MATA35-2021
Department of Computer & Mathematical Sciences
University of Toronto Scarborough
MATA35H3Y: Calculus II for Biological Sciences
Final Examination Practice Sheet
Examiner: Yun William Yu
Date: August 16, 2021, 8am
Duration: 2 hours
Please make sure you include the following information on your cover sheet, to be scanned to Crowdmark:
Student information:
Course information
A species of bird has three life stages, hatching (H), juvenile (J), and adult (A). After some study, you build a Leslie Age-Structured model capturing the yearly change in the population with the following Leslie diagram:
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The population one year into the future will be 160 hatchlings, 18 juveniles, and 70 adults.
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The population two years into the future will be 158 hatchlings, 32 juveniles, and 53 adults.
To estimate the population one year ago, you need to invert the matrix multiplication problem. One method is to find the matrix inverse, but this is probably overkill. Instead, we simply need to solve the following:
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We can do that either as a system of equations or using Gaussian elimination.
Gaussian elimination:
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System of equations:
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Either way, our estimate for the population in the previous year is 200 hatchlings, 50 juveniles, and 20 adults.
Let
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To classify the saddle points, we can either use the D-test or look at the eigenvalues of the Hessian. I personally prefer eigenvalues of the Hessian.
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Sincehas opposite sign eigenvalues, it is a saddle point.
Sincehas positive eigenvalues, it is a relative minimum.
You are an epidemiologist from the University of Toronto who was recently tasked with understanding the Cooties. To this end, you observe children at a local primary school, and come to several conclusions:
After weeks of painstaking observation, you gather enough information to build a two-compartment model:
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In the long-term, we have the endemic disease equilibrium, where asymptotically 85 students will have Cooties at any one time.
Solve each of the following problems as generally as possible.
Part 1 is separable.
Substitute in, and we get
Part 2 is inhomogeneous linear with constant coefficients.
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Part 3 is exact, so we can assume there exists some
such that
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Then, we simply need to integrate both pieces holding the other variable constant to get $f(x,y)
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Then, we need to substitute in the initial conditionswhen to solve for .
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The extra separable practice problem in part 4 resembles the logistic equation.
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Consider the first-order linear system
Find the general solution. Classify the type of equilibrium at the origin, and determine its stability.
Either the reduction method or the eigenbasis method will work for solving the equation. Using eigenvectors is somewhat easier whenever it is applicable.
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The only equilibrium is at the origin (0,0), and the equilibrium is a saddle point, which is unstable.
Reduction method
Note that
Characteristic equation
Thus,, .
Thus
And thus
So
In conclusion, we get the following:
Notice that theand are slightly different than the ones we got above using the eigenbasis method. However, the answers are actually the same, by just sending the here to , which you can do becuase it's an arbitrary constant.
Find a cubic approximation of the following function using the first 4 terms of its Maclaurin Series (Taylor Series at 0).
Hint:
We can solve this either directly by using the Taylor Series formula, or we more easily by plugging
in for in the formula for .
Hard direct method
Faster plugging-in method
Consider the following system
Real equilibria are at
and .
, which has eigenvalues , which are opposite signs, so this is an unstable saddle point.
, which has eigenvalues by the quadratic equation, which are both negative, so it is an asymptotically stable node.