math 155-004, quiz 1 january 27, 2008 nameshipman/math155/files/shipmanquizzesspri… · math...

Math 155-004, Quiz 1January 27, 2008 NAME:

Instructions: This quiz is closed books and closed notes. You may use calculators. Work that iserased or crossed out will not be graded.

1. (5 pts) Suppose that a baby grows (in volume) by 2 cm3/week. Given that 1 week = 7days, 1 day = 24 hours, and 1 in = 2.54 cm, how fast is the baby growing in in3/hour? Write outconversion factors and show your work for full and partial credit.

2. [Section 1.2, Ex. 53] (5 pts) The number of mosquitos (M) that end up in a room is afunction of how much the window is open (W , in cm2) according to M(W ) = 5W +2. The numberof bites (B) you get depends on the number of mosquitos by B(M) = 0.5M . Find the number ofbites you get as a function of how much the window is open.

Math 155-004, Quiz 2February 3, 2009 NAME:


1. [Section 1.26 Ex. 9,27] (6 pts) Let mt+1 = −0.5mt + 3 and m0 = 4. Graph the updatingfunction associated with this system and then cobweb for three steps. Clearly label your graph,including axes and cobwebbing points. What is the long-term behavior of this system?

2. (4 pts) Algebraically find the equilibria for the discrete-time dynamical system wt+1 =awt + 3, where a is a parameter. Is there a value of a for which there is no equilibrium?



1. [1.7 Ex. 29] (5 pts) Consider the function S(t) = 3e0.2t. (i) Is S(t) increasing or decreasing?(ii) Graph the function. (iii) Graph a semilog plot of the function. (iv) If the function is increasing,find the doubling time; if it is decreasing, find the half-life. (v) For what value of t does S(t) = 12?

2. (3 pts) Let f(x) = −2 + 3 cos(2t− 4). Give values for the following:

average value:

amplitude:

maximum value:

period:

phase:

3. (2 pts) Write a formula for the weighted average of two values x and y with weight q.



1. (7 pts) Let Vt represent the voltage of the AV node in the Heart Model.

Vt+1 ={

e−ατVt + u if Vt ≤ eατVce−ατVt if Vt > eατVc

Let e−ατ = 15 , u = 10, and Vc = 2.

a) If V0 = 6, calculate V1. Will the heart beat? Why or why not?

b) Does this system have an equilibrium? If so, find it algebraically; if not, explain why not.

c) Graph the updating function and cobweb from an initial value of V0 = 6 to determine if thisheart is

i) healthy, ii) has a 2:1 block, or iii) has the Wenckebach phenomenon.

2. (3 pts) Graphed below is an updating function f(mt), along with the diagonal. Cobwebstarting from an initial value of 0.6. Is the equilibrium m∗ = 1 stable or unstable?

Extra Credit: In the article, R(t) refers to the love (or hate if negative) of(someone’s name).

Math 155-004, Section 1.11

Let Vt+1 represent the voltage of the AV node in the Heart Model.

Vt+1 ={


A. Suppose that e−ατ = 0.6065, Vc = 25, u = 8, so that eατ ' 41.2180. Does this systemhave an equilibrium? Why or why not? Justify your answer. If it has an equilibrium, find italgebraically.

B. Cobweb from an initial value of Vt = 50. Diagnose this heart as healthy, having a 2:1 block,or as having the Wenkebach Phenomenon.

C. Repeat the above, replacing Vc = 20 with Vc = 5.



Vt+1 ={







Vt+1 ={







1. Evaluate the following limits.

a. limx→0−1x

b. limx→2x2−3x+2x−2

c. limx→3x2−3x+2x−2

2. Continuity.

a. Is the function f(x) =2x, if x < 1100, if x ≥ 1

continuous at x = 1? Why or why not?

b. Is the function g(x) = x2

ln(x) continuous at x = 0? Why or why not?

3. Let f(t) = 5t2.

a. Find the equation for the secant line between the points (1, 5) and (2, 20) on the graph ofthe function.

b. Find i) the average rate of change of f(t) as a function of ∆t near t0 = 1. ii) the instantaneousrate of change at time t0 = 1 (use the definition of the instantaneous rate of change as a limit).

