Questions 1. Special electrical circuits called LC circuits are critical to the operation of the tuning dial on a car radio. The charge Q(t) (measured in units of Coulombs C) in an LC circuit at any given time t (measured in seconds s) subject to a given voltage V(t) (given in Joules per Coulomb, J/C) obeys the expression LQ" (t) + + =—= Q(t) = = V(t) where L, c are given constants depending on the particular design of the circuit. What are the units of L and c? ⚫L and care unit-less. ⚫ The units of L are Js2/C2 and the units of c are J/C². • The units of L are Js2/C2 and the units of c are C²/J. ⚫ The units of L are J/C2 and the units of c are C²/J. ⚫ The units of L are J/C² and the units of c are J/C². 1 of 7 2. Matt collects vinyl records, but he doesn't always keep every record he buys. Let K(t) be the total number of vinyl records Matt has kept from the moment he started collecting up to week t, where t is the number of weeks since January 1, 2024. Which of the statements below is the best interpretation of K'(41) 9? Use the fact that October 7 is in the 41st week of 2024. From January 1, 2024 to October 7, 2024, Matt kept approximately 9 records. From January 1, 2024 to October 7, 2024 Matt kept approximately 9 records each week. We expect that Matt kept approximately 9 more records in the 41st week than in the 40th week of 2024. • We expect that Matt kept approximately 9 records between Oct 10 and Oct 7, 2024. 3. Consider an ecosystem containing a population of predator animals (say, foxes) and a population of prey animals (say, rabbits (poor rabbits)). Let p denote the density of predator organisms in the ecosystem (measured in organisms/km²) in October 2024. An ecologist has decided to model the fraction of prey that survive predation in October 2024 using the (unitless) function F(p)=eap, (where a 0 is an empirical constant with suitable units.) Let L(p) denote the linear approximation of F(p) near p = 0. Which of the statements below are true? L(p) has negative slope. F(p) and L(p) both tend to 0 as p→ ∞. L(p) predicts that all prey animals are eaten by the end of October if p = 1. L(p) gives a good approximation of F(p), even when p is much larger than p = 1. 4. Suppose that f(x) and g(x) are differentiable functions defined for all x such that f(-2) = g(-2) and f'(x) -2. • f(x) < g(x) for all x < -2. The graphs of ƒ and g do not intersect. The graphs of ƒ and g intersect at exactly one point. The graphs of ƒ and g intersect at more than one point. 5. Let f(x) be differentiable for all x, and let g(x) be the tangent line to f at a point a 0. Which of the expression(s) below are equal to f'(a)? • g'(a) • g(a + 1) − g(a) f(0+ h)-f(0) • lim 0+4 h f(x) − f(a) f(a+h) f(a - h) ⚫ lim h→0 2h

Questions 1. Special electrical circuits called LC circuits are critical to the operation of the tuning dial on a car radio. The charge Q(t) (measured in units of Coulombs C) in an LC circuit at any given time t (measured in seconds s) subject to a given voltage V(t) (given in Joules per Coulomb, J/C) obeys the expression LQ" (t) + + =—= Q(t) = = V(t) where L, c are given constants depending on the particular design of the circuit. What are the units of L and c? ⚫L and care unit-less. ⚫ The units of L are Js2/C2 and the units of c are J/C². • The units of L are Js2/C2 and the units of c are C²/J. ⚫ The units of L are J/C2 and the units of c are C²/J. ⚫ The units of L are J/C² and the units of c are J/C². 1 of 7 2. Matt collects vinyl records, but he doesn't always keep every record he buys. Let K(t) be the total number of vinyl records Matt has kept from the moment he started collecting up to week t, where t is the number of weeks since January 1, 2024. Which of the statements below is the best interpretation of K'(41) 9? Use the fact that October 7 is in the 41st week of 2024. From January 1, 2024 to October 7, 2024, Matt kept approximately 9 records. From January 1, 2024 to October 7, 2024 Matt kept approximately 9 records each week. We expect that Matt kept approximately 9 more records in the 41st week than in the 40th week of 2024. • We expect that Matt kept approximately 9 records between Oct 10 and Oct 7, 2024. 3. Consider an ecosystem containing a population of predator animals (say, foxes) and a population of prey animals (say, rabbits (poor rabbits)). Let p denote the density of predator organisms in the ecosystem (measured in organisms/km²) in October 2024. An ecologist has decided to model the fraction of prey that survive predation in October 2024 using the (unitless) function F(p)=eap, (where a 0 is an empirical constant with suitable units.) Let L(p) denote the linear approximation of F(p) near p = 0. Which of the statements below are true? L(p) has negative slope. F(p) and L(p) both tend to 0 as p→ ∞. L(p) predicts that all prey animals are eaten by the end of October if p = 1. L(p) gives a good approximation of F(p), even when p is much larger than p = 1. 4. Suppose that f(x) and g(x) are differentiable functions defined for all x such that f(-2) = g(-2) and f'(x) -2. • f(x) < g(x) for all x < -2. The graphs of ƒ and g do not intersect. The graphs of ƒ and g intersect at exactly one point. The graphs of ƒ and g intersect at more than one point. 5. Let f(x) be differentiable for all x, and let g(x) be the tangent line to f at a point a 0. Which of the expression(s) below are equal to f'(a)? • g'(a) • g(a + 1) − g(a) f(0+ h)-f(0) • lim 0+4 h f(x) − f(a) f(a+h) f(a - h) ⚫ lim h→0 2h