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All Textbook Solutions for Modeling the Dynamics of Life: Calculus and Probability for Life Scientists
Suppose you have a culture of bacteria, where the density of each bacterium is 2.0g/cm3. If each bacterium is 5m5m20m in size, find the number of bacteria if their total mass is 30 grams. Recall that 1m=106 meters. Suppose that you learn that the sizes of bacteria range from 4m5m15m to 5m6m25m What is the range of the possible number of bacteria making up the total mass of 30 grams?2SP3SPA lab has a culture of a new kind of bacteria where each individual takes 2 hours to split into three bacteria. Suppose that these bacteria never die and that all offspring are OK. Write an updating function describing this system. Suppose there are 2.0107 bacteria at 9 A.M. How many will there be at 5 P.M.? Write an equation for how many bacteria there are as a function of how long the culture has been running. When will this population reach 109?5SP6SP7SP8SP9SP10SPA person develops a small liver tumor. It grows according to S(t)=S(0)eat where S(0)=1.0 gram and =0.1/ day. At time t=30 days, the tumor is detected and treatment begins. The size of the tumor then decreases linearly with slope of -0.4 grams/day. Write the equation for tumor size at t=30. Sketch a graph of the size of the tumor over time. When will the tumor disappear completely?12SP13SP14SP15SP16SP17SP18SP19SP20SP21SP22SP23SP24SP25SP26SP27SP28SP29SPIdentify the variables and parameters in the following situations, give the units they might be measured in, and choose an appropriate letter or symbol to represent each. A scientist measures the mass of fish over the course of 100 days, and repeats the experiment at three different levels of salinity: 0,2 and 5.2ECompute the values of the following functions at the points indicated and sketch a graph of the function. f(x)=x+5 at x=0,x=1, and x=4.Compute the values of the following functions at the points indicated and sketch a graph of the function. g(y)=5y at y=0,y=1, and y=4.Compute the values of the following functions at the points indicated and sketch a graph of the function. h(z)=15z at z=1,z=2, and z=4.Compute the values of the following functions at the points indicated and sketch a graph of the function. F(r)=r2+5 at r=0,r=1, and r=4.Graph the given points and say which point does not seem to fall on the graph of a simple function. (0,-1),(1,1),(2,1),(3,5),(4,7).Graph the given points and say which point does not seem to fall on the graph of a simple function. (0,5),(1,10),(2,8),(3,6),(4,4).Graph the given points and say which point does not seem to fall on the graph of a simple function. (0,2),(1,3),(2,6),(3,11),(4,10).Graph the given points and say which point does not seem to fall on the graph of a simple function. (0,45),(1,25),(2,12),(3,12.5),(4,10).Evaluate the following functions at the given algebraic arguments. f(x)=x+5 at x=a,x=a+1, and x=4a.Evaluate the following functions at the given algebraic arguments. g(y)=5y at y=x2,y=2x+1, and y=2x.13E14E15E16E17E18E19E20E21E22E23E24E25EFind the inverses of each of the following functions. In each case, compute the output of the original function at an input of 1.0, and show that the inverse undoes the action of the function. g(x)=3x5.27E28E29EGraph each of the following functions and its inverse. Mark the given point on the graph of each function. g(x)=3x5. Mark the point (1,g(1)) on the graphs of g and g1 (based on Exercise 26 ).Graph each of the following functions and its inverse. Mark the given point on the graph of each function. G(y)=1/(2+y). Mark the point (1,G(1)) on the graphs of G and G1 (based on Exercise 27 ).Graph each of the following functions and its inverse. Mark the given point on the graph of each function. F(y)=y2+1 for y0. Mark the point (1,F(1)) on the graphs of F and F1 (based on Exercise 28 ).33EFind the compositions of the given functions. Which pairs of functions commute? f(x)=2x+3 and h(x)=3x12.35E36E37E38E39E40E41E42E43E44E45E46E47E48E49E50E51E52EThe following series of functional compositions describe connections between several measurements. The number of mosquitos ( M ) that end up in a room is a function of how far the window is open (W,incm2 .) according to M(W)=5W+2. The number of bites (B) depends on the number of mosquitos according to B(M)=0.5M. Find the number of bites as a function of how far the window is open. How many bites would you get if the window was 10cm2 open?The following series of functional compositions describe connections between several measurements. The temperature of a room (T) is a function of how far the window is open (W) according to T(W)=400.2W. How long you sleep ( S, measured