The Fitzhugh-Nagumo model for the electric impulse in a neuron states that, in the absence of relaxation effects, the electric potential, v(t), in a neuron obeys the differential equation dvdt = −v(v2−(1+a)v+a) where a is a constant such that 0 < a < 1. Find and classify all equilibrium solutions and provide a phase line summarizing behavior of v(t). Hint: While the quadratic formula can be used, you can proceed more quickly by factoring!
A cup of coffee has a temperature of 80∘C when it is placed outside where the temperature is 20∘C. Suppose the temperature of the coffee, T(t), changes according to Newton's Law of Cooling with proportionality constant k = 0.09. (a) Find and classify the equilibrium solution(s) using a phase line. (b) Find T(t). (c) Evaluate and interpret limt→∞ T(t). (d) How long will it take for the coffee to reach a temperature of 25∘C ? How long to reach a temperature of 20∘C ?
A corporation maintains a balance of B(t) (in millions of dollars) in an account, where t is measured in years. Interest with a nominal annual rate of 2% compounds continuously (that is, the rate at which interest is added to the account is proportional to B with proportionality constant 0.02 year −1 ). In addition, the corporation continuously withdraws money from the account at a rate of 4 million dollars per year. (a) Write a differential equation for the balance B(t). (b) Find all solutions to your differential equation. (c) Suppose that B(0) = 300. Determine limt→∞ B(t) and interpret your result. (d) Suppose instead that B(0) = 100. When will the account be empty?
Find a formula for the nth partial sum of the series 3 + 34 + 316 + 364+⋯+34 n−1+⋯ and use it to find the series' sum if the series converges. The formula for the nth partial sum, sn, of the series is If the series converges, what is the series' sum? Select the correct choice below and fill in any answer boxes within your choice. A. The series converges. The series' sum is . (Type an integer or a simplified fraction. ) B. The series diverges.
Determine if the geometric series converges or diverges. If a series converges, find its sum. (110) + (110)2 + (110)3 + (110)4 + (110)5+… Select the correct choice below, and fill in any answer boxes in your choice. A. (110) + (110)2 + (110)3 + (110)4 + (110)5 +… = (Simplify your answer. ) B. The series diverges.
Determine whether the series ∑n = 0∞ cos 7nπ converges or diverges. If it converges, find its sum. Select the correct choice below and, if necessary, fill in the answer box within your choice. A. The series diverges because it is a geometric series with |r| ≥ 1. The series converges because limn→∞ cos 7nπ = 0. The sum of the series is . B. (Type an exact answer, using radicals as needed. ) C. The series converges because limk→∞∑n = 0 kcos 7nπ fails to exist. The series converges because it is a geometric series with |r| < 1. The sum of the series is . D. (Type an exact answer, using radicals as needed. ) The series diverges because limn→∞cos 7nπ ≠ 0 or fails to exist.
Find the values of x for which the given geometric series converges. Also, find the sum of the series (as a function of x) for those values of x. ∑n = 0∞(−12)n(x−3)n Find the values of x for which the given geometric series converges. (Type an inequality or a compound inequality. Use integers or fractions for any numbers in the inequality. ) Find the sum of the series. ∑n = 0∞(−12)n(x−3)n =
Use the Integral Test to determine the convergence or divergence of the following series, or state that the test does not apply. ∑k = 4∞1 e2k Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The series converges. The value of the integral ∫4∞dxe2 x is. (Type an exact answer. ) B. The series diverges. The value of the integral ∫4∞dxe2 x is . (Type an exact answer. ) C. The Integral Test does not apply to this series.
Does the series ∑n = 1∞514 n−1 converge or diverge? Give a reason for your answer. Choose the correct answer below. A. The series converges because it is a p− series with p = 1. B. The series diverges because limn→∞514 n−1 ≠ 0. C. The series converges because the integral ∫N∞514 x−1 dx converges. D. The series diverges because the integral ∫N∞514 x−1 dx diverges.
Does the series ∑n = 1∞1(ln1.7)n converge or diverge? Choose the correct answer below. A. The series diverges because it is a geometric series with |r| ≥ 1. B. The nth-term test shows that the series converges. C. The series converges because it is a geometric series with |r| < 1. D. The integral test shows that the series converges.
Use the Comparison Test to determine if the following series converges or diverges. ∑n = 1∞17 n2+23 Choose the correct answer below. A. The series is divergent because 1 n2 < 17 n2+23 for all n and ∑n = 1∞1 n2 is divergent. B. The series is convergent because 17 n2+23 < 1 n2 for all n and ∑n = 1∞1 n2 is convergent. C. The series is divergent because 1 n < 17 n2+23 for all n and the harmonic series is divergent. D. The series is convergent because 17 n2+23 < 1 n3 for all n and ∑n = 1∞1 n3 is convergent.
