anonymous
  • anonymous
Dear, Math-talent! The slope of the tangent line to the graph of the function y=3x^3 at the point (381) is limx=>3 (3x^3−81)/(x-3.) By trying values of x near 3, find the slope of the tangent line
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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ZeHanz
  • ZeHanz
Take you calculator and try x=2.9, 2.99, 2.999, to get a good idea where this is going to...
anonymous
  • anonymous
ok,Thank you. I have a calculator with me.. but why x=2.9??
anonymous
  • anonymous
oh i get it beacuse its near 3....

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ZeHanz
  • ZeHanz
Well, you want to know the limit for x --> 3. You are not supposed to do all kinds of math trickery to do this, but just try a few values near 3. So 2.9, or 3.1 and then even closer: 2.99 or 3.01, or any other value, as long as it is "near" 3...
anonymous
  • anonymous
Once I gt all answers from equation. what shall I do?
ZeHanz
  • ZeHanz
Can you guess what the number is that you are getting closer and closer to?
anonymous
  • anonymous
hmmmm I will try put 2.9 , 2.999 in the equation first Please hold!
anonymous
  • anonymous
Ok, So I got 78.33 for x=2.9 and 80.7303 for x=2.99... I guess it gets closer to 81?
anonymous
  • anonymous
cuze at the point (3, 81)
ZeHanz
  • ZeHanz
It does, so it's a safe bet 81 is the slope of the tangent line!
ZeHanz
  • ZeHanz
(3,81) begin the point on the graph is in itself not of importance. By coincidence, the y-coordinate is also 81.
anonymous
  • anonymous
Omg, thats prety cool! thank you very much :) are you faimilar with Tagent Velocity:
ZeHanz
  • ZeHanz
You mean the velocity of a moving point along a curve is the tangent vector? Or is it a book? (havent't got a clue here..)
anonymous
  • anonymous
yes thats the one
ZeHanz
  • ZeHanz
I could be familiar with the stuff in the book ;)
anonymous
  • anonymous
cool. The displacement (in meters) of a particle moving in a straight line is given by s=3t^3 where t is measured in seconds. 1)Find the average velocity of the particle over the time interval [7,10]. 2)Find the (instantaneous) velocity of the particle when t=7
ZeHanz
  • ZeHanz
Do you know the difference between average velocity and instantaneous velocity?
anonymous
  • anonymous
mmm No. I have no idea. av velo =displacement/ time elapsed?
ZeHanz
  • ZeHanz
It is, so that one is easy. Just calculate (s(10-s(7))/(10-7).
ZeHanz
  • ZeHanz
For instanteneous velocity, you'll have to do the limit-stuff as in the first problem. Can you write down the limit?
anonymous
  • anonymous
so s(10)-s(7)/3? for the fist one?
ZeHanz
  • ZeHanz
Yes, because s(10-s(7) is the distance you've travelled.
anonymous
  • anonymous
and jusy plug s=3t^3 in s?
ZeHanz
  • ZeHanz
Don't get what you're saying there...
ZeHanz
  • ZeHanz
s(10)=3*10³=3000, s(7)=3*7³=1029 (3000-1029)/3=657 m/s (you may not be using meters, but feet...)
anonymous
  • anonymous
Oh! I see.
anonymous
  • anonymous
For instanteneous velocity, I have to do limit thing? x=>7?
anonymous
  • anonymous
limt S(t)-S(a)/T-a?
ZeHanz
  • ZeHanz
Write down the limit:\[v(7)=\lim_{t \rightarrow 7}\frac{ s(t)-s(7) }{ t-7 }\] and try numbers near 7.
anonymous
  • anonymous
OK, I tired to 6.99. s(6,99)=(6.99)^3*3=1024.59 S(7)= 1029 1024.59-1029/1029-7?
ZeHanz
  • ZeHanz
You have to divide by t-7, which is 6.99-7=-0.01
anonymous
  • anonymous
i got -7.630!
anonymous
  • anonymous
No I got 441
ZeHanz
  • ZeHanz
441 is good. Just use brackets...
anonymous
  • anonymous
what shall I do next?
ZeHanz
  • ZeHanz
Take a drink!
anonymous
  • anonymous
s(6.9999)?
anonymous
  • anonymous
haha!
ZeHanz
  • ZeHanz
Oh, now I get it ;) If you set t=6.999 you'll be even closer to 441, so that will be the answer
anonymous
  • anonymous
Oh! So limit is how you get closer to x or function?
ZeHanz
  • ZeHanz
Yes, it is the value of a formula as you get closer and closer to a certain number. Only, because of the special kind of formula, it is not possible to try that number directly. As in the formula above: if you want the speed at t=7, you can't just put t=7 and get the right answer. If you try it, you'll get 0/0, which is undefined.
ZeHanz
  • ZeHanz
It is possible however, to do more advanced calculations with the formula to get the limit. It is called differentiation... Once you know how to do that, you don't need to try all these cumbersome numbers "near" to a certain value!
anonymous
  • anonymous
Wow, thank you so much for your detailed explaination. It is really fun doing math with you!
ZeHanz
  • ZeHanz
Thanks for the compliment! I really appreciate it.

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