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calculus

Questions and Answers of Calculus

If E ˆˆ M [a, b], we define the (Lebesgue) measure of E to be the number m(E) := ˆ«ba 1E. In this exercise, we develop a number of properties of the measure function m : M [a, b]
Answer the questions posed in Exercise 1 for the following sequences (when properly defined): (a) kx/1 + k√x [0, 1]. (b) 1/√x(1 + xk) [0, 1]. (c) 1/√x(1 + xk) [1, 2]. (d) 1/√x(2 - xk) [0, 1].
Discuss the following sequences of functions and their integrals on [0, 1]. Evaluate the limit of the integrals, when possible. (a) e-kx. (b) e-kx/x. (c) kxe-kx. (d) k2xe-kx. (e) kxe-k2x2. (f)
(a) Show that(b) Show that
Let (fk) be a sequence on [a, b] such that each fk is differentiable on [a, b] and f1/k(x) → g(x) with |f1/k(x)| < K for all x ∈ [a, b]. Show that the sequence (fk(x)) either converges for all x
Show that a set G ⊂ R is open if and only if it does not contain any of its boundary points.
Show that a set F ⊂ R is closed if and only if it contains all of its boundary points.
If A ⊂ R, let Ao be the union of all open sets that are contained in A; the set Ao is called the interior of A. Show that Ao is an open set, that it is the largest open set contained in A, and that
Using the notation of the preceding exercise, let A, B be sets in R. Show that Ao ⊂ (Ao)o = Ao and that (A ∩ B)o = Ao ∩ Bo. Show also that Ao ⋃ Bo ⊂ (A ⋃ B)o, and give an example to show
If A ⊂ R, let A- be the intersection of all closed sets containing A, the set A- is called the closure of A. Show that A- is a closed set, that it is the smallest closed set containing A, and that
Using the notation of the preceding exercise, let A, B be sets in R. Show that we have A ⊂ A- (A-)- = A-, and that (A ∪ B)- = A- ∪ B-. Show that (A ∩ B)- ⊂ A- ⊂ B-, and give an example to
If in the notation used in the proof of Theorem 11.1.9, we have Ix ∩ Iy ≠ θ show that bx = by.
Show that each point of the Cantor set F is a cluster point of F.
Show that each point of the Cantor set F is a cluster point of C(F).
Write out the Induction argument in the proof of part (b) of the Open Set Properties 11.1.4.
Prove that (0, 1) = ∩∞n=1 (0, 1 + 1/n), as asserted in Example 11.1.6(a).
Let K ≠ θ, be a compact set in R. Show that inf K and sup K exist and belong to K.
Let K ≠ θ be compact in R and let c ∈ R. Prove that there exists a point b in K such that |c - b| = sup{|c - x| : x ∈ K}.
Use the notion of compactness to give an alternative proof of Exercise 5.3.18.
If K1 and K2 are disjoint nonempty compact sets, show that there exist ki ∈ Ki such that 0 < |k1 - k2| = inf{x1 - x2| : xi ∈ Ki}.
Use the Heine-Borel Theorem to prove the following version of the Bolzano-Weierstrass Theorem: Every bounded infinite subset of R has a cluster point in R. (If a set has no cluster points, then it is
Prove that the intersection of an arbitrary collection of compact sets in R is compact.
Let (Kn : n ˆˆ N) be a sequence of nonempty compact sets in R such that K1 Šƒ K2 Šƒ ˆ™ ˆ™ ˆ™ Kn Šƒ ˆ™ ˆ™
Let f : R → R be defined by f(x) = x2 for x ∈ R. (a) Show that the inverse image f-1(I) of an open interval I := (a, b) is either an open interval, the union of two open intervals, or empty,
Let I := [a, b] and let f : I → R and g : I → R be continuous functions on I. Show that the set {x ∈ I : f(x) = g(x)} is closed in R.
Show that if f : R → R is continuous, then the set {x ∈ R : f(x) = k} is closed in R for each k ∈ R.
Prove that f : R → R is continuous if and only if for each closed set F in R, the inverse image f-1(F) is closed.
Show that the functions d1 and d∞ defined in 11.4.2(c) are metrics on R2.
Prove Theorem 11.4.11.
Prove Theorem 11.4.12.
Show that the functions d1 and d∞ defined in 11.4.2(d, e) are metrics on C[0, 1].
Verify that the discrete metric on a set S as defined in 11.4.2(f) is a metric.
If Pn := (xn, yn) ∈ R2 and d∞ is the metric in 11.4.2(c), show that (Pn) converges to P := (x, y) with respect to this metric if and only if (xn) and (yn) converge to x and y, respectively.
Verify the conclusion of Exercise 4 if d∞ is replaced by d1.
Let S be a nonempty set and let d be the discrete metric defined in 11.4.2(f). Show that in the metric space (S, d), a sequence (xn) in S converges to x if and only if there is a K ∈ N such that xn
Let P := (x, y) and O := (0, 0) in R2. Draw the following sets in the plane: (a) {P ∈ R2 : d1 (O, P) < 1}, (b) {P ∈ R2 : d∞ (O, P) < 1
Prove that in any metric space, an ε-neighborhood of a point is an open set.
