Question

How to find the coordinates of the points on the graph where the gradient is 4 where f:(-pi/2 , pi/2) and f(x)=tanx

Answer 1

We want to find the location(s) where the slope of the line tangent to the function f(x)=tan x is 4?
So we're using derivatives for this, yes? :o

Answer 2

the derivative of f(x) is sec^2x

Answer 3

yes

Answer 4

Normally, for critical points, we're looking for a slope of zero.
(So we set our derivative equal to zero and solve).

Since we're looking for a slope of 4,
we'll set our derivative equal to 4, yes?

Answer 5

\[\Large\rm 4=\sec^2x\]

Answer 6

im with you

Answer 7

So just do some algebra stuffs :)
Shouldn't be too bad.

Square root, yes?

Answer 8

\[\Large\rm \pm2=\sec x\]Understand why I put the plus/minus?

Answer 9

because you square rooted both sides

Answer 10

Well, to be a little more specific, because we took the square root of a square.

\(\Large\rm \sqrt{4}\ne\pm2\)
\(\Large\rm \sqrt{4}=2\)

Minor detail though :)

Answer 11

could you please explain why?

Answer 12

\[\Large\rm \sqrt{x^2}=|x|=\pm x\]When you take the square root of a squared term, you get the plus or minus.
That's one way to define the absolute function.

So we had:\[\Large\rm \sqrt{(\sec x)^2}=\pm \sec x\]I was kind of skipping a step in there I guess.
The plus/minus starts on the secant side, that's where it's coming from,
then we multiply it over to the other side.

Answer 13

ok now i see

Answer 14

I was just trying to point out that it had nothing to do with the 4,
hope that's clear :3

Answer 15

so in this case we only take positive?

Answer 16

Ummm let's work some angles first, then we'll decide whether we want to plus or minus case.

So we have,\[\Large\rm \pm2=\frac{1}{\cos x}\]Solving for cosx gives us,\[\Large\rm \cos x=\pm\frac{1}{2}\]

So if we're stuck between -pi/2 and pi/2, yah I guess you're right, we can only produce positive values when we take the cosine of angles in that interval.

Answer 17

So there are angles in the 4th and 1st quadrants that will give us positive 1/2.

Do you remember what they are? :o