Math 155-004, Quiz 6March 3, 2009 NAME:


1. (8 pts)

(a) If f(x) = 2x113 + 13x3 + x+ 1313, find f ′(x).

(b) Find the derivative of f(t) = t−0.2.

(c) If f(x) = (x3 + 2)(4x− 3), find dfdx using the product rule.

(d) Calculate ddx

(1

4√x3

).

2. (2 pts) Suppose that the fraction of chickadee chicks that survive, P (N), as a function ofthe number N of eggs laid is given by

P (N) = 1− 0.05N.

The total number of offspring is S(N) = NP (N). Find S′(N). Plot S′(N). What is the beststrategy for this bird?

Extra Credit: “Whether water or air,” Dabiri says, “it all comes down to the same



1. (4 points) Find the derivatives of the following functions.

(a) f(x) = x2 ln(x)

(b) f(z) = z2

ez

2. (3 pts) Let p(t) = −2t2 + 8t + 20 give the position (in meters) at time t (in seconds) of arock being thrown from a tower on the planet Gigl̈ıglop.

(a) Find the velocity of the rock (with units).

(b) Find the acceleration of the rock (with units).

(c) How high was the tower?

3. (3 pts). Find the first and second derivatives of the function f(x) = 2x3 + 1 and use themto sketch a graph of the function.

Extra Credit: Whereas most neurons are thought to incoming

signals together to come up with the appropriate neural response, the owl’s auditory map neuronsappear to .


Instructions: This quiz is closed books and closed notes. You may use calculators. Work that iserased or crossed out will not be graded. Show your work for full credit.

1. (3 pts) x∗ = 35 is an equilibrium for the discrete-time dynamical system

xt+1 =52xt(1− xt).

Using the Stability Theorem, is this equilibrium stable, unstable, or neither?

2. (3 pts) Consider the Logistic Growth Model xt+1 = rxt(1−xt). What condition on r guaranteesthat the equilibrium r∗ = 0 is stable?

3. (4 pts) Identify all the local and global maxima and minima of the function f(x) = x3 − 3x+ 1on the interval 0 ≤ x ≤ 3. Give your answers as coordinates (as in “There is a global max at(2,5).”).

Extra Credit: In recent years, researchers studying the physics of the human circulatorysystem have moved toward computer models that use software tools handed down form the

industry to solve fluid mechanics equations.

Math 155-004, Quiz 9 Practice


xt+1 = 4xt(1− xt).


2. Consider the Ricker Model xt+1 = rxte−xt . (a) Verify that the equilibria are x∗ = 0 andx∗ = ln(r). (b) Show that the derivative of the updating function is re−x − rxe−x. (c) Whatcondition on r guarantees that the equilibrium r∗ = 0 is stable?

Math 155-004, Quiz 9 Practice


xt+1 = 4xt(1− xt).


2. Consider the Ricker Model xt+1 = rxte−xt . (a) Verify that the equilibria are x∗ = 0 andx∗ = ln(r). (b) Show that the derivative of the updating function is re−x − rxe−x. (c) Whatcondition on r guarantees that the equilibrium r∗ = 0 is stable?

3. (4 pts) Identify all the local and global maxima and minima of the function f(x) = x3+2x2−x−2on the interval −3 ≤ x ≤ 2. Give your answers as coordinates (as in “There is a global max at(2,5).”).

3. (4 pts) Identify all the local and global maxima and minima of the function f(x) = x3+2x2−x−2on the interval −3 ≤ x ≤ 2. Give your answers as coordinates (as in “There is a global max at(2,5).”).

Math 155-004, Quiz 10April 5, 2009 NAME:

Instructions: You may use calculators. Work that is erased or crossed out will not be graded.

1. For f(x) = 200 ln(x) + x−2 + 2000e−x, find f0(x) and f∞(x).

2. For f(x) = 2x2

4+x2 , find f0(x) and f∞(x). Use the Method of Matched Leading Behaviors tosketch a graph of f(x) on the interval [0,∞).

3. Evaluate the following limits. If you use leading behavior, show each step. If you use L’Hopital’sRule, justify its use and show each step.

