MA 261 - Spring 2010Study Guide # 21. Gradient vector for f(x, y): ∇f(x, y) =∂f∂x,∂f∂y, properties of gradients; gradient points indirection of maximum rate of increase of f; ∇f(x0, y0) ⊥ level curve f (x, y) = C and, in the caseof 3 variables, ∇f(x0, y0, z0) ⊥ level surface f (x, y, z) = C :0(x ,y )xyf(x,y,z)=Cxy(x ,y ,z )0 0 0n = ∆0n = ∆zf(x ,y )00f(x,y)=Cf(x ,y ,z )0002. Directional derivative of f (x, y) at (x0, y0) in the direction~u : D~uf(x0, y0) = ∇f (x0, y0) ·~u,where~u must be a unitvector; tangent planes t o level surfaces f (x, y, z) = C (a no rmal vectorat (x0, y0, z0) is~n = ∇f (x0, y0, z0)).3. Relative/local extrema; critical points (∇f =~0 or ∇f does not exist); 2ndDerivatives Test: Acritical points is a local min if D = fxxfy y− f2xy> 0 and fxx> 0, local max if D > 0 and fxx< 0,saddle if D < 0; absolute extrema; Max-Min Problems; Lagrange Multipliers: Extremize f(~x)subject to a constraint g(~x) = C, solve the system: ∇f = λ∇g and g(~x) = C.4. Double integrals; Midpoint Rule for rectangle :ZZRf(x, y) dA ≈mXi=1nXj=1f(xi, yj) ∆A;5. Type I region D :g1(x) ≤ y ≤ g2(x)a ≤ x ≤ b; Type II region D :h1(y) ≤ x ≤ h2(y)c ≤ y ≤ d;iterated integrals over Type I and II regions:ZZDf(x, y) dA =ZbaZg2(x)g1(x)f(x, y) dy dx andZZDf(x, y) dA =ZdcZh2(y)h1(y)f(x, y) dx dy, respectively; Reversing Order of Integration (regionsthat are both Type I and Type II); properties of double integrals.6. Integral inequalities: mA ≤ZZDf(x, y) dA ≤ MA, where A = area of D and m ≤ f (x, y) ≤ Mon D.17. Change of Variables Formula in Polar Coordinates: if D :h1(θ) ≤ r ≤ h2(θ)α ≤ θ ≤ β, thenZZDf(x, y) dA =ZβαZh2(θ)h1(θ)f(r cos θ, r sin θ) r dr dθ.↑8. Applications of double integrals:(a) Area of region D is A(D) =ZZDf(x, y) dA(b) Volume of solid under graph of z = f(x, y), where f(x, y) ≥ 0, is V =ZZDf(x, y) dA(c) Mass of D is m =ZZDρ(x, y) dA, where ρ(x, y) = density (per unit area); sometimes writem =ZZDdm, where dm = ρ(x, y) dA.(d) Moment a bout the x-axis Mx=ZZDy ρ(x, y) d A; moment about the y-axis My=ZZDx ρ(x, y) dA.(e) Center of mass (x, y), where x =Mym=ZZDx ρ(x, y) dAZZDρ(x, y) dA,y =Mxm=ZZDy ρ(x, y) d AZZDρ(x, y) dARemark: centroid = center of mass when density is constant (this is useful).9. Elementary solids E ⊂ R3of Type 1, Type 2, Type 3; triple integrals over solids E:ZZZEf(x, y, z) dV ;volume of solid E is V (E) =ZZZEdV ; applications of triple integrals, mass of a solid, momentsabout the coor dinate planes Mxy, Mxz, My z, center of mass of a solid (x, y,