Use any method to determine if the series converges or diverges. Give reasons for your answer. ∑n = 1∞(−3)n2n Select the correct choice below and fill in the answer box to complete your choice. A. The series converges because it is a geometric series with r = . B. The series diverges because the limit used in the Ratio Test is C. The series diverges because it is a p-series with p = . D. The series converges per the Integral Test because ∫1∞12 xdx =
Does the series ∑n = 1∞(−1)nn4 n6+4 converge absolutely, converge conditionally, or diverge? A. The series converges absolutely because the limit used in the nth-Term Test is B. The series diverges because the limit used in the Ratio Test is not less than or equal to 1. C. The series diverges because the limit used in the nth-Term Test is not zero. D. The series converges absolutely per the Comparison Test with ∑n = 1∞1 n2. E. The series converges conditionally per the Alternating Series Test and because the limit used in the Root Test is F. The series converges conditionally per the Alternating Series Test and the Comparison Test with ∑n = 1∞1 n2.
Consider the series ∑n = 2∞xnn(lnn)2. (a) Find the series' radius and interval of convergence. (b) For what values of x does the series converge absolutely? (c) For what values of x does the series converge conditionally? (a) Find the interval of convergence. x Find the radius of convergence. R = (b) For what values of x does the series converge absolutely? x
Use the Ratio Test to determine if the following series converges absolutely or diverges. ∑n = 1∞7 nn! Since the limit resulting from the Ratio Test is
Does the series ∑n = 1∞(−1)n+1(0.1)n converge absolutely, converge conditionally, or diverge? Choose the correct answer below and, if necessary, fill in the answer box to complete your choice. A. The series converges conditionally since the corresponding series of absolute values diverges, but the series passes the Alternating Series Test. B. The series converges conditionally since the corresponding series of absolute values is a geometric series with r = . C. The series diverges per the nth-Term Test. D. The series converges absolutely since the corresponding series of absolute values is a geometric series with r = E. The series converges absolutely since the corresponding series of absolute values is a p-series with p =
Does the sequence {an} converge or diverge? Find the limit if the sequence is convergent. an = ln(n+4)n3 Select the correct choice below and, if necessary, fill in the answer box to complete the choice. A. The sequence converges to limn→∞an = . (Simplify your answer. ) B. The sequence diverges.
Use an appropriate test to determine whether the series given below converges or diverges. ∑n = 1∞13 n−1+6 Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. A. The Comparison Test with ∑(13)n−1 shows that the series converges. B. The series diverges because the limit used in the nth-Term Test is . (Type an exact answer. ) C. The Comparison Test with ∑(13)n−1 shows that the series diverges.
Does the series ∑n = 1∞(−1)nn+n+2 converge absolutely, converge conditionally, or diverge? Choose the correct answer below and, if necessary, fill in the answer box to complete your choice. A. The series converges conditionally per the Alternating Series Test and because the limit used in the Root Test is not less than or equal to 1. B. The series diverges because the limit used in the nth-Term Test does not exist. C. The series converges conditionally per the Alternating Series Test and the Integral Test because ∫1∞f(x)dx does not exist. D. The series converges absolutely because the limit used in the Root Test is E. The series diverges per the Comparison Test with ∑1∞1 n. F. The series converges absolutely because the limit used in the Ratio Test is
Write out the first few terms of the series ∑n = 0∞(13 n+(−1)n5 n). What is the series' sum? The first term is . (Type an integer or a simplified fraction. ) The second term is . (Type an integer or a simplified fraction. ) The third term is . (Type an integer or a simplified fraction. ) The fourth term is. (Type an integer or a simplified fraction. ) If the series is convergent, what is the series' sum? Select the correct choice below and fill in any answer boxes within your choice. A. The series converges. The series' sum is . (Type an integer or a simplified fraction. ) B. The series diverges.
Find the radius of convergence of the following series. ∑n = 1∞(n!)25 n+6(2 n)!xn+2 The radius of convergence is (Simplify your answer. )
Determine whether the alternating series ∑n = 1∞(−1)n+13 nn3 converges or diverges. Choose the correct answer below and, if necessary, fill in the answer box to complete your choice. A. The series does not satisfy the conditions of the Alternating Series Test but diverges because it is a p-series with r = B. The series does not satisfy the conditions of the Alternating Series Test but diverges because the limit used in the Ratio Test is C. The series does not satisfy the conditions of the Alternating Series Test but converges because it is a geometric series with r = D. The series does not satisfy the conditions of the Alternating Series Test but converges because the limit used in the Root Test is E. The series converges by the Alternating Series Test.
