Solutions. 1. Use the table of integral formulas in Appendix B in the textbook to help compute the integrals below. u du 9 + 4u + C 48

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1 ams 11b Study Guide 3 econ 11b Solutions 1. Use the table of integral formulas in Appendi B in the tetbook to help compute the integrals below. 4 d a = ln C = 4 15 ln C Formula #8, with a = 3. b. e d = 9 + 4e e e d = 9 + 4e u du 9 + 4u c. = 4u u + C = e e + C 48 6 First, note that e = e e and substitute u = e, du = e d, then use formula #15 with a = 9 and b = 4. 1 t e.6t t e.6t 1 dt =.6 = 1e te.6t dt = 1e e e.6t.6t Formula #39 with n = and a =.6, followed by formula #38 with a =.6 again. 1 d. e. f. 3 dv v v + 16 = ln + v = ln3 + 5 ln + 4 = ln Formula #7, with a = ln d = 54 ln C Formula #4, with n = d = d d = 3 7 = ln 16 ln ln Formulas # and #3, with a = and b = 7. ln ln = ln 8 ln + 7 1

2 . Compute the present value of the continuous annuity that pays at the continuous rate ft = 5t for T = years, where the constant interest rate is r = 4.75%. See section 15.3 in the tet, and formula #38. Present Value = = T fte rt dt 5te.475t dt = 5e.475t.475t Let y = f satisfy i dy d = 3y and ii y1 =. Find the function f. Separate: dy = 3 d. y dy Integrate: y = 3 d = 1 y = 3 Solve for y: y = C 3. Solve for C: y1 = = = C 3 Solution: y = C. This is the implicit solution. = C 3 = 1 = C = The dead body of an eccentric socialite is found in a Las Vegas motel room. At 1: am, her body temperature was measured to be 9.4 F. Her body was left in the room for an hour and at 11: am her body temperature was 9.5 F. The room itself was kept at a constant temperature of 7 F. Use Newton s law of cooling to estimate the time of the socialite s death. * See Eample 3 in section Newton s law of cooling or heating, says that the the rate of change in the temperature of a body immersed in a medium of constant ambient temperature is proportional to the difference between the temperature of the body T and the ambient temperature A. If T t is the temperature of the body at time t, then Newton s law can be epressed as dt = kt A, where k is the unknown constant of proportionality. To solve this differential equation, The constant k depends on the physical properties of the body, in particular its heat conductance.

3 we separate, integrate and solve for T : Separate : T A = k dt Integrate : T A = k dt = ln T A = kt + C Solve for T : T A = e kt+c = e C e kt = T A = ±e C e kt = T = A + αe kt, where α = ±e C. Net, we use the given data. First, the constant ambient temperature is A = 7 F, so T = 7 + αe kt. Second, we ll let t = correspond to 1: am, so 9.4 = T = 7 + αe k = 7 + α = α = =.4. Net, measuring time in hours, 11: am corresponds to t = 1, so 9.5 = T 1 = 7 +.4e k 1 = 7 +.4e k = e k = = k = ln Finally, assuming that the socialite s temperature was normal 98.6 F at her time of death t d, we find that 98.6 = T t d = 7+.4e kt d = e kt d = = kt d = ln = t d = ln ln This means that the socialite died.714 hours before 1: am at roughly 7:17 am The population of a tropical island grows at a rate that is proportional to the third root 3 of its size. In 195, the islands population was 178 and in 198, the islands population was 744. What will the islands population be in? First, translate the description of the growth rate into a differential equation. If P t is the size of the population at time t in years, then the description above leads to the differential equation dp dt = k 3 P, where k is the unknown constant of proportionality. Separate the variables: Integrate both sides: dp = k dt. P 1/3 P 1/3 dp = k dt = 3 P /3 = kt + C. Solve for P : Multiplying by /3 gives P /3 = kt + C, because the factor of /3 is absorbed by both k and C. Net, raise both sides to the power 3/ to see that P = kt + C 3/. 3

4 Solve for the parameters k and C: First set t = for the year 195, so 178 = P = + C 3/ = C 3/ = C = 178 /3 = 144. Net, the year 198 corresponds to t = 3, so 744 = P 3 = 3k / = 3k = 744 /3 = 196 = k = Thus P t = 6 t / 15 and in the population will be 3/ 6 P 7 = The income-elasticity of monthly demand q for a price-controlled good is assumed to be proportional to the natural logarithm of average monthly disposable income y in the market for that good. When y = 5, the demand is q = 5 and when y =, the demand is q = 35. What is the predicted monthly demand for this good if monthly disposable income decreases to y = 15? The income-elasticity of demand is η = dq dy y, and if this is proportional to the natural q logarithm of income, we obtain the separable differential equation dq dy y q dq = k ln y = q = k ln y y dy, with k being the unknown constant of proportionality as usual. Integrating both sides gives dq ln y q = k ln y dy = ln q = k + C = kln y + C, y using the substitution u = ln y, du = 1 dy for the right-hand integral and absorbing the y factor of 1/ into k. Eponentiating both sides gives q = e kln y +C = e C e kln y = Ae kln y, and it remains to use the given data to solve for A and k. The data q5 = 5 and q = 35 gives a pair of equations for A and k: } Ae kln = 35 Aekln Ae kln = = 35 5 = 5 Ae kln 5 5 = ek[ln ln 5 ] =.7 Net, take the natural logarithm of both sides to find k: k [ ln ln 5 ] = ln.7 = k = To find A, use one of the two equations: ln.7 ln ln Ae kln = 35 = A = 35e kln.879. Finally, we can predict the demand when income decreases to y = 15: q15 = Ae kln e.1365ln Why did I not write ln q and ln y here? 4

5 7. Compute the indicated partial derivatives of the functions given below. a. f, y, z = 3 y z + 3 y 5 z 3 4y 3 + yz 4. f = 6 y z + 6y 5 z 3 4y 3 f y = 4 3 yz + 15 y 4 z 3 1y + z 4 f z = 3 y + 9 y 5 z + 4yz 3 b. w = u v u + v. w u = uvu + v u v 1 = u v + uv 3 u + v u + v w v = u u + v u vv = u3 u v u + v u + v c. F, y, z, λ = 1 ln y 5 z 3 λ5 + y + 8z. F = 1 y 5 z 3 y 5 z 3 5λ = 5λ If you use properties of the log function to simplify F before you differentiate, the differentiation is a little easier: F, y, z, λ = ln + 5 ln y + 3 ln z λ5 + y + 8z = F = 5λ F λ = 5 + y + 8z. d. z = y e y. z = y e y y = 4y 3 e y In this one, y behaves like a constant factor. z y = 4yey + y e y = 4y + y e y. We have to use the product rule here. 5