anonymous
  • anonymous
Find equations of all tangents to the curve f(x)=9/x that have slope -1
Mathematics
jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
At first you have to take the derivative of f. Do you understand why is that?
anonymous
  • anonymous
yes I do, I got -9/x2
anonymous
  • anonymous
Ok, now what is the meaning of the derivative? Knowing this meaning can you guess what the next step would be?

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anonymous
  • anonymous
that is when a function change as the input change, but I'm stuck at the 2nd step what shall I do next? plug in -1 slope as x?
anonymous
  • anonymous
Yes, in other words it is the rate of change of a function. But the meaning I was thinking of is the same, but a little diferent. When the input increases in a really small amount, the value changes too. Since those two changes are very small, the rate of change is approximately the change in the value divided by the increase in the input. Do you understand what I just said?
anonymous
  • anonymous
not really, can you lead me into solving this? what is the second step I should take?
anonymous
  • anonymous
You need to understand it before gettingg to the next step. But if you want to know, the derivative of a function at x is the slope of its tangent line at this point, that was what I was trying to show you. Then, you should put -9/x2=-1, that is the slope, to find in which points the line passes.
anonymous
  • anonymous
Yes I got to this step, when I was trying to solve for x I got +- 3
anonymous
  • anonymous
but the answer that it should be is y=-x-6 and y = -x+6
anonymous
  • anonymous
so I must have done something wrong right?
anonymous
  • anonymous
Wait, you just found out one point in wich the line passes, it is not the answer yet
anonymous
  • anonymous
What you did so far is correct
anonymous
  • anonymous
oh ic okay, so the other point is 3 since 9/x = y and x is 3 therefore y = 3 right?
anonymous
  • anonymous
or -3, but yes, thats correct
anonymous
  • anonymous
so now what's next?
anonymous
  • anonymous
Now, you know a point that belongs to the line, and its slope. The general formula for a line is y=ax+b, and you alrady have a, now put the point you know that belongs to it, and you will find b.
anonymous
  • anonymous
use slope intercept form now
anonymous
  • anonymous
y = mx+b --> -3=-1x+b
anonymous
  • anonymous
sorry, I'm kinda slow at this point, how do I set this = to 9?
anonymous
  • anonymous
You dont need to, you know x and y, from the points (-3, -3) and (3, 3) that this tangent passes through, and you know a. Now: -3=-1*(-3)+b1 3=-1*3+b2
anonymous
  • anonymous
oh okay
anonymous
  • anonymous
I got it from here
anonymous
  • anonymous
Before, you found out in which points the tangent have this slope, then, you found out the value of the function at this point, and now you use those information to get the formulas for the tangent lines. You seem a little bit confused with what the numbers you are getting mean.
anonymous
  • anonymous
I got it thanks so much for being so patience :)
anonymous
  • anonymous
Your welcome
anonymous
  • anonymous
can I ask you something else?
anonymous
  • anonymous
Of course
anonymous
  • anonymous
same type of problem but it's f(x) = square root of (x+9)
anonymous
  • anonymous
slope 1/2 I did the derivatives and solve for x I got x= -1
anonymous
  • anonymous
I plugged x= -1 back to original equation I got y = +/- 2 sqrt 2 then I use slope intercept to find equations, I got y= 1/4+9/4 sqrt 2
anonymous
  • anonymous
but the answer should be y=1/4+13/4 where did I do wrong?
anonymous
  • anonymous
You got the x wrong.
anonymous
  • anonymous
Or the derivative
anonymous
  • anonymous
original fx is sqrt(x=9) derivative is 1/2(x+9)^1/2
anonymous
  • anonymous
Where did I do wrong.
anonymous
  • anonymous
its (1/2)(x+9)^(-1/2)
anonymous
  • anonymous
Yes i got that.. But solving for x=-1
anonymous
  • anonymous
Well, thats wrong, did you see the minus, in the exponent?
anonymous
  • anonymous
No i said that mine is 1/{2(x+9)^1/2 same thing
anonymous
  • anonymous
\[\frac{ 1 }{ 2 }=\frac{ 1 }{ 2\sqrt{x+9} }\rightarrow \sqrt{x+9}=1\rightarrow x+9=\pm1\rightarrow x=-10 or -8\]
anonymous
  • anonymous
the slope is 1/4 not 1/2
anonymous
  • anonymous
sorry, didnt ssee you posting. Then the left side of the equation will be 1/4, and sqrt(x+9)=2 and x+9=+-4, wich gives us: x=-5, and x=-13
anonymous
  • anonymous
Can you go from there?
anonymous
  • anonymous
let me try to do calculation, I did the calculation but got +/- 2 sqrt 2.. give me 1 minute please
anonymous
  • anonymous
Sure
anonymous
  • anonymous
I got it, I think I got messed up when I was trying to square to get rid of sq rt. Thanks Do you have time to help me on other problems?
anonymous
  • anonymous
I'm almost leaving actually, but I'm here almost every day.
anonymous
  • anonymous
quick question on solving this problem \[z(4z+7) - x(4x+7) / (z-x)\]
anonymous
  • anonymous
I can combine (z-x) then cancel top and bottom right?
anonymous
  • anonymous
so what is have left is (4z+7) * (4x+7) right?
anonymous
  • anonymous
No, be carefull, think of what is being divided, when you have something that is not being divided you cannot sum them withou a common denominator
anonymous
  • anonymous
oh maybe that's why. thanks

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