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mathematics
calculus
Modeling the Dynamics of Life Calculus and Probability for Life Scientists 3rd edition Frederick R. Adler - Solutions
Cobweb the following discrete-time dynamical systems for three steps starting from the given initial condition. Compare with the solution found earlier. nt+1 = 0.5nt starting from n0 = 1200 For Information: (as in Section 1.5, Exercise 7).
Cobweb the following discrete-time dynamical systems for three steps starting from the given initial condition. Compare with the solution found earlier. Mt+1 =0.75 Mt + 2.0 starting from the initial condition M0 = 16.0 For Information: (as in Section 1.5, Exercise 8).
Graph the updating functions associated with the following discrete-time dynamical systems, and cobweb for five steps starting from the given initial condition. xt+1 = 2xt - 1, starting from x0 = 2.
Graph the updating functions associated with the following discrete-time dynamical systems, and cobweb for five steps starting from the given initial condition. zt+1 = 0.9zt + 1, starting from z0 = 3.
Graph the updating functions associated with the following discrete-time dynamical systems, and cobweb for five steps starting from the given initial condition. wt+1 = -0.5wt + 3, starting from w0 = 0.
Solve the following equation for x and check your answer. 7e3x = 21.
Solve the following equation for x and check your answer. 4e2x+l = 20.
Solve the following equation for x and check your answer. 4e-2x+l = 7e3x.
Solve the following equation for x and check your answer. 4e2x+3 = 7e3x-2.
Sketch graphs of the following exponential function. For each, find the value of x where it is equal to 7.0. For the increasing functions, find the doubling time, and for the decreasing functions, find the half-life. For what value of x is the value of the function 3.5? For what value of x is the
Sketch graphs of the following exponential function. For each, find the value of x where it is equal to 7.0. For the increasing functions, find the doubling time, and for the decreasing functions, find the half-life. For what value of x is the value of the function 3.5? For what value of x is the
Sketch graphs of the following exponential function. For each, find the value of x where it is equal to 7.0. For the increasing functions, find the doubling time, and for the decreasing functions, find the half-life. For what value of x is the value of the function 3.5? For what value of x is the
Sketch graphs of the following exponential function. For each, find the value of x where it is equal to 7.0. For the increasing functions, find the doubling time, and for the decreasing functions, find the half-life. For what value of x is the value of the function 3.5? For what value of x is the
Sketch graphs of the following updating function over the given range and mark the equilibria. h(z) = e-z for 0 ≤ z ≤ 2.
Sketch graphs of the following updating function over the given range and mark the equilibria. F(x) = ln(x) + 1 for 0 ≤ x ≤ 2. (Although this cannot be solved algebraically, you can guess the answer.)
Suppose M(t) = 43.2e5.1t and S(t) = 18.2e4.3t. Find the slope and intercept of ln(M(t)) as a function of in(S(t)).Find the equations of the lines after transforming the variables to create semi log or double-log plots.
Suppose L(t) = 0.72e-2.34t and K(t) = 4.23e0.91t. Find the slope and intercept of ln(L(t)) as a function of ln(K(t)).Find the equations of the lines after transforming the variables to create semilog or double-log plots.
A population follows the discrete-time dynamical system bt+1 = rbt with r = 1.5 and b0 = 1.0 x 106. When will the population reach 1.0 x 107?
A population follows the discrete-time dynamical system bt+l = rbt with r = 0.7 and b0 = 5.0 x 105. When will the population reach 1.0 x 105?
Cell volume follows the discrete-time dynamical system vt+1 = 1.5vt with initial volume of 1350 μm3 (as in Exercise 37). When will the volume reach 3250 μm3?
Gnat number follows the discrete-time dynamical system nt+1 = 0.5nt with an initial population of 5.5 x 104. When will the population reach 1.5 x 103?