(a) limx→0tanx

x+ex−1

(b) limx→∞x2+ex

ln(x)+e5x

(c) limx→3sin(x−3)x−3



1. For f(x) = 200 ln(x) + x−2 + 2000e−x, find f0(x) and f∞(x).

2. For f(x) = 2x2

4+x2 , find f0(x) and f∞(x). Use the Method of Matched Leading Behaviors tosketch a graph of f(x) on the interval [0,∞).

3. Evaluate the following limits. If you use leading behavior, show each step. If you use L’Hopital’sRule, justify its use and show each step.

(a) limx→0tanx

x+ex−1

(b) limx→∞x2+ex

ln(x)+e5x

(c) limx→3sin(x−3)x−3



1. (3 points) Use the tangent line approximation with base point a = 3 to estimate 3.032.

2. (3 points) Consider the process of using Newton’s Method to find an equilibrium of pt+1 =pt−p2

t +sin(pt). (a) Write Newton’s Method as a discrete-time dynamical system for this example;use xt+1 and xt.(b) Find x1 if x0 = 1.

3. (4 pts) Use Euler’s Method to approximate y(1) if dydt = t2 and y(0) = 2. Use ∆t = 0.5.

Extra Credit: Topology is the mathematical study of shapes. Circle the two of the followingthat are topologically equivalent:

trefoil knot coffee mug doughnut sphere



1. (4 points) Use the indefinite integral to solve the differential equation dMdt = t2 + 1

t2with

M(3) = 10.0.

2. (6 points) Evaluate the following integrals:(a)

∫3e2tdt

(b)∫x3 sin(x4 − 2)dx

(c)∫

5x sin(4x)dx (use integration by parts)

Extra Credit: The Road to gives you an Aha! experience mainly

because you a minimally aware of what you doing when you are solving a problem.

Math 155-004, Quiz 13May 5, 2009 NAME:


1. (2 points) Find the left-hand estimate to the definite integral of the function et from t = 0 tot = 1. Use n = 2.

2. (4 points) Evaluate the following integrals:(a)

∫ √π0 3t sin(t2)dt

(b)∫∞1

1x2 dx

3. (2 points) Find the area under the graph of the curve g(x) = xe2x for 1 ≤ x ≤ 2 (use integrationby parts).

4. (2 points) The population p(t) of bunny rabbits in Ingersoll Meadow follows the differentialequation

dp

dt= 4 + 2 sin(3t),

and the population at time t = 0 is 100. (a) Express the total change in the population betweentimes t = 0 and t = π

3 as a definite integral. (b) Evaluate the definite integral.

Extra Credit: The basic model used to analyze the dynamics of infectious disease is a systemof .

Math 155-004, Quiz 13 PracticeMay 5, 2009 NAME:

1. [4.4] Find the left-hand estimate to the definite integral of the function f(t) = 1 + t + t2 fromt = 0 to t = 1. Take n = 4.

2. Evaluate the following integrals:(a)

∫ 21 (x√x2 + 1− 2t−4 + 6π)dt

(b)∫ 42 x

3 sin(x4 − 2)dx;∫ 42 x

3(x4 − 2)14 dx

(c)∫∞1

1x2 dx

3. [4.6] Find the area between the graph of the curve g(x) and the x-axis on the given interval.

(a) g(x) = xe2x; 1 ≤ x ≤ 2 (use integration by parts).

(b) g(x) = 3 sin(2x); 0 ≤ x ≤ π2

4. [4.5] Archibald throws a rock off of a cliff. Suppose that the position p(t) of the rock follows thedifferential equation

dp

dt= −9.8t− 7

and that p(0) = 100.

(a) Write down a definite integral that represents the total change in position of the rock betweentimes t = 1 and t = 4. (b) Evaluate the integral.

5. [4.5] Suppose that number s(t) of cases of swine flu obeys the differential equation

ds

dt= 20(t− 2009)2,

and that s(2009) = 400.

(a) Write down a definite integral that represents the total number of new swine flu cases between2010 and 2012. (b) Evaluate the integral.

6. [4.6] Find the average value of the function sec2(x) on the interval [0, π4 ].

math 155-004, quiz 1 january 27, 2008 nameshipman/math155/files/shipmanquizzesspri… · math...

Documents