in hours) is a function of the temperature according to S(T)=14T5. Find how long you sleep as a function of how far the window is open. How long would you sleep if the window was 10cm2 open?The following series of functional compositions describe connections between several measurements. The number of viruses ( V, measured in trillions or 1012 ) that infect a person is a function of the degree of immunosuppression ( I, the fraction of the immune system that is turned off by stress) according to V(I)=5I2. The fever (F, measured in C ) associated with an infection is a function of the number of viruses according to F(V)=37+0.4V. Find fever as a function of immunosuppression. How high will the fever be if immunosuppression is complete (I=1)?56E57E58E59E60E61E62E63E64E65E66E67E68E69E70E1E2E3E4E5E6E7E8E9E10E11E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28E29E30E31E32EFind the mass in kilograms of the following objects. A coral colony of 3200 individuals cach weighing 0.45 g.34EChange the units in the following functions, and compare a graph in the new units with the original units. (Based on Section 1.2, Exercise 45 ) The number of bees b on a plant is given by b=2f+1 where f is the number of flowers. Suppose each flower has 4 petals. Graph the number of bees as a function of the number of petals.36E37E38E39E40E41E42E43E44E45E46E47E48E49E50EFor the following lines, find the slopes between the two given points by finding the change in output divided by the change in input. What is the ratio of the output to the input at each of the points? Which are proportional relations? Which are increasing and which are decreasing? Sketch a graph. y=2x+3, using points with x=1 and x=3.2E3E4E5E6E7E8E9E10E11E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28E29E30E31E32E33E34E35E36E37E38E39E40E41E42EThe following data give the elevation of the surface of the Great Salt Lake in Utah. Graph these data.44E45E46E47E48E49E50E51E52E53E54E55E56E57E58E59E60EWrite the updating function associated with each of the following discrete-time dynamical systems and evaluate it at the given arguments. Which are linear? pt+1=pt2, evaluate at pt=5,pt=10, and pt=15.Write the updating function associated with each of the following discrete-time dynamical systems and evaluate it at the given arguments. Which are linear? t+1=t2, evaluate at t=4,t=8, and t=12.Write the updating function associated with each of the following discrete-time dynamical systems and evaluate it at the given arguments. Which are linear? xt+1=xt2+2, evaluate at xt=0,xt=2, and xt=4.Write the updating function associated with each of the following discrete-time dynamical systems and evaluate it at the given arguments. Which are linear? Qt+1=1Qt+1, evaluate at Qt=0,Qt=1, and Qt=2.Compose the updating function associated with each discrete-time dynamical system with itself. Find the two-step discrete-time dynamical system. Check that the result of applying the original discrete-time dynamical system twice to the given initial condition matches the result of applying the new discrete-time dynamical system to the given initial condition once. Volume follows vt+1=1.5vt, with v0=1220m3.Compose the updating function associated with each discrete-time dynamical system with itself. Find the two-step discrete-time dynamical system. Check that the result of applying the original discrete-time dynamical system twice to the given initial condition matches the result of applying the new discrete-time dynamical system to the given initial condition once. Length obeys lt+1=lt1.7, with l0=13.1cm.7E8E9E10E11E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28E29E30EUse the formula for the solution to find the following, and say whether the results are reasonable. Using the solution for tree height ht=10.0+tm (Example 1.5 .13), find the tree height after 20 years.32E33E34E35E36E37E38E39E40E41E42E43E44E45E46E47E48E49E50E51E52E53E54E55E56E57E58E59E60E61E62EThe following steps are used to build a cobweb diagram. Follow them for the given discrete-time dynamical system based on bacterial populations. Graph the updating function. Use your graph of the updating function to find the point b0,b1. Reflect it off the diagonal to find the point b1,b1. Use the graph of the updating function to find b1,b2. Reflect off the diagonal to find the point b2,b2. Use the graph of the updating function to find b2,b3. Sketch the solution as a function of time. The discrete-time dynamical system bt+1=2.0bt with b0=1.0.2E3E4E5E6E7E8E9E10E11E12E13EFind the equilibria of the following discrete-time dynamical system from the graphs of their updating functions Label the coordinates of the equilibria.15E16E17E18E19E20E21E22E23E24E25E26E27E28E29E