About how many half-lives will occur in 50,000 years? Roughly what fraction will be left? How does this compare with the answer of Exercise 45? The amount of carbon-14 (14c) left t years after the death of an organism is given by Q(t) = Q0e-0.000122t where Q0 is the amount left at the time of
Use the laws of exponents to rewrite the following (if possible). If no law of exponents applies, say so. 43.27.2/43.26.2
The growing organism in Exercise 41 for 0 ≤ t ≤ 10. Mark where the organism has doubled in size and when it has quadrupled in size. Plot semi log graphs of the values.
The carbon-14 in Exercise 45 for 0 ≤ t ≤ 20,000. Mark where the amount of carbon has gone down by half. Plot semi log graphs of the values.
The population in Exercise 49 for 0 ≤ t ≤ 100. Mark where the population has doubled. Plot semi log graphs of the values.
Use the laws of exponents to rewrite the following (if possible). If no law of exponents applies, say so. 43.20.23 ∙ 43.20.77
The population in Exercise 53 for 0 ≤ t ≤ 100. Mark where the population has gone down by half. Plot semi log graphs of the values.
The antler size A(t) in centimeters of an elk increases with age t in years according to A(t) = 53.2e0.17t and its shoulder height L(t) increases according to L(t) = 88.5e0.1t. The following pairs of measurements can be described by ordinary, semi log, and double-log graphs. a. Graph each
Suppose a population of viruses in an infected person grows according to V(t) = 2.0e2.0t and that the immune response (described by the number of antibodies) increases according to I(t) = 0.01e3.0t during the first week of an infection. When will the number of antibodies equal the number of
The growth of a fly in an egg can be described all metrically. During growth, two imaginal disks (the first later becomes the wing and the second becomes the haltered) expand according to S1(t) = 0.007e0.1t and S2(t) =0.007e0.4t where size is measured in mm3 and time is measured in days.
While the imaginal disks are growing (Exercise 63), the yolk of the egg is shrinking according to Y(t) = 4.0e-1.2t. Create graphs comparing S1(t) and Y(t). The following pairs of measurements can be described by ordinary, semi log, and double-log graphs. a. Graph each measurement as a function of
For a cube with side length w. For each of the given shapes, find the constant c in the power relationship S = cV2/3 between the surface area S and volume V. By how much is does c exceed the value (36π)1/3 = 4.836 for the sphere (which is in fact the minimum for any shape).
For a cylinder with radius r and height 3r. For each of the given shapes, find the constant c in the power relationship S = cV2/3 between the surface area S and volume V. By how much is does c exceed the value (36π)1/3 = 4.836 for the sphere (which is in fact the minimum for any shape).
The -3/2 law of self-thinning in plants argues that the mean weight W of surviving trees in a stand increases while their number N decreases, related by W = cN-3/2. Suppose 104 trees start out with mass of 0.001 kg. Graph the relationship, and find how heavy the trees would be when only 100 remain
Suppose that the population density D of a species of mammal is a decreasing function of its body mass M according to the relationship D = cM-3/4. Suppose that an unlikely 1 g mammal would have a density of 104 per hectare. What is the predicted density of species with mass of 1000 g? A species
Use the laws of exponents to rewrite the following (if possible). If no law of exponents applies, say so. (43.2-1/8)16
Use the table or a calculator to find the values of sine and cosine for the following input (in radians), and plot them on a. A graph of sin(θ), b. A graph of cos(θ), c. As the coordinates of a point on the circle. θ = π/2.
The other trigonometric functions (tangent, cotangent, secant, and cosecant) are defined in terms of sin and cos byCalculate the value of each of these functions at the following angles (all in radians). Plot the points on a graph of each function. π/2
The other trigonometric functions (tangent, cotangent, secant, and cosecant) are defined in terms of sin and cos byCalculate the value of each of these functions at the following angles (all in radians). Plot the points on a graph of each function. 3Ï€/4
The other trigonometric functions (tangent, cotangent, secant, and cosecant) are defined in terms of sin and cos byCalculate the value of each of these functions at the following angles (all in radians). Plot the points on a graph of each function. π/9
The other trigonometric functions (tangent, cotangent, secant, and cosecant) are defined in terms of sin and cos byCalculate the value of each of these functions at the following angles (all in radians). Plot the points on a graph of each function. 5.0.
