,1],[1, 1, 1,1]} is a maximal linearly independent subset of S {[1, 4, 1,2],[1, 1, 1,1],[3, 2,1, 0],[2, 3, 0,1]}.
(b) Calculate dim(span(S)).
(c) Does span(S) R4? Why or why not?
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An elastic band is hung on a hook and a mass is hung on the lower end of the band. When the mass is pulled downward and then released, it vibrates vertically. The equation of motion is ? = 2 /3 cos(8?)− 1 /6 sin(8?), ? ≥ 0, where ? is measured in centimeters and ? in seconds. (Take the positive direction to be downward.)
g. How far from its equilibrium position does the mass travel?
h. Find the total distance traveled by the mass during the first 4? seconds.
i. When is the mass speeding up when heading downward?
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