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Richard P.'s Resources

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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...

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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...

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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...

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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)      ...

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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,  ...

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In(u)^3=7In(u) (answer)

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          ...

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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...

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

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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...

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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)...

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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...

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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...

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