The other trigonometric functions (tangent, cotangent, secant, and cosecant) are defined in terms of sin and cos byCalculate the value of each of these functions at the following angles (all in radians). Plot the points on a graph of each function.-2.0
Use the table or a calculator to find the values of sine and cosine for the following input (in radians), and plot them on a. A graph of sin(θ), b. A graph of cos(θ), c. As the coordinates of a point on the circle. θ = 3π/A.
The other trigonometric functions (tangent, cotangent, secant, and cosecant) are defined in terms of sin and cos byCalculate the value of each of these functions at the following angles (all in radians). Plot the points on a graph of each function.3.2
The following are some of the most important trigonometric identities. Check them at a. θ = 0, b. θ = π/4, c. θ = π/2, d. θ = π. cos (θ/2) = √1 + cos(θ)/2. Only check at points a, c, and d.
The following are some of the most important trigonometric identities. Check them at a. θ = 0, b. θ = π/4, c. θ = π/2, d. θ = π. sin2(θ) + cos2(θ) = l.
The following are some of the most important trigonometric identities. Check them at a. θ = 0, b. θ = π/4, c. θ = π/2, d. θ = π. cos(θ - π) = -cos(θ).
The following are some of the most important trigonometric identities. Check them at a. θ = 0, b. θ = π/4, c. θ = π/2, d. θ = π. cos (θ - π/2) = sin(θ).
The following are some of the most important trigonometric identities. Check them at a. θ = 0, b. θ = π/4, c. θ = π/2, d. θ = π. cos(2θ) = cos2(θ) - sin2(θ).
The following are some of the most important trigonometric identities. Check them at a. θ = 0, b. θ = π/4, c. θ = π/2, d. θ = π. sin(2θ) = 2sin(θ) cos(θ).
Convert the following sinusoidal oscillations to the standard form and sketch a graph. r(t) = 5.0[2.0 + 1.0cos(2πt)].
Convert the following sinusoidal oscillations to the standard form and sketch a graph. g(t) = 2.0 + 1.0sin(t).
Convert the following sinusoidal oscillations to the standard form and sketch a graph. f(t) = 2.0 - 1.0cos(t).
Convert the following sinusoidal oscillations to the standard form and sketch a graph. h(t) = 2.0 + 1.0cos (2πt - 3.0).
Graph the following function. Give the average, maximum, minimum, amplitude, period, and phase of each and mark them on your graph. f(x) = 3.0 + 4.0cos (2π x - 1.0/5.0).
Graph the following function. Give the average, maximum, minimum, amplitude, period, and phase of each and mark them on your graph. g(t) = 4.0 + 3.0cos [2π (t - 5.0)].
Graph the following function. Give the average, maximum, minimum, amplitude, period, and phase of each and mark them on your graph. h(z) = 1.0 + 5.0cos (2π z - 3.0/4.0).
Graph the following function. Give the average, maximum, minimum, amplitude, period, and phase of each and mark them on your graph. W(y) = -2.0 + 3.0cos (2π y + 0.1/0.2).
Oscillations are often combined with growth or decay. Plot graphs of the following function, and describe in words what you see. Make up a biological process that might have produced the result. f(t) = l + t + cos(2πt) for 0 < t < 4.
Oscillations are often combined with growth or decay. Plot graphs of the following function, and describe in words what you see. Make up a biological process that might have produced the result. h(t) = t + 0.2sin(2πt) for 0 < t < 4.
Oscillations are often combined with growth or decay. Plot graphs of the following function, and describe in words what you see. Make up a biological process that might have produced the result. g(t) = et cos(2πt) for 0 < t < 3.
Oscillations are often combined with growth or decay. Plot graphs of the following function, and describe in words what you see. Make up a biological process that might have produced the result. W(t) = e-t cos(2πt) for 0 < t < 3.
