3. Stationary Points & Small Increments in Calculations Flashcards
(18 cards)
How do you find the stationary point(s) of a multivariate function?
dz/dx=0 and dz/dy=0 then solve the equations to find values for ‘x’ and ‘y’ resepctively.
In brief, how do you determine the location of, and nature of stationary points for a multivariate function.
- Perform a derivation of the function with respect to x and y separately.
- Stationary points are found where the first derivation is equal to 0 (dz/dx=0 and dz/dy=0)
- Perform a second derivation
- Use the delta formula at the given points, values greater than zero indicate an extremum, values less than zero indicate no extremum.
- Check d^2z/dx^2 and d^2z/dy^2, greater than 0 indicates a local minimum, less than 0 indicates a local maximum
What methods are there for identifying your ‘x’ and ‘y’ values at stationary points after the first derivatives have been set to 0.
- Substitution
- Quadratic equation
What is the expression used at given stationary points to determine whether an extremum is present or not?
Delta (triangle) = (d^2z/dx^2) x (d^2z/dy^2) - (d^2z/dxdy)^2
How do you conclude whether a stationary point is an extremum or not following the delta expression?
Delta>0 indicates an extremum
Delta<0 indicates NO extremum
Delta=0 indicates an inconclusive test, max or min still possible
What do you do next if there is no extremum at the stationary point?
Nothing, proceed to the next stationary point to investigate it’s nature.
What do you do next if there is an extremum at the stationary point?
Check (d^2z/dx^2) and (d^2z/dy^2) using the appropriate x,y values for that stationary point.
> 0 indicates a local minimum
<0 indicates a local maximum
What is the approximate relation formula for small increments?
Assume the subject is delta y
δy ~ (dy/dx) x δx
What is the approximate relation formula for small increments in multivariate functions? (x,y,z)
δz ~ (dz/dx) x δx + (dz/dy) x δy
How do you estimate the volumetric change (δV) if both ‘r’ and ‘h’ are changes by a small amount and the formula for the volume is equal to:
V=PIr^2h
Take the first derivatives of the volume formula with respect to the variables that change only, i.e. ‘r’ and ‘h’
Use the approximate relation formula:
δV ~ (dV/dr) x δr + (dV/dh) x δh
Divide your answer by the original formula to get your % change in volume as a decimal
When the formula for a small increments question contains a fraction, what can you do to the denominator to make it more simple?
e.g. x/t^3
Set the denominator equal to superscript ^-(power)
e.g. x/t^3 becomes xt^-3
How do you consider numbers that come in front of each expression during the approximate relation formulas, i.e. if you had the following:
2/3 PIrh x 0.015r + 1/3 PIr^2 x (-0.06h)
Where the original formula is: V = 1/3 PIr^2h
How do you calculate the resultant change in volume?
Take all of the multiples out of the functions so that they equal the original formula, and multiply the δr or δh value by that same factor. E.g.
2/3 PIrh x 0.015r
becomes
1/3 PIrh x 0.030r
To calculate the resultant change in volume, you ignore the formula and sum only the δ values, in this case it would be 0.030 + (-0.060) = -0.030 (-3%)
How do you summarise your answer in a small increments question?
‘The relative change in the _ is +/- _%
Some small increment questions include a reference test and subsequent tests changing one variable at a time. You will be asked to approximate the total change to the system based on multiple variables changing at once.
What formula should be used? (w=calculated value, b and h = variables)
δw ~ (dw/db x δb) + (dw/dh x δh)
Some small increment questions include a reference test and subsequent tests changing one variable at a time. You will be asked to approximate the total change to the system based on multiple variables changing at once.
What are the steps for solving this problem?
W= calculated value
b,h= variables
- Estimate the first derivation of your first variable with respect to the calculated value (e.g. dw/db) by taking the change in ‘w’ (δw) from the reference to the 1st test and dividing it by the change in ‘b’ (δb) which is a decimal multiplied by the reference value for b.
- Repeat this for the remaining variables being careful to select the correct test types.
-Calculate the total change to the system using the approximation function and your calculated values:
δw ~ (dw/db x δb) + (dw/dh x δh)
Where ‘dw/db’ is your value calculated by division and ‘δb’ is the change in b specified in the new design in the question.
What is δx equal to in small increment questions with a reference test?
Where x= calculated value
The change in ‘x’ between the reference and the appropriate test
What is dx/dy equal to in small increment questions with a reference test?
dx/dy = δx/δy
It is equal to the change in one variable between the reference and current test, divided by the change in another between the same tests
What subscripts must you use in the following scenarios within small increment questions with reference tests?
Calculating the change between test 1 (ref) and test 2 (δb only)
Calculating the change between test 1 (ref) and test 4(δc only)
The superscripts would be:
(superscript) 1 2
(superscript) 1 4
