CPhill

100/10^-4  = 100*10^4  = 10^2*10^4   = 10^6  = 1 million grains

Melody....that was BRILLIANT!!!!  [ I'm still working it all out.....!!!  ]

I'll try my hand at this one........

For the first one......we are looking for a 2 digit number that is divisible by both 9 and 7.......since we are multipying this by 1000, the result will be divisible by the other integers 1-9.

This number, N, is 63

For the second one.......we need all the multiples of 63 between 100 - 1000.....these are

126, 189, 252, 315, 378, 441, 504, 567, 630, 693, 756, 819, 882, 945

For the third, there are 1000 possible outcomes for each dice roll.

For each of the above numbers - except 252, 441 and 882 - there are 3! = 6 outcomes that could produce that particular number......for example, the outcome 126 plus the rearrangement of the outcomes 162, 216, 261, 612, 621 would all produce 126  ......for 252, 441 and 882, there are just 3 identifiable outcomes associated with each of these numbers......for example, 252 , 225 and 552

So the probability that the dice could be rearranged to make one of the numbers is :

[11 * 6  +  3 *3]  / 1000  =    [66 + 9] / 1000   =  75/1000  = 3/40

That's my best shot.....!!!!!

All lines of the form    y = a    have a slope of 0.....to see why....look at at any two points on this line.....for example   (2, -12)  and (3 - 12)

The slope between these two points =  [  -12  - (-12) ] / [ 3 - 2]   =   [-12 + 12] / 1    =  0 / 1    = 0

lim x → 0   ((sin(x+1))-(sin2x))/(x-1)  =

lim x → 0  [ sinxcos1  - sin1cosx - 2sinxcosx] / [ x - 1]  =

lim x → 0  [ cosx(sin1 - 2sinx) - sinxcos1 ]  / [ x - 1 ] =

[ cos(0)(sin1 - 2sin(0)) - sin(0)cos1 ]  / -1   =

[1 * sin1]  / -1    ≈   -0.841

c(x)=200-10x+0.01x^2

The minimum number will be given by    -b/2a    where b = -10  and a = 0.01.....so we have

min =  -(-10)  / (2* 0.01)   = 10 / 0.02   =  500  will minimize the cost

f(x)=I x^2-9 I   can be written as

f(x)  =  [ ( x^2 - 9)^2 ]^(1/2)

So....using the Chain Rule twice, we have :

f ' (x)  =  (1/2) [ (x^2 - 9)^2] ^(-1/2) [ 2 (x^2 - 9) *2x]  =

2x [ x^2 - 9] / [(x^2 - 9)^2] ^(1/2)   =

2x[x^2 - 9] /  l x^2 - 9 l

Note that if x = -3   or if x = 3, the derivative is undefined, because we can't divide by 0

Notice the "sharp points" at x = -3 and x = 3  on the graph :  https://www.desmos.com/calculator/ee9uapzsmi

The derivative is undefined at these points.......so.......this funnction is differentiable at all x except at x = -3  and x = 3

You have the correct derivative.......what do you want to "solve?"

1. Find the volume of a cylinder with a radius of 6 and a height of 5

 [ Hint :V_cylinder  =  pi * (radius)^2 * (height)  ]

2. Find the volume of a cone with the same dimensions as above......

[ V_cone = (1/3) * pi * (radius)^2 * (height) ]

3. Find the volume of a sphere with a radius of 10  [ V_sphere  = (4/3) * pi *  (radius)^3 ]

No, definitely not a³ !!!!

a + a²    remains just that.........although it could be expressed as a product  = a ( 1 + a)