Oscillations are often combined with growth or decay. Plot graphs of the following function, and describe in words what you see. Make up a biological process that might have produced the result. H(t) = cos(et) for 0 < t < 3.
Oscillations are often combined with growth or decay. Plot graphs of the following function, and describe in words what you see. Make up a biological process that might have produced the result. b(t) = cos(e-t) for 0 < t < 3.
Find the formula and sketch the graph of sleepiness over the course of a day due to the circadian rhythm. Sleepiness has two cycles, a circadian rhythm with a period of approximately 24 hours and an ultradian rhythm with a period of approximately 4 hours. Both have phase 0 (starting at midnight)
Find the formula and sketch the graph of sleepiness over the course of a day due to the ultradian rhythm. Sleepiness has two cycles, a circadian rhythm with a period of approximately 24 hours and an ultradian rhythm with a period of approximately 4 hours. Both have phase 0 (starting at midnight)
Sketch the graph of the two cycles combined. Sleepiness has two cycles, a circadian rhythm with a period of approximately 24 hours and an ultradian rhythm with a period of approximately 4 hours. Both have phase 0 (starting at midnight) and average 0, but the amplitude of the circadian rhythm is 1.0
At what time of day are you sleepiest? At what time of day are you least sleepy? Sleepiness has two cycles, a circadian rhythm with a period of approximately 24 hours and an ultradian rhythm with a period of approximately 4 hours. Both have phase 0 (starting at midnight) and average 0, but the
1.0 L of water at 30°C is mixed with 2.0 L of water at 100°C. What is the temperature of the resulting mixture? Use the idea of the weighted average to find the above.
2.0 ml of water with a salt concentration of 0.85 mol/L, is to be mixed with 5.0 ml of water with a salt concentration of 0.70 mol/L, as in Exercise 2. Before mixing, however, evaporation leads the each concentration of each component to double. What is the concentration of the mixture? Is it
In a class of 52 students, 20 scored 50 on a test, 18 scored 75, and the rest scored 100. The professor suspects cheating, however, and deducts 10 from each score. What is the average score after the deduction? Is it exactly 10 less than the average found in Exercise 3? The above are similar to
In a class of 100 students, 10 score at 20, 20 score at 40, 30 score at 60, and 40 score at 80. Because students did so poorly, the professor moves each score half way up toward 100 (so the students with 20 are moved up to 60). What is the average score in the class? Is the new average the old
V = 2.0 L, W = 0.5 L, γ = 5.0 mmol/L, c0 = 1.0 mmol/L.Suppose that the volume of the lungs is V, the amount breathed in and out is W, and the ambient concentration is γ mmol/L. For each of the given sets of parameter values and the given initial condition, find the following:a. The amount of
The situation in Exercise 13.
The situation in Exercise 14.
The situation in Exercise 15.
2.0 ml of water with a salt concentration of 0.85 mol/L, is mixed with 5.0 ml of water with a salt concentration of 0.70 mol/L. What is the concentration of the mixture? Use the idea of the weighted average to find the above.
The situation in Exercise 16.
V = 2.0 L, W = 0.5 L, γ = 5.0 mmol/L, c0 = 1.0 mmol/L (as in Exercise 13). Compute the equilibrium of the lungs discrete-time dynamical system and check that c* = γ.
V = 1.0L, W = 0.1 L, γ = 8.0 mmol/L, c0 = 4.0 mmol/L (as in Exercise 14). Compute the equilibrium of the lungs discrete-time dynamical system and check that c* = γ.
V = 1.0L, W = 0.9 L, γ = 5.0 mmol/L, c0 = 9.0 mmol/L (as in Exercise 15). Compute the equilibrium of the lungs discrete-time dynamical system and check that c* = γ.
V = 10.0 L, W = 0.2 L, γ = 1.0 mmol/L, c0 = 9.0 mmol/L (as in Exercise 16). Compute the equilibrium of the lungs discrete-time dynamical system and check that c* = γ.
