Calculus: Can someone show me how to start this math problem?

Question

If a ball is thrown into the air with a velocity of 40 ft/s, its height in feet after t seconds is given by y=40t - 16t^2.
A) Find the average velocity for the time period beginning when t=2 and lasting
1) 0.5 second
2) 0.1 second
3)0.05 second
4)0.01 second
B) Find the instantaneous velocity when t=2

Please help!

Answer 1

Average velocity over an interval from t=a to t=b is given by
(y(b) - y(a)) / (b - a).
So, for instance, for A1) we get
y(2.5) = 100 - 16(25/4) = 0
y(2) = 80 - 64 = 16
Average velocity is (y(2.5) - y(2)) / (2.5 - 2)
= -16/0.5
= -32 ft/s (negative because the ball is heading downwards, and indeed hits the ground at time t = 2.5)

I'll let you do the remaining parts of question A yourself - it's exactly the same procedure. You should find the answers getting closer and closer to the answer for part B.

B) The instantaneous velocity is given by the derivative of the position function, i.e. y'(t) = 40 - 32t. So for t = 2 the instantaneous velocity is -24 ft/s.

Answer 2

I'll do A1 for you... you can use this to work your way through A2-A4...

We are told that height = 40t - 16t^2 where t=time

So to find the average velocity between 2 and 2.5 seconds we find how far it has travelled in that time and just divided this by the total time measured (0.5 seconds)

t=2 => height = 40*2 - 16*2^2 = 80-64 = 16 ft
t=2.5 => height = (40*2.5) - 16*(2.5^2) = 0 ft (i.e. ball hits the ground again)
travels 16 ft in 0.5 seconds - so average velocity is 32ft/second downwards.



To solve B we are basically trying to value of average velovity find the limit as the time period ->0
This is the same as the derivative of the function with respect to t

dy/dt = 40 - 16*2t = 40 - 32t when t=2 = 40-64 = -24 ft/sec (or 24 ft/sec in a downward direction.

You should have found that your values for A, B, C and D got closer and closer to this value as t got smaller.

Answer 3

On economic corporation investments or expenditures, each 3 hundred and sixty 5 days is a variable ( each 3 hundred and sixty 5 days would be distinctive, the quantity it self is a variable). occasion: 3 hundred and sixty 5 days a million: 3,000 * (a million.06) = 3,a hundred and eighty 3 hundred and sixty 5 days 2: 3,a hundred and eighty * (a million.06) = 3,370.8 3 hundred and sixty 5 days 3: 3,370.8 * (a million.06) = 3,573.048 etc... public double investment(){ double a = 3000.00; // preliminary money for(int i = a million; i <31; i++){ a = a * (a million.06); Sytem.out.println("3 hundred and sixty 5 days "+i+" : "+a); } return a; } gadget.out.println("R:"+a); result: **************************************... 3 hundred and sixty 5 days a million : 3180.0 3 hundred and sixty 5 days 2 : 3370.8 3 hundred and sixty 5 days 3 : 3573.0480000000002 3 hundred and sixty 5 days 4 : 3787.4308800000003 3 hundred and sixty 5 days 5 : 4014.6767328000005 3 hundred and sixty 5 days 6 : 4255.557336768001 3 hundred and sixty 5 days 7 : 4510.890776974081 3 hundred and sixty 5 days 8 : 4781.544223592526 3 hundred and sixty 5 days 9 : 5068.436877008077 3 hundred and sixty 5 days 10 : 5372.5430896285625 3 hundred and sixty 5 days 11 : 5694.895675006276 3 hundred and sixty 5 days 12 : 6036.5894155066535 3 hundred and sixty 5 days 13 : 6398.784780437053 3 hundred and sixty 5 days 14 : 6782.711867263277 3 hundred and sixty 5 days 15 : 7189.674579299074 3 hundred and sixty 5 days sixteen : 7621.055054057019 3 hundred and sixty 5 days 17 : 8078.318357300441 3 hundred and sixty 5 days 18 : 8563.017458738468 3 hundred and sixty 5 days 19 : 9076.798506262776 3 hundred and sixty 5 days 20 : 9621.406416638543 3 hundred and sixty 5 days 21 : 10198.690801636856 3 hundred and sixty 5 days 22 : 10810.612249735068 3 hundred and sixty 5 days 23 : 11459.248984719172 3 hundred and sixty 5 days 24 : 12146.803923802323 3 hundred and sixty 5 days 25 : 12875.612159230463 3 hundred and sixty 5 days 26 : 13648.148888784292 3 hundred and sixty 5 days 27 : 14467.03782211135 3 hundred and sixty 5 days 28 : 15335.060091438032 3 hundred and sixty 5 days 29 : 16255.163696924315 3 hundred and sixty 5 days 30 : 17230.473518739775 R=17230.473518739775 **************************************... the finished result, in terms of numbers is "17230.473518739775"; **in user-friendly terms interior the case that is specify that the annual interest income are on the unique quantity, then if may well be (3000*.06)*30 = income.

Answer 4

the instantaneous part is easy, just take the derivative and plug t=2
v = y' = 40 - 32t

avg velocity = distance over time
so plug each of those t's in
y(0.5) = 40t - 16t^2

and just divide by time

Answer 5

Hey guess what half of calculus is using derivatives