Approximate Solutions

Sometimes it is difficult to solve an equation exactly, but an approximate answer may be good enough.

What is Good Enough?

Well, that depends what we are working on!

So understanding what we are working on helps us know how accurate we should be.

Example: Cooking

For general cooking we can change things a bit and get away with it. Adding an extra 50g of carrot is fine. Just not too much salt, please!

cake

Example: Baking

When baking a cake, small changes in ingredients are often fine, a little extra cocoa or slightly less sugar is OK, but if we go too far the result may be a failure.

Adding 100ml extra milk will make the result wet and yucky.

Example: Aerospace Engineering

Calculations for the trajectory and fuel needs of spacecraft require extreme accuracy. Small errors can lead to going off-course and mission failure.

Solving Equations

We still want to be as accurate as possible, so try to handle any approximate data sensibly.

When solving equations:

Like this:

Example: Solve x/7 − 6.3068 + 2π = 0   (to 3 decimal places)

Start with:x/7 − 6.3068 + 2π = 0
Subtract −6.3068+2π from both sides:x/7 = +6.3068 − 2π
Multiply by 7:x = 7(6.3068 − 2π)
NOW do the calculations:x = 0.165

Why wait until the end to do the calculations? Well, every time we do a calculation we can introduce an error. If we do this several times our errors can accumulate to be large.

Checking

If our answer is approximate, then our checking will also be approximate.

Example: Check that x = 0.165 solves x/7 − 6.3068 + 2π = 0

Substitute 0.165 for x:0.165/7 − 6.3068+ 2π = 0
Calculate:−0.00004 = 0

Not quite right, but very close.

Graphical Estimation

We can make good approximations using graphs, particularly by using a zoom function, like on the Function Grapher.

Here is an example:

Example: estimate the solution to x3 − 2x2 − 1 = 0 (to 2 decimal places).

Solution: Plot it!

Here is my first attempt. I can see it crosses through y=0 at about x=2.2

graph

Let us zoom in there to see if we can see the crossing point better:

graph

It crosses between 2.20 and 2.21 ... slightly closer to 2.21. We are asked for 2 decimal places, so our answer is:

x3 − 2x2 − 1 = 0 at about x = 2.21

Check: (2.21)3 − 2(2.21)2 + 2 = approx 0.025, close to y=0

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