Question

Find all solutions of the following equations, correct to 4 decimal places.

Answer 1

\[tanx=\sqrt{4-x ^{2}}\]
\[\sqrt{x}=x ^{3}-3\]

Answer 2

What's your plan?

Can you solve directly?
Can you limit before searching?
Any particular tools you would like to use?

Answer 3

Whichever is the most practical way to solve it. How do I do this? Mere substitution?

Answer 4

There is a great deal of judgment in the determination of "practical". It MIGHT mean simply familiarity.

Please name some methods wish which you are familiar. Obviously, we can just guess, but that is generally not very helpful. We need a reasonable way to proceed. What section are you studying? What class are you in? If you are in Numerical Methods Class and studying Newton's Method, then we can get started with that.

More information.

Answer 5

Calculus I.

We have not done Newton's Method yet, but am willing to learn.

Answer 6

Well, we really can't be a primary source of lessons in this environment. What methods have you?

Fixed Point Iteration?
Bisection?
Regula-Falsi?

You have to give me a clue.

Answer 7

Nothing of that sort. Just Limits... The instructions mention to find all solutions and then graph.

Answer 8

Wow! I want to find your teacher and slap him or her. I will refrain from doing so, but I REALLY want to.

Okay, let's see if we can learn anything from the first one and you do the second.

\(\tan(x) = \sqrt{4-x^{2}}\)

Number One Thing to do. State the DOMAIN of each function. Go!

Answer 9

Haha. Wow.... XD

Sure, I'll have a go at it.

Domain for first: \[-2 \le x < -\frac{ \pi }{ 2 }\]

Domain for second: \[x \ge 0\]

Answer 10

Connection problems.

?? One at a time, please.

Domain of \(\tan(x)\) is all Real Numbers less a periodic vertical asymptote.
Domain of \(\sqrt{4-x^{2}}\;is\;[-2,2]\)

Can we agree on that?

Answer 11

Next, think about the Range.

The Range on \(\sqrt{4-x^{2}}\) is only \([0,2]\). This is very helpful. With a little thought, this leaves us only \(x \in [0,\pi/2]\) to look for an intersection.

Note1: The period of tan(x) through the Origin gives tan(x) > 0 only for x > 0.
Note2: Less obvious, but the next period of tan(x) (both directions) is too far away where it is positive.

We can narrow farther by noting that \(0 \le tan(x) \lt 2\), since that's the Range on the square root and the square root reaches 2 only for x = 0 where we know the tangent is zero (0). The tangent reaches 2 at about x = 1.107.

In case you lost track, we should look on \((0,atan(2) = 1.107)\). A little more thought, particularly the monotonic increasing nature of tangent, we should look closer to x = 1.

Like I said, considerable judgment might be required, both in how to proceed and where to start.

Answer 12

Hmmmmm

Answer 13

We're just thinking.... How close are you to following?