CPhill

We already know (1)  and (2)

3)   R(x)  - C( x)   =

10x - 0.0001x^2   - [ 50000 + 2x ]   =

-0.0001x^2 + 8x - 50000          this  the "profit" function

The derivative of this is

-0.0002x + 8

4)   Set the derivative to  0  and solve

-0.0002x  + 8  = 0

8  = 0.0002x      divide both sides by   0.0002

x = 40000    

.Here's the profit function graph : https://www.desmos.com/calculator/ejbztbmci4

The derivative will be positive from about  ( 6834, 40000) and negative from about (40000, 73,166)

5)   40000 books maximizes the profit......and, as the graph shows, the max profit  is $110,000

The maximum profit in this type of problem will always occur at the x value which makes the derivative = 0

I assume that you want to factor this????....if so, we have

x^2 - 4x + 4  =  (x - 2) (x -2)  =  (x - 2)²

(4^5*6^-8)^0     =  1

Proof

[4^5]^0 * [6^-8]^0  =

[1024]^0 * [1/1679616]^0  =

[1] * [1]   =

1

[sqrt ( x + h + 2)  - sqrt( x + 2)]  / h  =         multiply top/bottom by [ sqrt(x + h + 2) - sqrt ( x + 2)]

[sqrt ( x + h + 2) - sqrt (x + 2)] [ sqrt( x + h + 2) + sqrt(x + 2)] / ( h * [sqrt(x + h + 2) + sqrt (x + 2) ] )

[ (x + h + 2)  - ( x + 2)] /  ( h * [sqrt ( x + h + 2)  + sqrt(x + 2)] )

h /  ( h * [sqrt ( x + h + 2)  + sqrt(x + 2)] )

1 / [ sqrt ( x + h + 2) + sqrt (x + 2) ]        let h →  0

1 [ sqrt ( x + 2)  + sqrt(x + 2) ] =

1 / [ 2 sqrt (x + 2)]

And f ' (2)  =

1/ [2 sqrt (2 + 2)]  =

1 [ 2 sqrt (4) ] =

1/ [2 * 2] =

1/4

Sorry, Asinus.....we can't just "cancel" exponents in that fashion.......

Your answer ends up being =  99999/100000 =  .99999

However..........

99999^100000 / 100000^100000 =

[99999/10000] * [ 99999/100000] * [99999/100000].........100,000 times

[99999/100000]^100000  ≈  .3679

Note :

(a + 5)^2  - (a - 5)^2   factors as a difference of squares   =

[ (a + 5)  - (a - 5)]  [(a + 5) + (a - 5) ]  =

[ 10] [ 2a]  =

20a

Note that :

 sin^2(2x)   = [1 - cos(4x)]/ 2     ......so we have.....

[ 1 - cos4(x + h)] / 2] * [ 1/h]    -    [1 - cos(4x)] / 2 * [ 1/h ] =

[1 - cos(4x + cos4h)] / 2h   - [1 - cos(4x)] 2h  =

[cos(4x)  -  cos(4x + 4h) ] / 2h   =

[ cos(4x)  - [cos4xcos4h  - sin4xsin4h] ] / 2h  =

[ cos4x ][1 - cos4h] / 2h    +  sin4xsin4h/ 2h        [multiply top/bottom of both fractions by 2]

[2cos4x] [(1 - cos4h) / 4h ] +   [2sin4x] [sin4h/ 4h ]       let h →  0

[2cosx * 0]  +  [2sin4x * 1 ]  =

2sin4x  =

2sin(2x + 2x)  =

2[ sin2xcos2x + sin2xcos2x]  =

2 [ 2 * sin2xcos2x]  =

4[sin2x] [cos2x]

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Note that :   [sin(2x)]^2  =  [sin2x ] * [sin2x]     (1)

And using the Product Rule, the derivative  of (1)  =

[2cos2x] [sin2x] + [sin2x][ 2 cos2x]  =

[2cos2x] [ sin2x + sin2x]  =

[2cos2x] [2sin2x] =

4[sin2x][cos2x]

If the recipe is doubled....each ingredient amount is doubled....so

2 * 3.25 =    6.5  cups of flour

5x^2 + 3y^2 - 20x + 24y = -53       complete the square on x and y

5(x^2 - 4x  + 4)  +  3(y^2 + 8y + 16)  =   -53 + 20 + 48

5(x - 2)^2    +  3 (y + 4)^2  =   15     divide both sides by 15

(x - 2)^2 / 3   +  (y + 4)^2 / 5     =  1

This is an ellipse centered at ( 2 , -4)    with a major axis length  of 2sqrt(5)  and a minor axis length of 2sqrt(3)

Here's the graph : https://www.desmos.com/calculator/kklcz1gncn

3 / 50   = x /90     cross-multiply

3 * 90   =   x * 50     divide both sides by 50

3 *  90  / 50   = x

270 / 50   = x

27 / 5   = x

5 2/5  hrs   = x   = 5.4 hrs  =  5 hr  24 min