Math 155-004, Quiz 1 January 27, 2008 Instructions: This quiz is closed books and closed notes. You may use calculators. Work that is erased or crossed out will not be graded. 1. (5 pts) Suppose that a baby grows (in volume) by 2 cm 3 /week. Given that 1 week = 7 days, 1 day = 24 hours, and 1 in = 2.54 cm, how fast is the baby growing in in 3 /hour? Write out conversion 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 a function of how much the window is open (W, in cm 2 ) according to M(W ) = 5W + 2. The number of bites (B) you get depends on the number of mosquitos by B(M) = 0.5M. Find the number of bites you get as a function of how much the window is open.
Math 155-004, Quiz 2 February 3, 2009 Instructions: This quiz is closed books and closed notes. You may use calculators. Work that is erased or crossed out will not be graded. 1. [Section 1.26 Ex. 9,27] (6 pts) Let m t+1 = 0.5m t + 3 and m 0 = 4. Graph the updating function 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 w t+1 = aw t + 3, where a is a parameter. Is there a value of a for which there is no equilibrium?
Math 155-004, Quiz 3 February 10, 2009 Instructions: This quiz is closed books and closed notes. You may use calculators. Work that is erased or crossed out will not be graded. 1. [1.7 Ex. 29] (5 pts) Consider the function S(t) = 3e 0.2t. (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.
Math 155-004, Quiz 4 February 17, 2009 Instructions: This quiz is closed books and closed notes. You may use calculators. Work that is erased or crossed out will not be graded. 1. (7 pts) Let V t represent the voltage of the AV node in the Heart Model. { e V t+1 = ατ V t + u if V t e ατ V c e ατ V t if V t > e ατ V c Let e ατ = 1 5, u = 10, and V c = 2. a) If V 0 = 6, calculate V 1. 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 V 0 = 6 to determine if this heart 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(m t ), along with the diagonal. Cobweb starting 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 V t+1 represent the voltage of the AV node in the Heart Model. { e V t+1 = ατ V t + u if V t e ατ V c e ατ V t if V t > e ατ V c A. Suppose that e ατ = 0.6065, V c = 25, u = 8, so that e ατ 41.2180. Does this system have an equilibrium? Why or why not? Justify your answer. If it has an equilibrium, find it algebraically. B. Cobweb from an initial value of V t = 50. Diagnose this heart as healthy, having a 2:1 block, or as having the Wenkebach Phenomenon. C. Repeat the above, replacing V c = 20 with V c = 5. Math 155-004, Section 1.11 Let V t+1 represent the voltage of the AV node in the Heart Model. { e V t+1 = ατ V t + u if V t e ατ V c e ατ V t if V t > e ατ V c A. Suppose that e ατ = 0.6065, V c = 25, u = 8, so that e ατ 41.2180. Does this system have an equilibrium? Why or why not? Justify your answer. If it has an equilibrium, find it algebraically. B. Cobweb from an initial value of V t = 50. Diagnose this heart as healthy, having a 2:1 block, or as having the Wenkebach Phenomenon. C. Repeat the above, replacing V c = 20 with V c = 5. Math 155-004, Section 1.11 Let V t+1 represent the voltage of the AV node in the Heart Model. { e V t+1 = ατ V t + u if V t e ατ V c e ατ V t if V t > e ατ V c A. Suppose that e ατ = 0.6065, V c = 25, u = 8, so that e ατ 41.2180. Does this system have an equilibrium? Why or why not? Justify your answer. If it has an equilibrium, find it algebraically. B. Cobweb from an initial value of V t = 50. Diagnose this heart as healthy, having a 2:1 block, or as having the Wenkebach Phenomenon. C. Repeat the above, replacing V c = 20 with V c = 5.
Math 155-004, Quiz 5 February 24, 2009 Instructions: This quiz is closed books and closed notes. You may use calculators. Work that is erased or crossed out will not be graded. 1. Evaluate the following limits. a. lim x 0 1 x b. lim x 2 x 2 3x+2 x 2 c. lim x 3 x 2 3x+2 x 2 2. Continuity. a. Is the function f(x) = 2x, if x < 1 100, 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) = 5t 2. a. Find the equation for the secant line between the points (1, 5) and (2, 20) on the graph of the function. b. Find i) the average rate of change of f(t) as a function of t near t 0 = 1. ii) the instantaneous rate of change at time t 0 = 1 (use the definition of the instantaneous rate of change as a limit).
Math 155-004, Quiz 6 March 3, 2009 Instructions: This quiz is closed books and closed notes. You may use calculators. Work that is erased or crossed out will not be graded. 1. (8 pts) (a) If f(x) = 2x 113 + 13x 3 + x + 13 13, find f (x). (b) Find the derivative of f(t) = t 0.2. (c) If f(x) = (x 3 + 2)(4x 3), find df dx using the product rule. ( ) (d) Calculate d 1 dx 4. x 3 2. (2 pts) Suppose that the fraction of chickadee chicks that survive, P (N), as a function of the 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 best strategy for this bird? Extra Credit: Whether water or air, Dabiri says, it all comes down to the same
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Math 155-004, Quiz 7 March 10, 2009 Instructions: This quiz is closed books and closed notes. You may use calculators. Work that is erased or crossed out will not be graded. 1. (4 points) Find the derivatives of the following functions. (a) f(x) = x 2 ln(x) (b) f(z) = z2 e z 2. (3 pts) Let p(t) = 2t 2 + 8t + 20 give the position (in meters) at time t (in seconds) of a rock being thrown from a tower on the planet Giglïglop. (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) = 2x 3 + 1 and use them to 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 neurons appear to.
