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Find the expansions of the solutions of x^2 + (4+epsilon) x + 4 - epsilon = 0 around epsilon = 0.
Ignoring resistance, a sailboat starting from rest accelerates at a rate proportional to the difference between the velocities of the wind and the boat.
Explain how we arrive at the formula for Simpson's rule (standard formula) using the Lagrange Interpolating Polynomial of degree 2.
Cart A is being pulled away from Q at a speed of 2 ft/sec. How fast is cart B moving toward Q at the instant cart A is 5 feet from Q?
Let f: [a,b] --> R, a < b, twice differentiable with the second derivative continuous such that f(a)=f(b)=0.
Let f(x) be a continuous function of one variable. Give the definition of the derivative.
For y= 0.5, solve for x to get (2) x(t) then take a (partial) derivative of x(t) to get the rate of change of x which is the velocity of the wave.
The Maclaurin series for the inverse hyperbolic tangent is of the form x+x^3/3+x^5/5...x^7/7. Show that this is true through the third derivative term.
Prove that if f(x) = x^alpha, where alpha = 1/n for some n in N (the natural numbers), then y = f(x) is differentiable and f'(x) = alpha x^(alpha - 1).
This machine costs £C to lease each week according to the formula and t is the number of hours per week worked by the machine.
In the above example, let c1 be the objective function coefficient of . Determine the optimal z-value as a function of c1.
State the differential equation of the orthogonal family, and show your steps in obtaining a solution.
Use calculus to find the value of x so that V is as large as possible. Justify your answer. What is the largest possible value of the volume?
The optimal value function of a portfolio analysis problem solved using quadratic programming is __________________.
Let A€R mxn . Prove that one of the following systems has a solution but not both:
Solve the following two equations. In each case, determine dy/dx:Is this right? y'=x(-sin)(2x^2)(4x)
If f(x) = x^4 - 4x^3 + 10 find the relative extrema of the function and the points of inflection of its graph. Also, sketch the graph of the function.
A brick becomes dislodged from the top of the empire state building (at a height of 1250) and falls to the sidewalk below.
Suppose that the average yearly cost per item for producing x items of a business product is C(x)=10+(100/x) .
Consider a Lagrangian system, with configuration space R^n, given by (x^1, ... x^n); and Lagrangian L(x', ..., x^n; v^1, ... v^n).
Bob's objective is to mix these drinks in such a way as to make the largest possible number of drinks in advance. Formulate a LP model for this situation.
A container with a rectangular base, rectangular sides and no top is to have a volume of 2 subic meters. The width of the base is to be 1 meter.
What number of balls per hour should Acme produce to minimize the cost per hour of manufacturing these golf balls?
What is the optimal solution to this problem? How do you know?What is the dual of this problem?
What does Newton's method solve and how does it solve it? What is the underlying idea behind the method? Is it guaranteed to work?