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## Richard P.'s Resources

A TI-84 calculator can be used to find the inverse .  It is   first row:      1 -1 -.5 Second row:  0 -1  0 Third row:     0  0  .5

There are two real zeros of this polynomial.   Neither one of them is a rational number (which can be shown with aid of the rational zeros theorem).  A graphing calculator shows the two zeros to be   x = -.4331071   x = 3.3522244   If the last...

125 (cos 288 + i sin 288)   can be written as     125 ei 288   or 125  ei (288 +360)  or 125  ei( 288+ 720)    To take the cube root of each of these, take the cube root of the 125  ( which is 5) and divide the argument...

The best place to start with this problem is a side calculation of the area, A, of the surface of the water as a function of the water height, h.   Since the radius of the water surface is proportional to h,   A is proportional to h2.  One can write      ...

The reason that hydrogen bonding is weaker than ionic bonding is that in ionic bonding the charges involved are of order the charge of an electron, while in hydrogen bonding the effective charges involved are a small fraction of the electron charge.   Since the Coulomb forces is proportional...

This is a set of three linear equations in two unknowns.   The slopes of the three lines corresponding to the equations are all different ( 1, -1 and 3).  With this set of conditions, there are no points that lie on all three lines, and so there are no points in common to all three...

Let x = the needed number of liters of the 10% solution  and y = the needed number of liters of the 30% solution.   Then   (.1 x + .3 y) /(x + y) = .15    and                x +...

The integrand is the sum of two terms, each one of which is a power of x.        The anti derivative of  xn  is   xn+1 /(n+1)     So  ∫ x dx =  x2/2  and  ∫ x3 =  x4/4.      So...

Since   z = 1 - i      |z| = sqrt( (1- i)(1+i)) = √2     With this result    (1- i) = √2 exp(i θ) = √2( cos θ + i sin θ)         where θ = arg z.        ...

One approach to this problem is algebraic.    Let x = the needed volume of the 50% mixture and y = the needed volume of the 90% mixture.     Then        .75   =  (.5 x + .9 y) /( x+y)      ...

The molar mass of NaOH is 40 g,  so 6 grams is 6/40 = 0.15 moles.      Since NaOH is a strong base,   it dissociates 100% in solution.     NaOH  ==>    Na+  + OH-  Consequently, 0.15 moles NaOH...

The key is the P value.   If the parabola   is    f(x) = a ( x - v)2 +h,    the P value is  P =  1/(4a).   The P value is both the distance from the vertex (v,h) to the focus and the distance from the vertex to the directrix...

The distance from the vertex to the focus is called the P value.    The P value is related to the coefficient of the squared term, a, by   P =  1/(4a) for positive a or P = -1/(4a) for negative a.    For this problem the P value is  P = 2,  ...

If this means    [ ln(u) ]3  = 7 ln(u)      proceed as follows:      First divide both side by ln(u)   this leads to      [ln(u) ]2  = 7          ...

First factor out xy from the expression.    This leads to     xy [ 3 x2 -13 xy  +4 y2 ]     The expression inside the parenthesis is trinomial.  The discriminant is             169...

Pb3(PO4)2   This works because   the oxidation number of PO4  is -3     and  the oxidation state of lead (II)  is  +2

The disk method looks to be easier than the washer method for this problem.   The integral over x is from -2 to 2  over a stack of thin disks each  with  volume  π (4 - x2)2 dx   This integral can be carried out by expanding the squared binomial and...

This type of problem usually requires a numerical (calculator) solution.    However, this case is easy.      We want to find the value of x such that   ln(x) + arctan(x) =   π/4.   Start by remembering that tan(π/4)...

I think this one is easier using the washer method.   The volume element  is   2π (x -1) x   dx  .  The factor (x-1) appears instead of the usual x because the vertical rotation axis is at x =1 not at x =0.     The   integral...

The average slope on (1,10)  is   the same as on the closed interval [1,10] so      mavg =    (1/9) ( f(10) - f(1)) =   (2/3) ( √10  -1 )   To find c  set  f'(c) =  mavg     (2/3)( √10...