Using the discrete-time dynamical system and the derivation of Equation 1.10.7, find pt, mt+1, bt+1 and pt+1 in the following situations.s = 1.8, r = 1.8, mt = 1.2 × 105, bt = 3.5 × 106.
Solve for the equilibria of the following discrete-time dynamical systems
Solve for the equilibria of the following discrete-time dynamical systems
xt+1 = xt / 1 + axt where a is a positive parameter. What happens to this system if a = 0?Find all non-negative equilibria of the above mathematically elegant discrete-time dynamical systems.
xt+1 = xt / a + xt where a is a positive parameter. What happens to this system if a = 0?Find all non-negative equilibria of the above mathematically elegant discrete-time dynamical systems.
Identify stable and unstable equilibria on the following graphs of updating functions.
Identify stable and unstable equilibria on the following graphs of updating functions.
Identify stable and unstable equilibria on the following graphs of updating functions.
Identify stable and unstable equilibria on the following graphs of updating functions.
Find and graph the updating functions for the following cases of the selection model (Equation 1.10.7). Cobweb starting from p0 = 0.1 and p0 = 0.9. Which equilibria are stable?s = 1.2, r = 2.0.
Find and graph the updating functions for the following cases of the selection model (Equation 1.10.7). Cobweb starting from p0 = 0.1 and p0 = 0.9. Which equilibria are stable?s = 1.8, r = 0.8.
Find and graph the updating functions for the following cases of the selection model (Equation 1.10.7). Cobweb starting from p0 = 0.1 and p0 = 0.9. Which equilibria are stable?s = 0.3, r = 0.5.
Find and graph the updating functions for the following cases of the selection model (Equation 1.10.7). Cobweb starting from p0 = 0.1 and p0 = 0.9. Which equilibria are stable?s = 1.8, r = 1.8.
For each of the following discrete-time dynamical systems, indicate which of the equilibria are stable and which are unstable.
For each of the following discrete-time dynamical systems, indicate which of the equilibria are stable and which are unstable.
Begin with 1.0 x 106 wild type and 1.0 x 105 mutants.This section has ignored the important evolutionary force of mutation. This series of problems builds models that consider mutation without
Begin with 1.0 x 105 wild type and 1.0 x 106 mutants.This section has ignored the important evolutionary force of mutation. This series of problems builds models that consider mutation without
Begin with bt wild type and m, mutants. Find the discrete-time dynamical system for the fraction pt of mutants. Find the equilibrium fraction of mutants. Cobweb starting from the initial condition in
Begin with bt wild type and mt mutants, but suppose that a fraction 0.1 mutate and a fraction 0.2 revert. Find the discrete-time dynamical system and the equilibrium fraction of mutants.This section
Begin with 1.0 x 106 wild type and 1.0 x 105 mutants.This series of problems combines mutation with selection. In one simple scenario, mutations occur in only one direction (wild type turn into
The population of red birds is multiplied by a factor of r, and the population of blue birds remains the same. A population consists of 200 red birds and 800 blue birds. Find the fraction of red
Begin with 1.0 x 105 wild type and 1.0 x 106 mutants.This series of problems combines mutation with selection. In one simple scenario, mutations occur in only one direction (wild type turn into
Begin with bt wild type and mt mutants. Find the discrete-time dynamical system for the fraction pt of mutants. Find the equilibrium fraction of mutants. Cobweb starting from the initial condition in
Begin with bt wild type and mt mutants, but suppose that a fraction 0.2 mutate and that the per capita production of mutants is 1.0. Find the discrete-time dynamical system and the equilibrium
Suppose there are 100 butterflies on each island at time t = 0. How many are on each island at t = 1? At t = 2?The model of selection studied in this section is similar to a model of migration.
Suppose there are 200 butterflies on the first island and none on the second at time t = 0. How many are on each island at t = 1? At t = 2?The model of selection studied in this section is similar to
Find equations for xt+1 and yt+1 in terms of xt and yt.The model of selection studied in this section is similar to a model of migration. Suppose two nearby islands have populations of butterflies,
Divide both sides of the discrete-time dynamical system for xt by xt+1 + yt+1 to find a discrete-time dynamical system for the fraction pt on the first island. What is the equilibrium fraction?The
Each butterfly that begins the year on the first island produces one offspring after migration (whether they find themselves on the first or the second island). Those that begin the year on the
Now suppose that the butterflies that do not migrate reproduce (making one additional butterfly each) and those that do migrate fail to reproduce from exhaustion. No butterflies die.The above problem
Suppose that fraction used = 0.5 / 1.0 + 0.1Mt.Write the discrete-time dynamical system and solve for the equilibrium and compare with M* = 2.0 for the basic model.The model describing the dynamics
The population of blue birds is multiplied by a factor of s, and the population of red birds remains the same A population consists of 200 red birds and 800 blue birds. Find the fraction of red birds
Suppose that fraction used = 0.5 / 1.0 + 0.4Mt. Write the discrete-time dynamical system, solve for the equilibrium, and compare with M* = 2.0 for the basic model. The model describing the dynamics
Suppose that fraction used = β / 1.0 + 0.1Mt. for some parameter β ≤ 1. Write the discrete-time dynamical system and solve for the equilibrium. Sketch a graph of the equilibrium as a function of
Suppose that fraction used = 0.5 / 1.0 + αMt for some parameter α. Write the discrete-time dynamical system and solve for the equilibrium. Sketch a graph of the equilibrium as a function of α.
