Question

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

Answer 1

Take you calculator and try x=2.9, 2.99, 2.999, to get a good idea where this is going to...

Answer 2

ok,Thank you. I have a calculator with me.. but why x=2.9??

Answer 3

oh i get it beacuse its near 3....

Answer 4

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...

Answer 5

Once I gt all answers from equation. what shall I do?

Answer 6

Can you guess what the number is that you are getting closer and closer to?

Answer 7

hmmmm I will try put 2.9 , 2.999 in the equation first Please hold!

Answer 8

Ok, So I got 78.33 for x=2.9 and 80.7303 for x=2.99...
I guess it gets closer to 81?

Answer 9

cuze at the point (3, 81)

Answer 10

It does, so it's a safe bet 81 is the slope of the tangent line!

Answer 11

(3,81) begin the point on the graph is in itself not of importance. By coincidence, the y-coordinate is also 81.

Answer 12

Omg, thats prety cool! thank you very much :) are you faimilar with Tagent　Velocity:

Answer 13

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..)

Answer 14

yes thats the one

Answer 15

I could be familiar with the stuff in the book ;)

Answer 16

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

Answer 17

Do you know the difference between average velocity and instantaneous velocity?

Answer 18

mmm No. I have no idea. av velo =displacement/ time elapsed?

Answer 19

It is, so that one is easy. Just calculate (s(10-s(7))/(10-7).

Answer 20

For instanteneous velocity, you'll have to do the limit-stuff as in the first problem.
Can you write down the limit?

Answer 21

so s(10)-s(7)/3? for the fist one?

Answer 22

Yes, because s(10-s(7) is the distance you've travelled.

Answer 23

and jusy plug s=3t^3 in s?

Answer 24

Don't get what you're saying there...

Answer 25

s(10)=3*10³=3000, s(7)=3*7³=1029
(3000-1029)/3=657 m/s (you may not be using meters, but feet...)

Answer 26

Oh! I see.

Answer 27

For instanteneous velocity, I have to do limit thing? x=>7?

Answer 28

limt S(t)-S(a)/T-a?

Answer 29

Write down the limit:\[v(7)=\lim_{t \rightarrow 7}\frac{ s(t)-s(7) }{ t-7 }\]
and try numbers near 7.

Answer 30

OK, I tired to 6.99. s(6,99)=(6.99)^3*3=1024.59
S(7)= 1029
1024.59-1029/1029-7?

Answer 31

You have to divide by t-7, which is 6.99-7=-0.01

Answer 32

i got -7.630!

Answer 33

No I got 441

Answer 34

441 is good. Just use brackets...

Answer 35

what shall I do next?

Answer 36

Take a drink!

Answer 37

s(6.9999)?

Answer 38

haha!

Answer 39

Oh, now I get it ;)
If you set t=6.999 you'll be even closer to 441, so that will be the answer

Answer 40

Oh! So limit is how you get closer to x or function?

Answer 41

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.

Answer 42

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!

Answer 43

Wow, thank you so much for your detailed explaination. It is really fun doing math with you!

Answer 44

Thanks for the compliment! I really appreciate it.