Midterm 2 ReviewMary RadcliffeMath 10CFall 2010November 11, 2010Chapter 13 (Section 1)• Know the definitions of displacement vector, magnitude, anddirection.• Know how to add and subtract vectors, and multiply byscalars.• Know how to write displacement vectors with~i,~j,~kcomponents and calculate sums, scalar products, andmagnitude with this expression.• Know the definition of a unit vector(~u is a unit vector if ||~u|| = 1).Chapter 13 (Section 2)• Given the speed and direction of an object, be able to resolveinto a vector with~i,~j, and~k components.• Be able to find the total displacement of an object affected byoutside forces (e.g. wind, currents, etc.)• Know the properties of addition and scalar multiplication (seep. 697) for vectors.• Be able to manipulate n-dimensional vectors and understandhow to use them to organize information.Chapter 13 (Section 3)• Know the algebraic definition of dot product:(v1~i + v2~j + v3~k) · (w1~i + w2~j + w3~k) =v1w1+ v2w2+ v3w3• Know the geometric definition of dot product:~v ·~w =||~v|| ||~w|| cos(θ)• Know basic properties of the dot product:I~v ·~w =~w ·~vI~v · (λ~w) = λ(~v ·~w) = (λ~v) ·~wI(~v +~w) ·~u =~v ·~u +~w ·~u.I~v ·~v = ||~v||2• Two nonzero vectors~v and~w are perpendicular if and only if~v ·~w = 0.• Equation of a plane: if~n = a~i + b~j + c~k is perpendicular to aplane containing the point P0= (x0, y0, z0), then the equationof the plane is 0 = a(x − x0) + b(y − y0) + c(z − z0).Chapter 13 (Section 4)• Know the algebraic definition of cross product:(v1~i + v2~j + v3~k) × (w1~i + w2~j + w3~k) =(v2w3− v3w2)~i − (v1w3− v3w1)~j + (v1w2− v2w1)~k• Know the geometric definition of cross product: If~n is normalto~v and~w, found by the right hand rule, then~v ×~w =(||~v|| ||~w|| sin(θ))~n• Know basic properties of the cross product:I~v ×~w = −~w ×~vI~v × (λ~w) = λ(~v ×~w) = (λ~v) ×~wI~u × (~v +~w) =~u ×~v +~u ×~w.• Equation of a plane: If P, Q, and R are points on a plane, wecan find the equation as follows:IForm vectors~PQ and~QR. These vectors both lie in the plane.IFind~n =~PQ ×~QR. This will be perpendicular to the plane.IUse equation as in 13.3 with~n and P0= P.Chapter 14 (Section 1, 2)• Know the definition and notations for the partial derivatives ofa function f of more than one variable.• Be able to interpret the meaning of a partial derivative in agiven scenario.• Be able to estimate partial derivatives from contour diagrams.• Be able to compute partial derivatives algebraically.Chapter 14 (Section 3)• Know what “local linearity” means, and be able to construct alinear approximation (a.k.a tangent plane) to a functionf (x, y) at a point (a, b). The tangent plane is given byf (x, y) ≈ z = f (a, b) + fx(a, b)(x − a) + fy(a, b)(y − b).• Know how to use the tangent plane to approximate values off .• Know what the differential of z = f (x, y ) is and how to use it:dz = fxdx + fydy.• Know how to use the differential to approximate values of f .Chapter 14 (Section 4)• Know the definition of directional derivative: if~u = u1~i + u2~jis a unit vector, then the directional derivative of f at a point(a, b) in the direction~u isf~u(a, b) = fx(a, b)u1+ fy(a, b)u2.• Know the definition of the gradient vector: for a functionf (x, y ), the gradient is grad f =−→∇f = fx~i + fy~j.• Properties of the gradient vector:If~u(a, b) =~u ·−→∇f (a, b).I−→∇f (a, b) is perpendicular to the contour through (a, b)I−→∇f (a, b) gives the direction of fastest increase of f at (a, b)I||−→∇f (a, b)|| gives the fastest rate of change at (a, b), and islarger when contours are closer togetherChapter 14 (Section 6)• Know common chain rules:IIf z = f (x, y ), x = g (t), and y = h(t), thendzdt=∂z∂xdxdt+∂z∂ydydtIIf z = f (x, y ), x = g (u, v ), and y = h(u, v ), then∂z∂u=∂z∂x∂x∂u+∂z∂y∂y∂uand∂z∂v=∂z∂x∂x∂v+∂z∂y∂y∂v• Know how to diagram a chain rule for any given function.Chapter 14 (Section 7)• Know the definitions and notations of the second-order partialderivatives of a function z = f (x, y):I∂2z∂x2= fxx= (fx)xI∂2z∂x∂y= fyx= (fy)xI∂2z∂y ∂x= fxy= (fx)yI∂2z∂y2= fyy= (fy)y• The two mixed partials are equal, i.e. fxy= fyx• Be able to find the sign of partial derivatives from a contourdiagram.• Be able to construct a Taylor polynomial of degree 1 or 2 fora function z = f (x, y) at a point (a, b):IDegree 1: Tangent plane.f (x, y) ≈ P1(x, y ) = f (a, b) + fx(a, b)(x − a) + fy(a, b)(y − b)IDegree 2:f (x, y) ≈ P2(x, y ) = f (a, b)+fx(a, b)(x −a) + fy(a, b)(y −b) +12fxx(a, b)(x − a)2+ fxy(a, b)(x − a)(y − b) +12fyy(a, b)(y −