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A lesson in understanding definite integrations

Think about what you are finding when you get the area under a curve.

In this instance from t=0 to t=1 you are finding the area of a triangle which is 4.  But 4 what?

Look at the units.  They are     t  time (seconds)   in the horizonal directions  and  velocity (feet/sec) in the vertical direction.

When you find the area you can find the units the same way.  The unit of the area will be      $seconds\times \frac{feet}{sec}=feet$

So that is why the area under the velocity curve gives displacement.

And youcan find the area with integration.

NOW

Lets think about integration for a moment.

The integration sign.    $\int$    is a stylized S and it stands for   SUM

What you are doing with any definite integral is finding the sum of all the rectangles (with infinitely small width) that lie between the curve and the axis that you are integrating with respect to (usually the horizonal axis).

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A lesson in understanding differentiation

With differentiation the  d   stands for difference.  So    $\frac{dx}{dt}=\frac{\text{difference in x}}{\text{difference in t }}$

Since this is the answer as t tends to 0, dx/dt results in the instantaneous velocity.

The gradient of the tangent to a curve IS the differential to the curve at that point.

If you have questions then please ask them    

It depends on how much security your school has, also if you have chromebooks or macbooks or actuall desktop pc's

18 + 19 + 20 + 21 + 22 =

18 + 22 + 19 + 21 + 20 =

40 + 40 + 20 =

100

So   ( a, n)  =  (18, 5)

Nice! Keep going! :)

I agree.....the result should =  pi

I do not know......maybe alan does

here is hint: when n is even, you can pair up numbers, when n is odd, find the middle term, because it will be the average.

HOPE THIS HELPED!

0.5! isnt defined ..

I can do a few

a) The velocity is positive on  0 < t  < 1

This means the ball is  rising during this time

b)  The antiderivative is     s (t) =  32t - 16t^2  + C

At  s(0)   we have    32(0) - 16(0)^2 + C = 0

So C  = 0

c)  s(1) - s(1/2)  =    [ 32(1) - 16(1)^2 ] - [ 32(1/2) - 16(1/2)^2 ]  =  [ 16] - [ 16 - 4] =  4

This represents the change in postion from t = 1/2  to t = 1

[ It rises 4 ft in this interval  = the displacement on this interval ]

d) At t = (1/2).....the height of the triangle formed is   32 - 32(1/2) =  32 - 16  = 16

So....the area =  (1/2) base * height =  (1/2) (1/2)(16) =  4 

This area also represents the displacement on t =1/2 to t =1

f) s(2) - s(0)  =  [ 32(2) - 16(2)^2] - 0   =   0

This means that the total displacement = 0 ......it falls as far as it rises

Note that the velocity at t = 0 = 32 ft/s

And at t = 2  = -32 ft/s

This makes sense.....the terminal velocity = the beginning velocity....only in the opposite direction

Thank you very much 

                 x   + 3

x^2 + 1   [ x^3 + 3x^2 - 2Ax + 3 ]

                x^3             +  1x

               ______________________

                         3x^2  - (2A + 1)x  + 3

                         3x^2                      +3

                       ____________________

                                 - (2A + 1)x

So

-(2A + 1) x = 5x

-(2A + 1)  = 5

-2A  - 1  =  5

-2A = 6

A = - 3

The  common difference is 3

So...we can solve this

11,111 = 2 + 3 (n -  1)

11,111 = 2 + 3n - 3

11,111 = 3n - 1

11,112 = 3n    

3704 = n

So.....11,111  is the 3704 term

And it appears on   page  ....  ceiling ( 3704 /100)   =  page 38

Note however.....if we drop the provision that Kylie's points be evenly divisible by 4.....at least one integer solution exists

(14, 19)

This satisfies 

3(14)  - 2(19)   =  5      which is  < 6

14  ≥  19 /4 +  9

14  ≥   13.75

And

y < 20

19 < 20

Glad we have more newcomers!! Hope you enjoy your time here! Always feel free to ask any questions!! :)

Here's a hint for you to solve this! VVV
 

In an algebra problem like this, one of the best strategies is to translate the sentences into equations.

Let Kylie's points be K and Shaniqua's points be S.

The first sentence says: If you subtract 2 times Kylie's points from 3 times Shaniqua's points, the difference is less than 6

We can translate this into an equation of: 3S-2K <6 

The second sentence says: Shaniqua scored at least as many points as nine more than 1/4 of Kylie's points

We can translate this into an equation of: K/4 + 9 ≥ S

And for the third sentence: Kylie scored fewer then 20 points 

We can translate this into an equation of: K < 20

From this sentence, "Remember that in basketball you can only score whole number of points." we know that the answers are positive integers. 

So, our translations get us this systems of equations:

1) 3S-2K <6 

2) K/4 + 9 ≥ S

3) K < 20

You can solve these system of equations by separating them into separate cases: in other words, you can still do substitution with the variables except try doing it so that K/4 + 9 is EQUAL TO S and then K/4 + 9 IS GREATER THAN S. 

Just a suggestion! Hope this helped!

Thank you so much!

Let x = Shaniqua's points

Let y = Kylie's points

If you subtract 2 times Kylie's points from 3 times Shaniqua's points, the difference is less than 6

This translates as

3x - 2y < 6

Shaniqua scored at least as many points as nine more than 1/4 of Kylie's points

This translates as

x ≥   y/4 +  9

Kylie scored fewer then 20 points

This translates as

y < 20

Note that Kylie's number of points must be divisible by 4

And since she scoes < 20, she must have scored either 16, 12, 8 or 4    points

But....you can test for yourself.....there are no integer solutions

As close as we can be is if she scores 16

Then as few as Shaniqua could score is  16/4 + 9   =  13

But....these values make the first inequality     3(13) - 2(16)  =  7

But...the constraint is that this is < 6

So.....no integer solutions exist

Is that the original question? It's a bit hard to understand, my bad. 

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81 + 100 - 80

Because it's just subtraction and addition, we can go ahead and do 81-80 and then add 100:

1 + 100 = 101

Gingerale you are out of your league again!!. My proprietary programme works and finds "prime" or "composite" numbers all the time.  Go back to school and learn Number Theory and maybe then you might understand how my programme works. Stick to eating bananas and climbing trees until you learn how to factor large numbers. 

Hello!

Can you please answer my question?

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