Math 225 - Exam Solutions

Odd-numbered Problems Practice Exam the Second 1,a) (f(x,y)=\frac{y}{x^{2}} f=-2,) -1,0,1,2 \[z=1\] \[z\ne2\] \[\frac{y}{x^{2}}=-2\] \[y=-2x^{2}\] \[\frac{y}{x^{2}}=-1\] \[y=-x^{2}\] \[\frac{u}{x^{2}}=0\] \[z=0\] \[y=0\] \[\frac{y}{x^{2}}=1\] \[y=x^{2}\] \[\frac{y}{x}=2\] \[y=2x^{2}\] \[z=-11\] \[z=-2\] b) \(f(0,0)\) is undefined, so the apparent intersection does not happen. \(lim_{(x,y)\rightarrow(0,0)}\frac{cos(x^{2}+y^{2})-1}{(x^{2}+y^{2})^{2}}\) Convert to polar coordinates \[lim_{r\rightarrow0}\frac{cos(r^{2})-1}{r^{4}}\] Use L'Hospital's rule \[lim_{r\rightarrow0}\frac{-sin(r^{2})2r}{4r^{3}}\] \[=lim_{r\rightarrow0}\frac{-sin(r^{2})}{2r^{2}}\rightarrow1\] \[=\frac{1}{2}\] The limit of \(\frac{sin~\theta}{\theta}\) goes to 1 (f(x,y)=x^{4}y-x^{2}y^{3}) a) Looking for (-\nabla f) at the point (2,-3) (\nabla f = <4x^{3}y-2xy, x^{4}-3x^{2}y^{2}>) (\nabla f(2,-3)=<-96+108, 16-108>) =<12,-92> Direction of fastest decrease: <-12, 92> b) z = (z_{0}+f_{x}(x-x_{0})+f_{y}(y-y_{0})) Equation for tangent plane = (60+12(x-2)-92(y+3)) \(f(2,-3)=-48+108=60\) \(f(2.02,2.97) \approx 60+12(2.02-2)-92(-2.97+3)\) \(\approx 60+12(.02)-92(.03)\) \(\approx 60+.24-2.76\) ≈60-2.52 ≈57.48 \(x^{4}y-x^{2}y^{3}=60\) 1 degree of freedom. (\frac{dy}{dx}=\frac{-F_{x}}{-F_{y}}) (Fy≠0) Implicit Function Theorem. 2xy²) =-12-12 x-3x²y (2,-3) -92 92 "The "slope" of (\nabla f) = (\frac{-12}{92}), which is the negative reciprocal of (\frac{dy}{dx}), which is because they are perpendicular. (f(x,y)=\frac{x^3}{3}+\frac{y^3}{3}-\frac{x^2}{2}-y^2) (f_x=x^2-x=0) (x=1) (0,0) (0,2) (1,0) (1,2) (f_y=y^2-2y=0) (y=2) saddle saddle max min (D=(f_{xx})(f_{yy})-(f_{xy})^2) (D=(2x-1)(2y-2)-0^2) (v=xyz) subject to (x+2y+3z=10) (v_x=yz=\lambda) (v_y=xz=2\lambda) (v_z=xy=3\lambda) (x=3z) (xz=2yz) (x=2y) (x+x+x=10) (3x=10) (x=\frac{10}{3}) (y=\frac{10}{6}) (z=\frac{10}{9}) a) (\iint_R xy^2+xy+y^2 dydx) [0,2]x[0,4] Now calculate! b) (\iint_R y\cos(xy)dydx) Flip (\iint_R \cos(xy)xdydx) Now calculate! use u-sub! c) (\int_0^4 \int_{\frac{y^2}{64}}^x x+2y dxdy) Calculate!