Question

Find equations of all tangents to the curve f(x)=9/x that have slope -1

Answer 1

At first you have to take the derivative of f. Do you understand why is that?

Answer 2

yes I do, I got -9/x2

Answer 3

Ok, now what is the meaning of the derivative? Knowing this meaning can you guess what the next step would be?

Answer 4

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?

Answer 5

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?

Answer 6

not really, can you lead me into solving this? what is the second step I should take?

Answer 7

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.

Answer 8

Yes I got to this step, when I was trying to solve for x I got +- 3

Answer 9

but the answer that it should be is y=-x-6 and y = -x+6

Answer 10

so I must have done something wrong right?

Answer 11

Wait, you just found out one point in wich the line passes, it is not the answer yet

Answer 12

What you did so far is correct

Answer 13

oh ic
okay, so the other point is 3 since 9/x = y and x is 3 therefore y = 3 right?

Answer 14

or -3, but yes, thats correct

Answer 15

so now what's next?

Answer 16

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.

Answer 17

use slope intercept form now

Answer 18

y = mx+b --> -3=-1x+b

Answer 19

sorry, I'm kinda slow at this point, how do I set this = to 9?

Answer 20

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

Answer 21

oh okay

Answer 22

I got it from here

Answer 23

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.

Answer 24

I got it thanks so much for being so patience :)

Answer 25

Your welcome

Answer 26

can I ask you something else?

Answer 27

Of course

Answer 28

same type of problem but it's f(x) = square root of (x+9)

Answer 29

slope 1/2
I did the derivatives and solve for x I got x= -1

Answer 30

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

Answer 31

but the answer should be y=1/4+13/4
where did I do wrong?

Answer 32

You got the x wrong.

Answer 33

Or the derivative

Answer 34

original fx is sqrt(x=9)
derivative is 1/2(x+9)^1/2

Answer 35

Where did I do wrong.

Answer 36

its (1/2)(x+9)^(-1/2)

Answer 37

Yes i got that.. But solving for x=-1

Answer 38

Well, thats wrong, did you see the minus, in the exponent?

Answer 39

No i said that mine is 1/{2(x+9)^1/2 same thing

Answer 40

\[\frac{ 1 }{ 2 }=\frac{ 1 }{ 2\sqrt{x+9} }\rightarrow \sqrt{x+9}=1\rightarrow x+9=\pm1\rightarrow x=-10 or -8\]

Answer 41

the slope is 1/4 not 1/2

Answer 42

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

Answer 43

Can you go from there?

Answer 44

let me try to do calculation, I did the calculation but got +/- 2 sqrt 2.. give me 1 minute please

Answer 45

Sure

Answer 46

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?

Answer 47

I'm almost leaving actually, but I'm here almost every day.

Answer 48

quick question on solving this problem
\[z(4z+7) - x(4x+7) / (z-x)\]

Answer 49

I can combine (z-x) then cancel top and bottom right?

Answer 50

so what is have left is (4z+7) * (4x+7) right?

Answer 51

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

Answer 52

oh maybe that's why. thanks