Use substitution to find the Taylor series at x = 0 of the function e−2x. What is the general expression for the nth term in the Taylor series at x = 0 for e−2x ? ∑n = 0∞ (Type an exact answer. )
Find the Maclaurin series for the function. coshx = ex+e−x2 The Maclaurin series for the function is ∑n = 0 ( )
Use the Root Test to determine if the following series converges absolutely or diverges. ∑n = 1∞(−1)n(1+10 n)n2 (Hint: limn→∞(1+x/n)n = ex) Since the limit resulting from the Root Test is , (Type an exact answer. )
Use the limit comparison test to determine if ∑n = 1∞6 n45+3 n+5 n14−4 converges or diverges, and justify your answer. Answer Attempt 1 out of 3 Apply the comparison test with the series ∑n = 1∞1 np where p = . If an = 6 n45+3 n+5 n14−4 and bn = 1 np, then limn→∞an bn = . Since an, bn > 0 and the limit is a finite and positive (non-zero) number, the limit comparison test applies. ∑n = 1∞1 np since a p-series will if and only if . Therefore, ∑n = 1∞6 n45+3 n+5 n14−4
Test each of the following series for convergence by either the Comparison Test or the Limit Comparison Test. If at least one test can be applied to the series, enter CONV if it converges or DIV if it diverges. If neither test can be applied to the series, enter NA. (Note: this means that even if you know a given series converges by some other test, but the comparison tests cannot be applied to it, then you must enter NA rather than CONV. ) ∑n = 1∞7 n9−n7+3 n5 n11−n6+6 2. ∑n = 1∞cos2(n)nn6 3. ∑n = 1∞3 n6 n9+9 4. ∑n = 1∞(ln(n))6 n+6 5. ∑n = 1∞cos(n)n3 n+9
You are given the shape of the vertical ends of a trough that is completely filled with water. Find the force exerted by the water on one end of the trough. (The weight density of water is 62.4 lb/ft3. ) 666 lb 1,331 lb 2,662 lb 333 lb
A vertical plate is submerged in water (the surface of the water coincides with the x-axis). Find the force exerted by the water on the plate. (The weight density of water is 62.4 lb/ft3. ) 2,246 lb 1,123 lb 374 lb 749 lb
A cylindrical drum of diameter 2 ft and length 3 ft is lying on its side, submerged in water 3 f deep. Find the force exerted by the water on one end of the drum to the nearest pound. (The weight density of water is 62.4 lb/ft3. )
A swimming pool is 10 ft wide and 36 ft long and its bottom is an inclined plane, the shallow end having a depth of 3 ft and the deep end, 12 ft. If the pool is full of water, find the hydrostatic force on the shallow end. (Use the fact that water weighs 62.5 lb/ft3. )
A trough is filled with a liquid of density 855 kg/m3. The ends of the trough are equilateral triangles with sides 7 m long and vertex at the bottom. Find the hydrostatic force on one end of the trough. 2.5×105 N 6.5×105 N 4.5×105 N 3.5×105 N 5.5×105 N
You are given the shape of the vertical ends of a trough that is completely filled with water. Find the force exerted by the water on one end of the trough. (The weight density of water is 62.4 lb/ft3. )
You are given the shape of the vertical ends of a trough that is completely filled with water. Find the foree exerted by the water on one end of the trough. (The weight density of water is 62.4 lb/ft3. )
An aquarium is 3 ft long, 1 ft wide, and 1 ft deep. If the aquarium is filled with water, find the force exerted by the water (a) on the bottom of the aquarium, (b) on the longer side of the aquarium, and (c) on the shorter side of the aquarium. (The weight density of water is 62.4 lb/t3, )
A large tank is designed with ends in the shape of the region between the curves y = x^2/2 and y = 15, measured in feet. Find the hydrostatic force on one end of the tank if it is filled to a depth of 9 ft with gasoline. (Assume that the density of the gasoline is 42.0 lb/ft3. ) 5,785 lb 4,850 lb 6,918 lb 2,683 lb 7,698 lb 3,855 lb
A trough has vertical ends that are equilateral triangles with sides of length 2 ft. If the trough is filled with water to a depth of 1 ft, find the force exerted by the water on one end of the trough. Round to one decimal place. (The weight density of water is 62.4 lb2 /ft3. ) 6.0 Ib 31.2 lb 12.0 lb 62.4 lb
A manufacturer of light bulbs wants to produce bulbs that last about 500 hours but, of course, some bulbs burn out faster than others. Let F(t) be the fraction of the company's bulbs that burn out before t hours. F(t) lies between 0 and 1. Let r(t) = F′(t). What is the value of ∫0∞r(t)dt ? divergent ∫0∞r(t)dt = 1 ∫0∞r(t)dt = 2 ∫0∞r(t)dt = 0 ∫0∞r(t)dt = 500
Determine whether the integral converges or diverges. If it converges, find its value. ∫1∞ dx xlnx
Determine whether the improper integral converges or diverges, and if it converges, find its value: ∫−∞∞5 exb5+e2x dx Diverges 5 π55 π52
The region ((xy)∣x ≥ -10, 0 ≤ y ≤ e−x/8} is represented below. Find the area of this region to two decimal places. 15.89 27.92 17.89 16.08 15.87
Use the Comparison Theorem to determine whether the integral is convergent or divergent. ∫0π2 sin2(x)xdx divergent, because 2 sin2(x)x > 2 x and ∫0π2 xdx is divergent. convergent, because 2 sin2(x)x < 2 x and ∫0π2 xdx is convergent. divergent, because 2 sin2(x)x < 2 x and ∫0π2 xdx is divergent. convergent, because 2 sin2(x)x > 2 x and ∫0π2 xdx is convergent.