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Lab notes for math 1170 - calculus for biologists, covering weeks 1-7. The notes include goals for each week, commands used, and examples of functions and their derivatives. The document also includes instructions for students on how to use maple to compute derivatives and solve equations.
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Lab room: LCB 115 Lab instructor: Sean Laverty - laverty@math.utah.edu, sean.laverty@utah.edu
Section 02: 9:40 am Section 03: 10:45 am Section 04: 12:55 am
Lab 8 Review: We’ll spend some extra time reviewing the first half of the course, look at new commands we learned each week and how we used them, and how everything fits together.
Week one: The goal for week one was to look at some graphs
of simulated data. We looked at a summary of the data (final vs initial) then we looked at the full population trajectories in time. There were a lot of commands hidden in the lab, but the most important command was plot(). We also defined a line by guessing a slope and an intercept.
restart; my_first_line:=x->slope*x+interc ept; slope:=5; intercept:=1; plot(my_first_line(x),x=0..10,y= 0..10,labels=["function output","input variable"],color=green); my_first_line := x → slope x + intercept slope := 5 intercept := 1
slope:=5; intercept:=1;
my_first_line := x → slope x + intercept slope := 5 intercept := 1
my_other_line:=x->other_slope*x+ other_intercept; other_slope:=1/4; other_intercept:=-1; plot([my_first_line(x),my_other_ line(x)],x=0..10,y=0..10,labels= ["function output","input variable"],color=[green,black]);
my_other_line :=
x → other_slope x + other_intercept
other_slope :=
other_intercept := -
Week three: The goal for week three was to explore
discrete-time dynamical systems. We found equilibria using the solve() command. Then we used new user-defined commands to cobweb solutions and print or plot the iterated values. If our discrete time dynamical system were defined by M[t+1]=my_first_line(M[t]), we would do the following to find the equilibrium point and prepare to cobweb.
restart; my_first_line:=x->slope*x+interc ept; solve(my_first_line(x)=x,x); slope:=1.25; intercept:=-1; plot([x,my_first_line(x)],x=0.. 0,y=0..10,color=[red,black]);
then we could install the
package with cobwebbing functions my_first_line := x → slope x + intercept
intercept slope − 1
your lines as updating functions of the simulated and real experiment. We cobwebbed some more, but this time we were allowed to think about nonlinear updating functions. The important difference in this lab was the existence of multiple equilibrium points.
nothing really new happened
here... whew!
Week five: In this lab, we reviewed the material for your first in-class test. This was a tricky lab since we had to translate a few times from word problem to math problem to Maple problem back to word problem. This was what you had to do for your test except in the lab we could use Maple to solve some equations and make some graphs. One concept we highlighted was the difference between solving for the value of an input variable for which a function takes on a particular value (solving for the x for which f(x)=5) and evaluating a function at a particular input (plugging in an input to our function).
suprisingly, nothing really
new happened here... whew!
solve(my_first_line(x)=50,x); ## at which x does the function have the value of 50? my_first_line(5); ## what is the value of the function when x=5?
Week six: In this lab, we used some old methods to define linear functions. Now we were looking for particular forms of lines (computing lines in point-slope form) and not guessing slopes and intercepts. We graphed a few secant and tangent lines and also computed limits using the new command limit(). Here is a limit:
limit(1/x,x=0,right); limit(1/x,x=0,left); limit(1/x,x=0);
∞ −∞
1
2
3
4
5
–1 0 1 2 3 4 x
g := x →− 1 + x − +
x^2
x^3
2
4
–1 1 2 3 4 x
We computed derivatives by hand last time, but Maple can compute derivatives for us using a new command. With the function f(x) defined above, we’ll use the new command to take a derivative and check to see that both methods agree.
diff(f(x),x);
− 1 + x − +
x^2
x^3
The command diff() takes in two arguments, the first is the expression of function that you wish to
derivative of f_prime(x)
f_prime_prime:=unapply(diff(f(x) ,x,x),x); ## take two derivatives of f(x)
f_prime_prime := x → 1 − x +
x^2
f_prime_prime := x → 1 − x +
x^2
I’ve argued before that most Maple commands have sensible names. This argument breaks down with unapply(). It doesn’t make any sense, but you’ll have to try to remember it as "the command with the name that makes no sense." When you need it, I’ll try to remind you. As an example, the code G:=unapply(expression with x, x) says "make ‘G’ a function of ‘x’, whose right hand side is ‘the expression with x’. We can take lots of derivatives by hand, but the diff() command is definitely useful. In the homework, you’ll be asked to take some derivatives and plot functions and their
derivative(s). Using the solve command you can solve for important values of your function that will help you give detailed descriptions of the function.
Paper saving tips:
click and shrink your figures so they are large
enough to read, but small enough so that I don’t
feel guilty about making you print them.
1). Enter the function q(x)=x^4+exp(x)*(88x^3-76x^2-65x+25).
use this space to enter the
function
q:=...
2). Find q’(x), the derivative of q(x), using the diff() command.
use this space to compute the
the graph that you just made, describe (as completely as possible) how the function and its derivatives are related. Think of things like increasing, decreasing, and concavity.
8). Take the 5th derivatives of q(x) - depending on your method you can jump straight to the 5th derivative or take the 3rd and 4th along the way.
Plot q(x) and its fifth derivative and describe what this derivative looks like.
use this line to compute the
derivative
fifth_derivative:=x->
plot it
11). Take a few minutes and lines to remind yourself of any old commands as well as any new commands we used today.
we know a few commands new
