Writing the equation d² dt2 in the form of the system +x², = x³ X = dt V, d ·V= x³ + x4 dt x = x (t), x = x(t), v = v(t), dx (a) Find all the stationary points (x, v) (the points where d = 0, dt dv dt (1) (2) = 0).
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- Find the linearization L(x,y) of the function at each point. f(x,y) = x² + y? + 1 а. (2,1) b. (0,2) а. L(x,y) %3 b. L(x,y) =Find the linearization at x = a. f(x) = ) = — - ₁a = 1 x2, = 7Applying variation of parameters to the DE: " – y = e* + e-*, we get the following system of equations: uj eº + u,e-² = 0 u e" – u,e-* = eª + e¬* What are u1 and u2? (U1 and uz are functions of x.) u1 = -e"; u2 %3D O u1 = ;a – ÷e-2*; u2 = = -e2 - æ 2 O uj = In sin a (e*) ; u2 = sec x (e") i+e ; uz = - fee 1 -2r 1 O uj = 4 2r 8 8
- Find the linearization L(x,y) of the function at each point. f(x,y) = x? + y? + 1 а. (4,1) b. (3,3)Find the linearization L(x,y) of the function at each point. f(x,y) = x² + y² + a. (3,0) b. (4,3)Two chemical engineers, A and B, are working independently to develop a model to predict the viscosity of a product (y) from the pH (x1) and the concentration of a certain catalyst (x2). Each engineer has fit the linear model y =A +hr, + B;x; +r The engineers have sent you output summarizing their results: Engineer A Predictor Constant тР Coef SE Coef 199.2 0.5047 394.7 0.000 -1.569 pH Concent. -4.730 0.5857 -8.08 0.000 0.4558 -3.44 0.007 Engineer B Predictor Coef Constant 199.0 0.548 SE Coef T 363.1 0.000 pH Concent. -3.636 1.952 -1.256 1.983 -0.63 0.544 -1.86 0.112 The engineers have also sent you the following scatterplots of pH versus concentration, but forgot to put their names on them. Concentration Concentration Which plot came from which engineer? How do you know? Which engineer's experiment produced the more reliable results? Explain. a. b.