determine where each function is defined and continuous. Investigate the behavior as x too. Use a...

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determine where each function is defined and continuous. Investigate the behavior as x too. Use a graph- ing calculator to sketch the corresponding graphs. 1. f(x) = e- 2. f(x)= 1csi X 0 3. f(x)= 4. f(x)=√²-1 e +ex 5. Sketch the graph of a function that is discontinuous from the left and continuous from the right at x =1. if x #0 if x=0 6. Sketch the graph of a function f(x) that is continuous on [0.2], except at x = 1, where f(1)= 3, lim, f(x) = 2, and lim,-1+ f(x) = 4. 7. Sketch the graph of a continuous function on [0, 00) with f(0)=0 and lim,-. f(x) = 1. 8. Sketch the graph of a continuous function on (-∞0,00) with f(0)= 1. f(x) ≥ 0 for all x e R, and lim,- f(x) = 0. 9. Show that the floor function f(x) = [x] is continuous from the right, but discontinuous from the left at x = -2. k+t 10. Suppose f(x) is continuous on the interval [1. 3]. Iƒ(1) = 0 and f(3) =3 explain why there must be a numberce (1, 3) such that f(c) = 1. 11. Population Size Assume that the size of a population at time tis N(t)= 120 139 where a and k are positive constants. Suppose that the limiting population size is lim N(t) = 1.24 x 10°. 118 and that, at time = 5, the population size is half the limiting population size. Use the preceding information to determine the constants a and k. 12. Population Size Suppose that N(t)=10+2e sint, 120 describes the size of a population (in millions) at timer (mea- sured in weeks). (a) Use a graphing calculator to sketch the graph of N(r), and describe in words what you see. (b) Give lower and upper bounds on the size of the population; that is, find N, and N₂ such that, for all 120, N₁ ≤ N(1) ≤ N₂ (c) Use the sandwich theorem to find lim,... N(1). 13. Physiology Suppose that an organism reacts to a stimulus only when the stimulus exceeds a certain threshold. Assume that the stimulus is a function of time and that it is given by s(t)= sin(xt). t 20 The organism reacts to the stimulus and shows a certain reaction when s(t) ≥ 1/2. Define a function g(t) such that g() = 0 when the organism shows no reaction at timer and g(t) = 1 when the organism shows the reaction. (a) Plot s(t) and g(t) in the same coordinate system. (b) Is s(t) continuous? Is g(t) continuous? 14. Quorum Sensing For some species of growth and other be- haviors change depending on how many bacteria are present in determine where each function is defined and continuous. Investigate the behavior as x too. Use a graph- ing calculator to sketch the corresponding graphs. 1. f(x) = e- 2. f(x)= 1csi X 0 3. f(x)= 4. f(x)=√²-1 e +ex 5. Sketch the graph of a function that is discontinuous from the left and continuous from the right at x =1. if x #0 if x=0 6. Sketch the graph of a function f(x) that is continuous on [0.2], except at x = 1, where f(1)= 3, lim, f(x) = 2, and lim,-1+ f(x) = 4. 7. Sketch the graph of a continuous function on [0, 00) with f(0)=0 and lim,-. f(x) = 1. 8. Sketch the graph of a continuous function on (-∞0,00) with f(0)= 1. f(x) ≥ 0 for all x e R, and lim,- f(x) = 0. 9. Show that the floor function f(x) = [x] is continuous from the right, but discontinuous from the left at x = -2. k+t 10. Suppose f(x) is continuous on the interval [1. 3]. Iƒ(1) = 0 and f(3) =3 explain why there must be a numberce (1, 3) such that f(c) = 1. 11. Population Size Assume that the size of a population at time tis N(t)= 120 139 where a and k are positive constants. Suppose that the limiting population size is lim N(t) = 1.24 x 10°. 118 and that, at time = 5, the population size is half the limiting population size. Use the preceding information to determine the constants a and k. 12. Population Size Suppose that N(t)=10+2e sint, 120 describes the size of a population (in millions) at timer (mea- sured in weeks). (a) Use a graphing calculator to sketch the graph of N(r), and describe in words what you see. (b) Give lower and upper bounds on the size of the population; that is, find N, and N₂ such that, for all 120, N₁ ≤ N(1) ≤ N₂ (c) Use the sandwich theorem to find lim,... N(1). 13. Physiology Suppose that an organism reacts to a stimulus only when the stimulus exceeds a certain threshold. Assume that the stimulus is a function of time and that it is given by s(t)= sin(xt). t 20 The organism reacts to the stimulus and shows a certain reaction when s(t) ≥ 1/2. Define a function g(t) such that g() = 0 when the organism shows no reaction at timer and g(t) = 1 when the organism shows the reaction. (a) Plot s(t) and g(t) in the same coordinate system. (b) Is s(t) continuous? Is g(t) continuous? 14. Quorum Sensing For some species of growth and other be- haviors change depending on how many bacteria are present in

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1 fxex is defined and continuous for all x in R As x goes to infinity fx approaches 0 2 fx is define... View the full answer