One widely used nonlinear model of competition is the "logistic" model, where per capita production is a linearly decreasing function of population size. Suppose that the per capita production isOur
In an alternative model, the per capita production decreases as the reciprocal of a linear function. Suppose that the per capita production isOur models of bacterial population growth neglect the
In another alternative model, called the Ricker model, the per capita production decreases exponentially. Suppose that per capita production isOur models of bacterial population growth neglect the
In a model with an Allee effect, organisms reproduce poorly when the population is small. In one case, per capita production followsOur models of bacterial population growth neglect the fact that
Sketch graphs of the above function. f(x) = x / x + 1 for 0 ≤ x ≤ 2 For Information: (the updating function in Section 1.5, Exercise 23).
Sketch graphs of the above function. g(x) = 3x / 2x + 1 for 0 ≤ x ≤ 2.
Using the discrete-time dynamical system and the derivation of Equation 1.10.7, find pt, mt+1, bt+1 and pt+1 in the following situations. s = 1.2, r = 2.0, mt = 1.2 × 105, bt = 3.5 × 106.
Using the discrete-time dynamical system and the derivation of Equation 1.10.7, find pt, mt+1, bt+1 and pt+1 in the following situations.s = 1.2, r = 2.0, mt = 1.2 × 105, bt = 1.5 × 106.
Using the discrete-time dynamical system and the derivation of Equation 1.10.7, find pt, mt+1, bt+1 and pt+1 in the following situations.s = 0.3, r = 0.5, mt = 1.2 × 105, bt = 3.5 × 106.
Suppose n = 2. Show that Vt = 1 is an equilibrium. Sketch a graph and cobweb with c = 1/4. Does the equilibrium seem to be stable?Consider the following continuous system that approximates the
Suppose n = 4. Show that Vt = 1 is an equilibrium. Sketch a graph and cobweb with c = 1 /4. Does the equilibrium seem to be stable?Consider the following continuous system that approximates the
Suppose h = 1000 and Nc = 1000 and r = 1.5. Investigate some solutions starting with different values of N0 < 1000. What is happening? Population models with thresholds can also have unusual
Find the equilibrium when h = 1000 and Nc = 1000 and r = 1.5. What would happen to solutions starting with values greater than the equilibrium? Use this information, and that in the previous problem,
Redo Exercise 15 with r = 1.65. How do the results differ from those in Exercise 15?Data from exercise 15 Population models with thresholds can also have unusual behavior. Evaluate the following
Find the equilibrium when h = 1000 and Nc = 1000 and r = 1.65. Can you explain why solutions that start below the equilibrium can shoot off to infinity?Population models with thresholds can also have
The case in Exercise 1. Describe the long-term dynamics in each of the given cases. Find which ones will beat every time, which display 2:1 AV block, and which show some sort of Wenckebach
The case in Exercise 2. Describe the long-term dynamics in each of the given cases. Find which ones will beat every time, which display 2:1 AV block, and which show some sort of Wenckebach
The case in Exercise 3. Describe the long-term dynamics in each of the given cases. Find which ones will beat every time, which display 2:1 AV block, and which show some sort of Wenckebach
The case in Exercise 4. Describe the long-term dynamics in each of the given cases. Find which ones will beat every time, which display 2:1 AV block, and which show some sort of Wenckebach
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%. Identify the variables and parameters in the above
(0, 45), (1, 25), (2, 12), (3, 12.5), (4, 10). Graph the given points and say which point does not seem to fall on the graph of a simple function.
f(x) = x + 5 at x = a, x = a + 1, and x = 4a. Evaluate the above functions at the given algebraic arguments.
g(y) = 5y at y = x2, y = 2x + 1, and y = 2 - x. Evaluate the above functions at the given algebraic arguments.
h(z) = 1/5z at z = c/5, z = 5/c, and z = c + l. Evaluate the above functions at the given algebraic arguments.
F(r) = r2 + 5 at r = x + l, r = 3x, and r = 1/x. Evaluate the above functions at the given algebraic arguments.
A function whose argument is the name of a state and whose value is the highest altitude in that state. State Highest Altitude (ft) California .......................... 14,491 Idaho
A function whose argument is the name of a bird and whose value is the length of that bird. Bird Length Cooper's hawk ............... 50 cm Goshawk ..................... 66 cm Sharp-shinned hawk
f(x) = 2x + 3 and g(x) = 3x - 5. For each of the above sums of functions, graph each component piece. Compute the values at x = -2, x = -1, x = 0, x = 1, and x = 2 and plot the sum.

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