CPhill

2 ^{log ₄x}

Note  that  we can write   2  as  (4)^(1/2)

So we have

[ 4 ^(1/2)]^(log ₄ x)  =

4 ^ [ (1/2) log ₄ x ]   =      [ remember that   a log b   =  log b ^a  ]

4^ [ log ₄ x ^1/2 ]   =        [   remember that    a ^ [ log _a b ]   =  b   ]

x^1/2  =

√x

Note that, using the change-of-base theorem  we can write

y =  [ log r / log 2 ]  * [ log r / log 4 ]

y  =  ( 1  / [ log 2 *log 4 ])  * [ log r ] ^2       

Note  that  ( 1 / [ log 2 * log 4] )  is a constant

Apply the chain rule  to   [ log r ] ^2

Remember  that the derivative  of log r   =  1 / r

y ' =   ( 1/ [ log 2 * log 4]  *  2 * [ log r ] *  [ 1/r]        

               2  log r

y '  =    _____________

         r [ log 2 * log 4 ]

You are correct ......the last table   !!!!

Last one

a)  Making the same assumption as in the estimated mean in the  last problem  we have

[  2(7.5)  + 9(12.5)  + 5(17.5)  + 5 (22.5)  + 3(27.5) ]  /  [ 2 + 9 + 5 + 5 + 3]  ≈  17  min

b) Fraction finishing  in less than 20 min   =   [ 5 + 9 + 2 ]   / [ 2 + 9 + 5 + 5 + 3 ]  =  16 / 24  = 2/3  

Second one 

a) class interval that contains the median

We have  30  values

The  median  occurs  between the 15th and 16th value

The interval where this occurs  is     2 ≤ h < 3

b)  mean .....I'm less sure about this one because we don't actually know how many hours any particular individual spends on his/her tablet

But let's assume that each individual spends the midpoint of their class interval as a representative time on their tablet

[ For instance....assume that  every one in the first class interval spends   1/2  hr on their phone.....everyone in the second class interval spens 1 + 1/2  hrs on their phone, etc.]

So......one estimate  of the mean is

[ 4 (1/2)  +  8 (1 + 1/2)  + 11(2 + 1/2)  + 7(3 + 1/2) ]   / 30   ≈   2.2 hrs

First one

Mean height of team  =

[ (1)(198)  + (3)(199)  + (2)(200) + (5)(201)  + (2)(202) ]  / [ 1 + 3 + 2 + 5 + 2 ]  =

[2604/ 13]  =

200.3 cm

Let x  be the  height of the new player.....and we  have

[ 2604 + x ] / 14   = 201    multiply both sides by 14

2604 + x   =  2814        subtract 2604 from both sides

x  = 210  cm

Thanks, alan......that's a very crafty solution   !!!!!

We have  that

b =  k/a          so

9  = k/ 4       multiply both sides by 4

36  = k

And when  b = -72  we have that

-72  = 36 / a           multiply both sides by  a, divide both sides by -72

a =  36 / -72   =   - 1 / 2

Note  one thing...in inverse proportions....the product of ab  is a constant value

So

ab  =    ab

4*9  =  (-1/2)(-72)

36  =  36

Let the numberator  =  x

Then the denominator  = 2x

So...we have this

 x + 2                3

_______  =     ____           cross-multiply  

2x  - 2               4

4 ( x + 2)  = 3 (2x - 2)     simplify

4x + 8  =  6x  - 6         add 6 to both sides, subtract 4x from both sides

2x  = 14        divide both sides by 2

x  = 7

The denominator of the original fraction = 2x  = 2(7)   =  14

The easiest way  { in my opinion)  to do this is with a graph

We  want  three circles

One with a center of (-2, 5)  and a radius of 2

One with a center of (2,3)  and a radius of 4

One with a center of (-2, -4) and a radius of 7

See the graph of these here :   https://www.desmos.com/calculator/ah1hubg7wr

The point where the  three circles  intersect is the epicenter

As the graph shows....this point is  ( -2,3)