MATH 1170: Calculus for Biologists I (Fall 2010)
Lab Meets: September 28, 2010
Report due date: October 5, 2010
Section 002: Tuesday 9:40−10:30am
Section 003: Tuesday 10:45−11:35am
Lab location − LCB 115
Lab instructor − Erica Graham, graham@math.utah.edu
Lab webpage − www.math.utah.edu/~graham/Math1170.html
Lab 05
General Lab Instructions
In−class Exploration
Review: Last week, we analyzed a discrete−time dynamical system from start to finish.
Background: Today, we will use Maple to explore the relationships between functions and their secant or
tangent line approximations.
restart;
Let’s start with a function that describes a population of bacteria.
b:=t−>4*(0.5^t);
The "growth" in the population over time can be seen in the following graph. We will also specify the y−
axis limits within the plot by an option of the form ymin..ymax, placed directly after the x−axis limits.
Below, this is used to confine the graph of b(t) within the interval from ymin=0 to ymax=4 along the y−axis.
plot(b(t),t=0..10,0..4,labels=["t","b(t)"]);
b := t 4 0.5t
4
3
b(t) 2
1
0
0
2
4
6
8
10
t
An important question we could ask about our population is, "What’s the instantaneous rate of growth at
time t = 2?"
Point & slope
line: a mini−review
The point−slope form of a line can be calculated as follows:
Given a slope m and a point (x1, y1), the equation of the line passing through the point (x1,y1) in the (x,y)−
plane is represented by (y − y1) = m*(x − x1). To define y in terms of x, this becomes, y = m*(x − x1) + y1.
If y = f(x), this turns into f(x) = m*(x−x1) + y1.
Lab 05 Homework Problems
Please copy this entire section into a new worksheet, and save it as something
you’ll remember.
Your Full Name:
Your (registered) Lab Section:
Useful Tip #1: Read through the assignment to get a clear idea of what’s being asked of you. When you’ve
finished your assignment, review it to make sure you’ve completed each exercise and answered each
question.
Useful Tip #2: Save your work early and often.
Useful Tip #3: When in doubt, restart.
Useful Tip #4: Make the size of your output graphs smaller to save paper when you print them. (Try this
even if you ’print to file.’ You can see how much paper you’d use beforehand by going to File
Print
Preview.)
(1)(a) Re−enter the function b(t) used in class.
## Define b(t) here.
(b) What is the initial population of the bacterial population described by b(t)?
(c) What is the per capita growth rate of the same population?
(2)(a) The secant line connects two points on a curve. We will use the points (2,b(2)) and (4,b(4)) to define
the secant line.
Compute the slope of the line that connects the two points (2,b(2)) and (4,b(4)), and call it ’slope1.’
(Remember "rise over run"?)
Slope1 indicates the average rate of growth of the bacterial population between t=2 and t=4.
slope1:= ## use this space to compute slope1
(b) Define a linear function called ’sec_line1(t)’ in point−slope form using the point (2,b(2)) and the slope1
you computed above. (See the mini−review for help. To view its contents, click the side−facing triangle to
the left of the heading.)
sec_line1:=t−> ## use this space to enter your line
(c) Plot both the original function b(t) and its secant line approximation sec_line1(t) on the same graph for t
values between 0 and 10 and for y−axis values between 0 and 4.
Required: [1] Specify a different line style for each function (using the linestyle option) and [2] include a
legend to help identify each function. Use the help index and/or your old labs as a reference.
## use this space to plot the 2 functions
(3)(a) Compute the slope of the line that connects the two points (2,b(2)) and (0,b(0)), and call it ’slope2.’
slope2:= ## use this space to compute slope2
(b) Now, define the linear function sec_line2(t) using (2,b(2)) and slope2.
sec_line2:=t−> ## use this space to enter your line
(c) Plot b(t), sec_line1(t), and sec_line2(t) on the same graph for t values between 0 and 10 and for y−axis
values between 0 and 4.
Required: [1] Specify a different line style for each function (using the linestyle option) and [2] include a
legend to help identify each function. Use the help index and/or your old labs as a reference.
## use this space to plot the 3 functions
(d) Describe at least two interesting things you notice about the curves you plotted. (Describe means use
full sentences.)
## Thing 1
## Thing 2
(4)(a) Thinking of dt as a small increment of time, compute the slope between the points (2,b(2)) and (2 +
dt, b(2+dt)), and store it as a function of dt called ’slope.’
slope:=dt−> ## use this space to define the slope as a function of
dt
(b) Plot the slope as a function of dt values between −2 and 2 and y−axis values between −1.5 and 0.5.
## use this space to plot slope(dt)
(c) Describe what is happening in the graph from part (b). Also, approximately what value does slope(dt)
take on when dt=0?
(5)(a) Using a new Maple command, limit( ), take the limit of slope(dt) as dt approaches zero. Assign it a
name.
## the command looks like:
## limit(function(dt),dt=0),
## where ’function’ is whatever function you care about
(b) In words, describe the limit you just computed with respect to the calculations you made in previous
problems.
(c) Use your newly named limit from part (a) to define the tangent line at the point (2,b(2)) as a function,
and call it ’tan_line(t).’
tan_line:=t−> ## use this space to enter your line
(d) Plot b(t), tan_line(t) and just one of your secant lines from above for t values between 0 and 10 and for
y−axis values between 0 and 4.
Required: [1] Specify a different line style for each function (using the linestyle option) and [2] include a
legend to help identify each function. Use the help index and/or your old labs as a reference.
## use this space to plot
(e) How does the tangent line compare to the secant line?
(6)(a) Investigate your plot in (5)(d) around the point t = 2 by changing the interval of t−values in your plot
to t=1..3. DO NOT specify y−axis values here! Do it again for t=1.5..2.5. Try it again with the even
smaller window t=1.99..2.01. (You should have three new plots when you’re done.)
Required: [1] Specify a different line style for each function (using the linestyle option) and [2] include a
legend to help identify each function. Use the help index and/or your old labs as a reference.
## first plot
## second plot
## third plot
(b) Compare the graphs you generated in part (a). What similarities/differences do you notice?
(7)(a) Given what you now know about tangent and secant lines, what would you guess is the
instantaneous rate of growth of the bacterial population at time t=2?
(b) Trick question (yes, you still have to answer it): What do you think the derivative of b(t) at time t=2
is?
(8) Take a few minutes and lines to remind yourself of any old commands as well as any new ones we used
today. Required: list and describe them.