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8. It is thought that the period T of a planet's year is related to r, its mean distance from the sun, by an equation of the form T = Kr" where k and n constants. If T is measured in Earth-years and r in astronomical units of distance, the values for the six inner planets are: r T Mercury 0.3871 0.2408 Venus 0.7233 0.6152 Earth 1.000 1.000 Mars 1.524 1.881 Jupiter 5.203 11.86 Saturn 9.539 29.46 Use natural logarithms to plot these values as an approximate straight line. From your graph of In T against In r find the gradient of the graph and hence the values of n and k.

What are the roots of the auxiliary equation for a homogeneous linear equation with real, constant coefficient that has the given function: y = x² e¯* + 3 e3* A -1, -1, -1, 3 2, -1, -1, 3 с) -1, 3, 0 -1, -1, 3

The questions in 1-5 are concerned with functions of two variables.1. Let f(x, y) = x2y + 1. Find(a) f(2, 1)(b) f(1, 3)(c) f(3a, a)(d) f(ab, a ab)(e) f(0, 0)2. Let g(x) = x sin x. Find(a) g(x/y)(b) g(xy)(c) g(x xy)3. Find g(u(x, y), v(x, y)) if g(x, y) = y sin(x2y), u(x, y) = x2y3, and v(x, y) =πxy4. Find F(g(x), h(y)) if F(x, y) = xexy, g(x) = x3, and h(y) = 3y + 1.5. Let g(x, y) = ye3x, x(t) = ln(t2 + 1), and y(t) = √t. Find g(x(t), y(t)).6. Suppose that the concentration C in mg/L of medication in a patient’sbloodstream is modeled by the function C(x, t) = 0.2x(e0.2teet), wherex is the dosage of the medication in mg and t is the number of hours sincethe beginning of administration of the medication.(a) Estimate the value of C(25, 3) to two decimal places. Include appro￾priate units and interpret your answer in a physical context.(b) If the dosage is 100 mg, give a formula for the concentration as afunction of time t.(c) Give a formula that describes the concentration after 1 hour in…

1.1

1. Two substances A and B are combined to form a product C. The formation of the product is proportional to the time the reactants are combined. The final product is composed of two parts of B for every part of A. If initially A is 30 kg and B is 20 kg, and 5 kg of the product is formed after 30 mins., find the function of product formed at any given time.

Consider the two ODES dx dy dx +y=0 + y² = 0 (a) How are the two equations different in structure? (b) Show that, for any constant C, the function Ce is always a solution of 1, while Ca-¹ is a solution of 2 only when C = 0 or 1. (c) Show that, for any linear equation of the form dy dx eqil eq.2 eq.3 if z(x) is a solution, then, for any constant C, the function Cz(x) is also a solution. Hint: Plug Cz(x) into 3. (d) What happens if you try to obtain the same result for an equation of the form dy dx + P(x)y = 0, + P(x)y² = 0? eq.4 Does the same result hold? (e) If you have found one solution of a linear ODE of the form 3, how many solutions do you have? Consult your result from question 2c.

4.B and C and Section 3.9 N Find the most general antidernivativer. of the frunction. (Check your Df&) = {xZ + x Jx anver by differentiation)N %3D

2 1. At all times, the length of the long leg of a right triangle is 3 times the length x of the short leg of the triangle. Find an equation that relates area A to x. #3 E D 80 F3 $ 4 R F Q F4 % 5 T G 9 F5 6 MacBook Air Y H & 7 F7 U * 00 8 J DII F8 I ( 9 K F9 O L F10 | ' P F11 11 + WE { = [