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

NameDate

Figure — use these to answer the problems

Move the starting point around. Each choice gives one curve, and no two of them ever cross.

the rate1

starting x-2

starting y1

Figure 1

Straightforward practice. Get the method working first.

5 problems

Under uniqueness, how many solution curves pass through one point?
Answer
2
Why can solution curves not cross where uniqueness holds?
- The crossing point would have two different futures, which uniqueness forbids.
- It would be hard to draw.
f is continuous near the start. Does a solution exist? 1 yes, 0 no.
Answer
Continuity of f alone. Does it guarantee uniqueness? 1 yes, 0 no.
Answer
Which derivative must also be continuous for uniqueness? Enter 1 for the partial by y, 2 for the partial by x.
Answer

The same ideas with more to keep track of.

3 problems

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

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

4 problems

9
Match each condition to what it guarantees.
Draw a line from each item on the left to its match on the right.
- f is continuous near the point
- The partial of f by y is also continuous
- Neither condition holds
- A solution exists
- That solution is the only one
- Nothing is guaranteed
A numerical method run on a problem with no solution. Does it still produce numbers? 1 yes, 0 no.
Answer
The Crossing
Two distinct solution curves cross at a point where uniqueness holds. How many such crossings are possible?
Answer
The Polynomial
f(x, y) = 3x + 2y. Is it continuous everywhere? 1 yes, 0 no.
Answer