AP Physics
posted by AP Physics .
can you please explain this to me...
Were given this eqaution
x = At^2 + B
were A = 2.10 m/s^2
B = 2.80 m
ok and this is an example problem and it's wlking us through it
find the instantaneous velocity at t=t2=5.00s equlas the slope of the tangent to the curve at point P2 whoen in Fig. 212b and we could measure this slope off the gropah to obtain v2. we can calculate v more precisely, and for any time using the give formula
x = At^2 + B,
which is the engiene's position x at time t. Using the calculus formula for derivitves (THIS STEP I'M LOST)
(d/(dt))(Ct^n)= nCt^(n1) and (dC)/(dt) = 0,
where C is any given constant, then
v = (dx)/(dt) = (d/(dt))(At^2 + B) = 2At.
We are given A = 2.10 m/s^2, so for t=t2=5.00s,
v2 = 2At = 2(2.10 m/s^2)5.00s = 21.0 m/s
ok it has been a long time sense I have taken calculus. I understand why instanteous velociy is equal to (dx)/(dt) but I do not understand this problem. Can you please explain it to me. I got lost were I wrote (THIS STEP I'M LOST)
I don't understand this
"(d/(dt))(Ct^n)= nCt^(n1) and (dC)/(dt) = 0,
where C is any given constant, then
v = (dx)/(dt) = (d/(dt))(At^2 + B) = 2At."
Please explain to me the constant thing and what d is all by itself and what is dC and why they divide it by dt and how they got 2At
Thank you!

In general
if f(x) = x^n
then d f(x)/dx = n x^(n1)
so
if f(x) = x^2
then d f(x)/dx = 2 x
so if f(t) = t^2
df(t)/dt = 2 t
the constant multiple remains (three times the function has three times the slope) .
if f(t) = 2.1 t^2
then
d f(t)/dt = 4.2 t
Now about that constant B
The slope of a constant is zero.
so
d/dt (A t^2 + B)
= 2 A t + 0
the end except to prove that d/dx (x^n) = n x^(n1)
f(x+dx) = (x+dx)^n
binomial expansion
(x+dx)^n = x^n + C(n,1) x^(n1) dx + C(n,2) x^(n2) (dx)^2 ....... +dx^n
C(n,k) = binomial coefficient
= n!/[(nk!) k!]
so for Cn,1)
C(n,1) = n!/[(n1)!] = n (lo and behold)
so
f(x+dx) = x^n + n x^(n1)dx + terms in dx^2 through dx^n
so
f(x+dx) f(x) = n x^(n1) dx + terms in dx^2 to dx^n
so
df(x)/dx = [f(x+dx)f(x)]/dx
as dx > 0
= n x^(n1)
f(x) = x^n
f 
Ignore the last two lines, I was thinking out loud.
You may not need that proof at the end that d x^n/dx = n x^(n1) but it is not a bad idea to know where the derivative comes from.
Respond to this Question
Similar Questions

CALCULUS
I got this problem but its unrelated to any material that we worked on. Can some please help me to solve it and please explain it. Thanks Problem states: Given: x+(1/x)= square root of 3 PROVE: (x^13)+(1/(x^13))= square root of 3 
Physics
PLEASE HELP!! The x and y cooridinates of an object are given by x(t) = 10t + 5t^2 + 3t^3 y(t) = 20t  10t^2 + 2t^3 where x and y are in meters and t is in seconds. How fast is the object traveling at t = 1 s? 
Physics
PLEASE HELP!! The x and y cooridinates of an object are given by x(t) = (10 m/s)t + (5 m/s2)t2 + (3 m/s3)t3 y(t) = (20 m/s)t  (10 m/s2)t2 + (2 m/s3)t3 where x and y are in meters and t is in seconds. What is the object's acceleration … 
Math
In solving the eqaution(x+4)(x4),Eric stated thatthe solution would be x+4=33=>x=29 or (x4)=33=>x=37 However, at least one of these solutions fails to work when substituted back into the original eqaution. Why is that? 
algebra
i do not understand how to solve for x and y can someone please explain it to me. an example of the problem i am workin on is 6x4y=72 
Trig
Find an equation for the hyperbola described. Foci at (4,0) and (4,0); asymptote the line y= x. Please explain the steps of this problem. My book does not get an example of this type of problem and I'm not having any luck working … 
algebra
Explain in your own words what it means for an equation to model a real world situation an give an example. it is right to say that when given a problem, we want to identify the quality of the problem we want to find, translate form … 
geometry/calculus
Use Cavalieri's Principle to find the area of the region. In the example we see a "leaning tower" with base 'a' and height 'b'. The left side of the leaning rectangle is given by y = x^2, the right side given by y = (x  a)^2 the answer … 
Math Help
could someone please give me an example not an example using coins, marbles, spinner, like something different that would explain the difference b/w theoretical and experimental probability. And please explain. 
PreCalculus
The question is "Find parametric equations for the line with the given properties: Slope 2, passing through (10, 20)." I don't even know where to begin, or what to do. Please help! This is an example problem, not the main homework …