PREP Quickstart, Multivariable Calculus Sage Quickstart for ...

More documents

Recommendations

Info

2/20/13 PREP Quickstart, Multivariable Calculus -- Sagedone.# import matplotlib.cm; matplotlib.cm.datad.keys()# 'Spectral', 'summer', 'blues'g(x,y)=e^-x*sin(y)contour_plot(g, (x, -2, 2), (y, -4*pi, 4*pi), cmap = 'Blues',contours=10, colorbar=True)Partial DifferentiationThe following exercise is from Hass, Weir, and Thomas, University Calculus, Exercise 12.7.35. It has a localminimum at .(4,−2)f(x, y) = x^2 + x*y + y^2 - 6*x + 2test.sagenb.org/home/Carlos_Rodriguez/20/print 6/9
↦ 2 x + y − 6(x,y)↦ x + 2 y(x,y)f xxD = 3f xxf xx2/20/13 PREP Quickstart, Multivariable Calculus -- SageQuiz: Why did we not need to declare the variables in this case?fx(x,y)= f.diff(x)fy(x,y) = f.diff(y)fx; fyf.gradient()solve([fx==0, fy==0], (x, y))(x,y) ↦ (2 x + y − 6, x + 2 y)H = f.hessian()H(x,y)[[x = 4,y = (−2)]]2112And of course if the Hessian has positive determinant andis positive, we have a local minimum.( )Notice how we were able to use many things we've done up to now to solve this.MatricesSymbolic functionsSolvingDifferential calculusSpecial formatting commandsAnd, below, plotting!html("$f_{xx}=%s$"%H(4,-2)[0,0])html("$D=%s$"%H(4,-2).det())= 2And of course if the Hessian is positive definite andis positive, we have a local minimum.Notice how we were able to use many things we've done up to now to solve this.test.sagenb.org/home/Carlos_Rodriguez/20/print 7/9
Page 2 and 3: 25, )1254(6,2 t, )34 t2 (2,2/20/13
Page 4 and 5: 2/20/13 PREP Quickstart, Multivaria
Page 8 and 9: 2/20/13 PREP Quickstart, Multivaria

PREP Quickstart, Multivariable Calculus Sage Quickstart for ...

Create successful ePaper yourself

Delete template?

Save as template?