Susie

x*y = 100, minimize x+y. Thank you!

Let f(x) be a continuous function on [0,a], where a>0, such that f(x)+f(a-x) does not vanish on [0,a]. Evaluate the integral integral from 0 to a of f(x)/{((x)+f(a-x)),dx.

(a) The base of a solid is the region between the parabolas x = y^2 and 2y^2 = 3 - x. Find the volume of the solid if the cross-sections perpendicular to the x-axis are equilateral triangles.

(b) Let f be a continuous real-valued function such that (f'(x))^2 = f(x)f"(x). Suppose f(0) = 1 and f^(4)(0) = 9. Find all possible values of f'(0).

(c) SUppose f(x) is such that

Integral from - infinity to infinity of e^(tx) f(x) dx = sin^-1 (t - sqrt(1/2))

For all t where the right side expression is defined. Compute

integral from -infinity to infinity of xf(x) dx

The graph of the polar equation r = a cos theta + b sin theta, where a and b are positive real numbers, is a circle. Where (in rectangular coordinates) is its center?

(a) Define a sequence {a_n)_n=1^infinity a_1= sqrt(2) , a_n = sqrt(2a_(n-1)) for all n > 1. Compute Lim[n->infinity] a_n

(b) Find a real number A>0 such that Sum^infinity_(n=1) of n!/((cn)^n) converges if c > A and diverges if c < A.

Find the surface area of the torus of radius a with cross-sectional radius b.

integral from 1 to 2 of (9x+4)/(x^5+3x^2+x) dx as a log please

Acceleration due to gravity is actually slightly lower at higher altitudes. In particular, g = \frac{GM}{r^2}, where G is a constant (called the gravitational constant), M is the mass of the Earth, and r is the distance from the center of the Earth.

A plane that is initially on the ground takes off, and climbs to an altitude of 10 km above the surface of the Earth. Estimate the change in gravity that the plane experiences, as a percentage. Assume that the radius of the Earth is 6400 km.

evaluate integral from 1 to infinity of (log(x)/x)^2011

luded it here. It is a "predator-prey" problem of a different sort.) Juliet is in love with Romeo, but Romeo is more fickle. The more Juliet loves him, the more he hates her, and the more she hates him, the more he loves her. On the other hand, Juliet is more sensible: the more Romeo loves her, the more she loves him, and the more he hates her, the more she hates him.

(a) Explain why a reasonable model for their love for each other is \frac{dj}{dt} = kr,\; \frac{dr}{dt} = -kj, where j and r denote their love (if positive) or hate (if negative) for each other, and where k is a positive constant.

(b) Solve this system for r and j. Your answer will depend on k and two other arbitrary constants. (Hint: write a differential equation involving only one of r or j.)

(c) Suppose at time t=0, Romeo is fully in love with Juliet (so that r(0) = 1), but Juliet is indifferent to Romeo (so that j(0) = 0). Write equations for the solution in terms of k.

(d) Sketch the solution from (c), and explain in words what is going on.

Juliet is in love with Romeo, but Romeo is more fickle. The more Juliet loves him, the more he hates her, and the more she hates him, the more he loves her. On the other hand, Juliet is more sensible: the more Romeo loves her, the more she loves him, and the more he hates her, the more she hates him.

(a) Explain why a reasonable model for their love for each other is \frac{dj}{dt} = kr,\; \frac{dr}{dt} = -kj, where j and r denote their love (if positive) or hate (if negative) for each other, and where k is a positive constant.

(b) Solve this system for r and j. Your answer will depend on k and two other arbitrary constants. (Hint: write a differential equation involving only one of r or j.)

(c) Suppose at time t=0, Romeo is fully in love with Juliet (so that r(0) = 1), but Juliet is indifferent to Romeo (so that j(0) = 0). Write equations for the solution in terms of k.

(d) Sketch the solution from (c), and explain in words what is going on.

A not uncommon calculus mistake is to believe that the product rule for derivatives says that (fg)'=f'g'. If f(x)=e^{x^2}, determine, with proof, whether there exists an open interval (a,b) and a nonzero continuous function g defined on (a,b) such that this wrong product rule is true for x in (a,b).

Juliet is in love with Romeo, but Romeo is more fickle. The more Juliet loves him, the more he hates her, and the move she hates him, the move he loves her. On the other hand, Juliet is more sensible: the more Romeo loves her, the more she loves him, and the more he hates her, the more she hates him.

(a) Explain why a reasonable model for their love for each other is dj/dt = kr, dr/dt = -kj, where j and r denote their love (if positive) or hate (if negative) for each other, and where k is a positive constant.

(b) Solve this system for r and j. Your answer will depend on k and two other arbitrary constants. (Hint: write a differential equation involving only one of r or j.)

(c) Suppose at time t=0, Romeo is fully in love with Juliet (so that r(0) = 1), but Juliet is indifferent to Romeo (so that j(0) = 0). Write equations for the solution in terms of k.

(d) Sketch the solution from (c), and explain in words what is going on.

Let f be a continuous function whose domain includes [0,1], such that 0 \le f(x) \le 1 for all x \in [0,1], and such that f(f(x)) = 1 for all x \in [0,1]. Prove that \int_0^1 f(x)\,dx > \frac34.

Let f be a continuous function whose domain includes [0,1], such that 0 \le f(x) \le 1 for all x \in [0,1], and such that f(f(x)) = 1 for all x \in [0,1]. Prove that \int_0^1 f(x)\,dx > \frac34.

In 1798, Rev. Robert Malthus proposed that the rate of change of a population is proportional to the actual population at any given time.

If the world population was 3.712 billion in 1970 and 4.453 billion in 1980, then what does Malthus's law predict the world population to be in 2008?

Find the general solution of the differential equation y' = xy.

Let y be the function that satisfies y' = x + y^2. and y(0) = 1. If the Taylor series of y is given by y = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + ...., then find a_0, a_1, a_2, and a_3.

Mr. Chickenorfish prepares a cup of coffee, then promptly forgets about it. Initially, the coffee is at 210 F. Fifteen minutes later, the coffee is at 190 F. If room temperature is 72 F, when will the coffee be at 150 F?