Suppose q = 0.4 and α = 0.1. Why is the equilibrium concentration higher than with q = 0.2 even though the person is breathing more? Compare the equilibrium and total amount absorbed per breath for different values of q. Use an ambient concentration of γ = 0.21 and a volume of V = 6.0L.
Suppose q = 0.1 and α = 0.05. Think of this as a person gasping for breath. Why is the concentration nearly the same as in Example 1.9.9? Does this mean that gasping for breath is OK? Compare the equilibrium and total amount absorbed per breath for different values of q. Use an ambient
Oxygen concentration is reduced by 2% each breath (that is, if the concentration before absorption were 18%, it would be 16% after absorption). Find the discrete-time dynamical system and the equilibrium. Are there values of ct for which the system does not make sense? The above problem
Oxygen concentration is reduced by 1% each breath. Find the discrete-time dynamical system and the equilibrium. Are there values of ct for which the system does not make sense? The above problem investigates absorption that is not proportional to the concentration in the lungs. Assume γ = 0.21 and
The amount absorbed is 0.2(ct - 0.05) if ct ≥ 0.05. This models a case where the only oxygen available is that in excess of the concentration in the blood, which is roughly 5%. The above problem investigates absorption that is not proportional to the concentration in the lungs. Assume γ = 0.21
In a class of 52 students, 20 scored 50 on a test, 18 scored 75, and the rest scored 100. What was the average score? Use the idea of the weighted average to find the above.
The amount absorbed is 0.1 (ct - 0.05) if ct ≥ 0.05. The above problem investigates absorption that is not proportional to the concentration in the lungs. Assume γ = 0.21 and q = 0.1, and find the equilibrium concentration.
Oxygen concentration is reduced by an amount A (generalizing the case in Exercises 27 and 28). How does the amount absorbed with this value of A compare with the amount of oxygen absorbed in Example 1.9.9? Find the value of the parameter that produces an exhaled concentration of exactly 0.15.
The amount absorbed is α(ct - 0.05) (generalizing the case where only available oxygen is absorbed in Exercises 29 and 30). How does the amount absorbed with this value of a compare with the amount of oxygen absorbed in Example 1.9.9?Find the value of the parameter that produces an exhaled
Suppose S = 0.001. Write the discrete-time dynamical system and find its equilibrium. Compare the equilibrium with the ambient concentration.The above problem investigates production of carbon dioxide by the lungs. Suppose that the concentration increases by an amount S before the air is exchanged.
The actual concentration of carbon dioxide in exhaled air is about 0.04, or 100 times the ambient concentration. Find the value of S that gives this as the equilibrium.The above problem investigates production of carbon dioxide by the lungs. Suppose that the concentration increases by an amount S
A population of bacteria has per capita production r = 0.6, and 1.0 x 106 bacteria are added each generation. A bacterial population that has per capita production r < 1 but that is supplemented each generation follows a discrete-time dynamical system much like that of the lungs. Use the following
A population of bacteria has per capita production r = 0.2, and 5.0 x 106 bacteria are added each generation. A bacterial population that has per capita production r < 1 but that is supplemented each generation follows a discrete-time dynamical system much like that of the lungs. Use the following
A population of bacteria has per capita production r = 0.6, and 1.0 x 106 bacteria are added each generation (as in Exercise 35). Find the equilibrium population of bacteria in the above case with supplementation.
A population of bacteria has per capita production r = 0.2, and 5.0 x 106 bacteria are added each generation (as in Exercise 36). Find the equilibrium population of bacteria in the above case with supplementation.
A population of bacteria has per capita production r = 0.5, and S bacteria are added each generation. What happens to the equilibrium when S is large? Does this make biological sense? Find the equilibrium population of bacteria in the above case with supplementation.
In a class of 100 students, 10 score at 20, 20 score at 40, 30 score at 60, and 40 score at 80. What is the average score in the class? Use the idea of the weighted average to find the above.
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