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Answers by Richard P.

For the first one,   x4 - y4  factors as (x2 - y2)(X2 + y2)   the second factor cancels out leaving x2 + y2    which is zero in the limit indicated   For the second one,   change top polar coordinates:    x = r cos(θ) ...

The case b >1  is easiest to think about.    With b >1   , the argument of function f  changes more rapidly (as x increases)  than would be the case with b =1.   This means that anything that f is going to "do" happens more rapidly.  More rapid...

The anti-derivative of 1/(6 x3) is -(1/12)/ x2     So the area under the curve is (1/12) ( 1 - 1/t2 )   So  for t = 10    area =  (1/12) .99      for  t = 100   area = (1/12) (.9999)   The entire area corresponds...

This triangle is a right triangle with the right angle at vertex C    The orthocenter is therefore at C  This is a property of all right triangles:  The three altitudes intersect at the right angle vertex.   The orthocenter is where the three altitudes intersect...

Because  Lines BD and CE are parallel,   BCDE is a trapezoid.  Solving for the intersection of DE and CE  is easy because DE is parallel to the x axis.  The result is E: (-16,10) Similarly for DE and BD        D:  (-16,10) The...

It has a minimum value of 4 located at x = 2.    The expression is in what is called vertex form.

Please help me. (answer)

A bit more information is needed to close the loop on this problem.  However, some essential parts can be worked out.   First, rearrange the equation to read  dy/dt - a y = b     This is called the inhomogeneous equation.   The associated homogeneous...

The usual answer to this question is to try some experimentation with 3 x 3 matrices.  A few examples will show that two 3x3 matrices,  M and N, do not commute for most choices of M and N.  The only way to give a better answer is to provide a technical discussion in terms of eigenvalues...

complex geometry (answer)

An interesting problem.    The two lines intersect at the point   1 + i  (where r and s are both zero)   It would seem useful to visualize the evolution of the lines keeping r = s.  The real parts are both  1 + 3r , whereas the imaginary...

A way to work this type of problem is to set up two 'railroad track' computations - one assuming that the iron III is limiting and the other assuming that the chlorine is limiting.   Whichever results in the smaller amount of Iron III chloride being produced is the correct calculation...

The derivative of f(x)) can be worked out to be    f' = 1/(x+e)2 - 2 x /(x +e)3     Setting this equal to zero and multiplying by (x +e)2  results in   1 - 2x /(x+e) =  0.   This equation is easily solved to get x =...