jamie's bird got poisoned

THE CALCULUS CRUSADERS

Differential Equations: Poisoned Bird Questions

Mushroom Eater by Flickr user: Just Emi

a) Oh no! Jamie’s pet duck ate the poisonous mushroom! Luckily, Bench knows what type of mushroom it is. The Fungus Differentius has a very dangerous toxin. When eaten, it can cause mutations. (Don’t you think the duck has had enough torture?) The rate at which the poison is spreading throughout the duck is defined as dp/dt. Sketch a slope field for at the points indicated.

b) Use Euler’s Method to approximate the solution of dp/dt with the initial condition of with 5 steps of size 0.2.

c) Find a particular solution to dp/dt with the initial condition .

PART A

Differential Equations: Mushroom Scene

If we take a coordinate that’s given on the graph and plug the coordinate’s x- and y-values into the differential equation, we obtain the slope at that point

Example:

By creating a display of lines (the slope field), where each line indicates the slope at that point, we can see the parent functionThat is, the solution

of a derivative via slope fields is the parent function

Therefore, by shortening the distance between the points, a smoother line is generated, making the function we’ve created further resemble the parent function

Allow ∆x to be infinitesimally small, we have the parent function!

The values on the grid below correspond to the position of the coordinates on the graph:The value in cell A1 represents the

coordinates (-4, 4) on the graph

PART B

Differential Equations: Mushroom Scene

Using Euler’s Method, we start at the initial coordinates,

In this case, at (3, 3.0642)

Note that all values in the graphs above are rounded to four decimal places for simplicity

By plugging the P0 coordinates into the differential equation, y’, we obtain y’ at P0

In this case, we plugged (3, 3.0642) into to obtain dp/dt = 29.4975

We know the definition of a slope as the rise (the change in the dependent variable) over the run (the change in the independent variable)

By multiplying y’ (the slope) by ∆x (the run), we obtain ∆y (the rise)

We’re given that ∆x = 0.2 Adding ∆y to y0, we obtain y1

We repeat this process until we reach the number of desired steps

Note that all values in the table above are given in their decimal form, rounded to four decimal places for simplicity

PART C

Differential Equations: Mushroom Scene

We can separate the variables, that is, in this case, antidifferentiating t on one side with respect to dt and p on one side with respect to dp

Separating the variables is analogous to antidifferentiating after solving for ∆y when given the definition of a slope

Integration by parts is an antidifferentiation technique we can use when we have to antidifferentiate two factors

We’re undoing the product rule

Formula for integration by parts:

L I A T EOGARITHMIC

NVERSE

TRIGONOMETRIC

LGEBRAIC

RIGONMETRIC

XPONENTIAL

LIATE is a mnemonic used to determine which of the factors should be selected for f. LIATE tells us the order of preference for f.

Bench says: I have discovered a rule for differentiating products involving et without using the whole process of integration by parts or LIATE! MWAHAHA!

The power rule says that the derivative of any variable to an exponent can be found by multiplying the term by the exponent and decrease the exponent by 1

BENCH’S RULE•Differentiate the algebraic factor until we get a constant•Note that the signs alternate: minus, plus, minus, plus, etc.• In this case: t2 – 2t + 2

We’re given the initial value We can use this fact to determine C* Remember, C is a constant!

Note: Since we’re antidifferentiating, we’d expect C’s on both

sides Let’s group the C’s to one side of the equation for simplicity

Putting it all together, we now have a general solution for p

Bench:Now that I know how the poison reacts in

the bird, I can cure it.

Jamie:Hurry up! Save my bird!

I think the bird is happy and healthy now…

Was I a Cockatoo at

the beginning?

Happy Bird 1 by Flickr user: Calsidyrose