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) ...

Find multivariable limits (answer)

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 midpoint of AB is MP = (0,0,-3) A unit vector from MP to C is u =(3,-1,2)/sqrt(14) In coordinate form, the parametric form of the line is C + t u or (3,-1,-1) + t (3,-1,2)/sqrt(14) The parametric...

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...

One possible answer is C4 H6 with structural formula H3 - C- C Ξ C - C - H3 That is a dimethyl substituted acetylene

The domain of this function is (-∞ , -3] ∪ [5 , ∞ ) At x = -3 or x =5 the argument of the sqrt is zero. Thus the range is [0, ∞ )

Find the equation of the third side of the triangle given two equations of sides and orthocenter. (answer)

I did not find an easy way to solve this problem. I did find a solution via the standard diagramatic proof that the three altitudes of a triangle are concurrent coupled with a brute force application of coordinate geometry. The method requires pages of algebra - too much...

The distance 562.5 Gm is called the semilatus rectum. In terms of the usual parameters the semilatus rectum = b2/a The perihelion is sqrt(a2 + b2) - a So the equations for a and b are b2/a ...

This problem can be done by consulting a tabulated form of the cumulative normal distribution, or by using a calculator such as the TI-84. The calculator methods is much easier. Touch 2nd DISTR; select Draw then the first option :Shadenorm Enter zero for lower and...

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...

Minimum or maximum? (answer)

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...

The function f(x) = ln(x + sqrt(x2 +1)) has another name. It is arsinh(x) or sinh-1(x) that is, the inverse of the hyperbolic sine function : sinh(x) = (exp(x) - exp(-x) )/2 So the answer to part b) is sinh(x) It is easy...

To develop the equation, let x = the miles driven by the first car and let f1 = the fuel efficiency of car 1 (mi/gal) and f2 = the fuel efficiency of car 2. Then f1 = x/25 and f2 = (1750 -x) / 35. But...

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...

I think the ARC is just [ f(3+h) -f(3) ] /h this works out to 2 h + 11 The rational is that the average value of a function, g, on an interval [ a, b] is ∫ g(x) dx / (b -a) {upper limit = b , lower limit = b} For...

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...

Find the maximum of f(x)= x/(x+e)^2 (answer)

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 =...