Existence and Uniqueness

When you can trust that an answer exists, and that it is the only one.

12 problems · about 23 minutes · F-IF.A.1

What this lesson teaches

The student states the existence and uniqueness conditions and recognises where they fail.

Continuity of f near the starting point guarantees a solution exists there.
Continuity of the partial derivative with respect to y makes that solution unique.
Where uniqueness holds, solution curves can never cross.

Straightforward practice. Get the method working first.

5 problems

Under uniqueness, how many solution curves pass through one point?
Answer 1
Why 1.
Why can solution curves not cross where uniqueness holds?
Answer The crossing point would have two different futures, which uniqueness forbids.
Why Two futures from one point is impossible.
f is continuous near the start. Does a solution exist? 1 yes, 0 no.
Answer 1
Why That is the existence condition.
Continuity of f alone. Does it guarantee uniqueness? 1 yes, 0 no.
Answer 0
Why A second condition is needed.
Which derivative must also be continuous for uniqueness? Enter 1 for the partial by y, 2 for the partial by x.
Answer 1
Why The one involving the unknown.

The same ideas with more to keep track of.

3 problems

Under uniqueness, how many solutions pass through a given point?
Answer 1
Why Exactly one.
f = x + y. Is the partial derivative with respect to y continuous? 1 yes, 0 no.
Answer 1
Why It is the constant 1.
Does the theorem guarantee a solution for all time? 1 yes, 0 no.
Answer 0
Why Only a small interval.

Mixed problems. Work out what kind of question it is before you start.

4 problems

Match each condition to what it guarantees.
Answer f is continuous near the point → A solution exists; The partial of f by y is also continuous → That solution is the only one; Neither condition holds → Nothing is guaranteed
Why One condition gives existence, the other adds uniqueness.
A numerical method run on a problem with no solution. Does it still produce numbers? 1 yes, 0 no.
Answer 1
Why It has no way of knowing.
The Crossing: Two distinct solution curves cross at a point where uniqueness holds. How many such crossings are possible?
Answer 0
Why 0.
The Polynomial: f(x, y) = 3x + 2y. Is it continuous everywhere? 1 yes, 0 no.
Answer 1
Why Yes.