HELIXPrep › Engineering Mathematics › Change
Module 1.1 · Topic 5
Derivative
A derivative asks:
How does something change?
Partial Derivative
A partial derivative asks:
How does something change
if I vary only ONE factor?
Enzyme Activity Puzzle
Enzyme Activity depends on:
If enzyme activity changes,
which variable caused it?
The discovery
You've just discovered why
ordinary derivatives aren't enough.
| Temperature | pH | Activity |
|---|---|---|
| 25 | 6 | 40 |
| 30 | 6 | 60 |
| 30 | 7 | 90 |
When activity increased,
what caused it?
Discovery Activity
Reading the Contour Map
Lines closer together = steeper terrain. You stand at the base of the mountain.
If you move east,
what happens to the elevation?
Fermentation Example
Activity depends on:
Input 1
Temperature
Input 2
pH
Imagine standing on a surface.
Every point on the base represents a combination of Temperature and pH.
Every point = (Temperature, pH)
Height = Enzyme Activity
If we move only along the temperature direction,
how rapidly does activity change?
Explore the Surface
You stand at a specific point where Temperature and pH are fixed. Choose a direction to move.
The Partial Derivative
The Function
A = f(T, pH)
A = Enzyme Activity
T = Temperature
pH
means:
Change temperature.
T is the variable we vary.
Freeze pH.
pH = constant — held fixed.
Observe what happens.
How does A respond to the change in T?
means:
Change pH.
pH is the variable we vary.
Freeze temperature.
T = constant — held fixed.
Observe what happens.
How does A respond to the change in pH?
Activity depends on Temperature and pH. To compute ∂Activity/∂Temperature, what remains fixed?
The Optimization Question
Suppose you want maximum enzyme activity.
Which temperature and pH should you choose?
Can you guess which corner of the surface holds the maximum?
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