Math 155-004, Quiz 9 March 31, 2009 Instructions: This quiz is closed books and closed notes. You may use calculators. Work that is erased or crossed out will not be graded. Show your work for full credit. 1. (3 pts) x = 3 5 is an equilibrium for the discrete-time dynamical system x t+1 = 5 2 x t(1 x t ). Using the Stability Theorem, is this equilibrium stable, unstable, or neither? 2. (3 pts) Consider the Logistic Growth Model x t+1 = rx t (1 x t ). What condition 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) = x 3 3x + 1 on 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 circulatory system 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 1. (3 pts) x = 3 4 is an equilibrium for the discrete-time dynamical system x t+1 = 4x t (1 x t ). Using the Stability Theorem, is this equilibrium stable, unstable, or neither? 2. Consider the Ricker Model x t+1 = rx t e xt. (a) Verify that the equilibria are x = 0 and x = ln(r). (b) Show that the derivative of the updating function is re x rxe x. (c) What condition on r guarantees that the equilibrium r = 0 is stable? Math 155-004, Quiz 9 Practice 1. (3 pts) x = 3 4 is an equilibrium for the discrete-time dynamical system x t+1 = 4x t (1 x t ). Using the Stability Theorem, is this equilibrium stable, unstable, or neither? 2. Consider the Ricker Model x t+1 = rx t e xt. (a) Verify that the equilibria are x = 0 and x = ln(r). (b) Show that the derivative of the updating function is re x rxe x. (c) What condition 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) = x 3 +2x 2 x 2 on 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) = x 3 +2x 2 x 2 on the interval 3 x 2. Give your answers as coordinates (as in There is a global max at (2,5). ).
Math 155-004, Quiz 10 April 5, 2009 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 f 0 (x) and f (x). 2. For f(x) = 2x2 4+x 2, find f 0 (x) and f (x). Use the Method of Matched Leading Behaviors to sketch 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 s Rule, justify its use and show each step. (a) lim x 0 tan x x+e x 1 (b) lim x x 2 +e x ln(x)+e 5x (c) lim x 3 sin(x 3) x 3
Math 155-003, Quiz 10 April 5, 2009 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 f 0 (x) and f (x). 2. For f(x) = 2x2 4+x 2, find f 0 (x) and f (x). Use the Method of Matched Leading Behaviors to sketch 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 s Rule, justify its use and show each step. (a) lim x 0 tan x x+e x 1 (b) lim x x 2 +e x ln(x)+e 5x (c) lim x 3 sin(x 3) x 3
Math 155-004, Quiz 11 April 21, 2009 Instructions: This quiz is closed books and closed notes. You may use calculators. Work that is erased or crossed out will not be graded. 1. (3 points) Use the tangent line approximation with base point a = 3 to estimate 3.03 2. 2. (3 points) Consider the process of using Newton s Method to find an equilibrium of p t+1 = p t p 2 t + sin(p t ). (a) Write Newton s Method as a discrete-time dynamical system for this example; use x t+1 and x t. (b) Find x 1 if x 0 = 1. 3. (4 pts) Use Euler s Method to approximate y(1) if dy dt = t2 and y(0) = 2. Use t = 0.5. Extra Credit: Topology is the mathematical study of shapes. Circle the two of the following that are topologically equivalent:
trefoil knot coffee mug doughnut sphere
Math 155-004, Quiz 12 April 28, 2009 Instructions: You may use calculators. Work that is erased or crossed out will not be graded. 1. (4 points) Use the indefinite integral to solve the differential equation dm dt = t 2 + 1 t 2 with M(3) = 10.0. 2. (6 points) Evaluate the following integrals: (a) 3e 2t dt (b) x 3 sin(x 4 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 13 May 5, 2009 Instructions: You may use calculators. Work that is erased or crossed out will not be graded. 1. (2 points) Find the left-hand estimate to the definite integral of the function e t from t = 0 to t = 1. Use n = 2. 2. (4 points) Evaluate the following integrals: (a) π 0 3t sin(t 2 )dt (b) 1 1 x 2 dx
3. (2 points) Find the area under the graph of the curve g(x) = xe 2x for 1 x 2 (use integration by parts). 4. (2 points) The population p(t) of bunny rabbits in Ingersoll Meadow follows the differential equation dp dt = 4 + 2 sin(3t), and the population at time t = 0 is 100. (a) Express the total change in the population between times 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 system of.
Math 155-004, Quiz 13 Practice May 5, 2009 1. [4.4] Find the left-hand estimate to the definite integral of the function f(t) = 1 + t + t 2 from t = 0 to t = 1. Take n = 4. 2. Evaluate the following integrals: (a) 2 1 (x x 2 + 1 2t 4 + 6π)dt (b) 4 2 x3 sin(x 4 2)dx; 4 2 x3 (x 4 2) 1 4 dx (c) 1 1 x 2 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) = xe 2x ; 1 x 2 (use integration by parts). (b) g(x) = 3 sin(2x); 0 x π 2
4. [4.5] Archibald throws a rock off of a cliff. Suppose that the position p(t) of the rock follows the differential 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 between times 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 and that s(2009) = 400. ds dt = 20(t 2009)2, (a) Write down a definite integral that represents the total number of new swine flu cases between 2010 and 2012. (b) Evaluate the integral. 6. [4.6] Find the average value of the function sec 2 (x) on the interval [0, π 4 ].