Final Exam - Review. Cumulative Final Review covers sections and Chapter 12

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1 Final Exam - eview Cumulative Final eview covers sections and Chapter 12 The following is a list of important concepts from each section that will be tested on the Final Exam, but were not covered by Midterm Exam 1 or 2. This is not a complete list of the material that you should know for the course, but it is a good indication of what will be emphasized on the exam. A thorough understanding of all of the following concepts will help you perform well on the exam. Some places to find problems on these topics are the following: in the book, in the slides, in the homework, on quizzes, and WebAssign. Tangent Planes and irectional erivatives: Sections 11.4 and 11.6 The tangent plane to the graph z = f (x,y at the point P = (a,b,c is the plane containing tangent lines to any curve on the graph passing through P. The scalar equation of the tangent plane is z = f (a,b + f x (a,b(x a + f y (a,b(y b. The total differential of z = f (x,y is dz = f x (x,ydx + f y (x,ydy. If both partial derivatives f x and f y are continuous at (a,b, then f (x,y is differentiable at (a,b. If f is differentiable at (a, b, then the tangent plane and the total differential can be used to approximate values on the graph or differences in the height of points on the graph, respectively. The gradient vector of z = f (x,y at (a,b is f (a,b = f x (a,b, f y (a,b. If f is differentiable at (a,b, then the rate of change of f in the u-direction, u = 1, is the irectional erivative u f (a,b = f (a,b u The largest directional derivative is in the direction of f (a,b and equal to f (a,b. The smallest directional derivative is in the direction of f (a,b and equal to f (a,b. If f (a,b 0, then u = f (a,b is the direction with the largest directional derivative. f (a,b f (a,b is orthogonal to any vector that is tangent to the level curve L k ( f at (a,b. 1. Use a linear approximation of f (x,y,z = x 2 + y 2 + z to estimate ( ( Compare your approximation to the value provided by your calculator. 2. The volume V of a cone is computed using the values 3.5cm for diameter of the base and 6.2cm for the height. Estimate the maximum error and the relative error in V if each of these values has a possible error of at most 5%. 3. Suppose that you are hiking on a terrain modeled by z = xy + y 3 x 2. You are at the point (2,1, 1. (a etermine the slope you would encounter if you headed due West from your position. What angle of inclination does this correspond to? (b etermine the slope you would encounter if you headed due North-West from your position. What angle of inclination does this correspond to? (c etermine the slope you would encounter if you headed due South-West from your position. What angle of inclination does this correspond to? (d etermine the steepest slope you could encounter from your position and the direction of that slope (as a unit vector. 4. Find an equation of the tangent plane to the surface xz + 2x 2 y + y 2 z 3 = 11 at the point (2,1,1.

2 The Chain ule: Section 11.5 The Chain ule for 2-Variables: Suppose z = f (x,y and let x = g(s,t and y = h(s,t be differentiable 2- variable functions in s and t. Then z = f (x,y can be viewed as a function of s and t which is differentiable. That is, z = f (g(s,t,h(s,t and s = x x s + y y s t = x x t + y y t The General Chain ule: If z = f (x 1,x 2,...,x n, x 1 = g 1 (t 1,t 2,...,t m,..., x n = g n (t 1,t 2,...,t m. = f x 1 x 1 + f x 2 f f x n x n Implicit ifferentiation: If z = z(x,y is defined implicitly by F(x,y,z = 0 then x = F x(x,y,z F z (x,y,z y = F y(x,y,z F z (x,y,z 1. Use the Chain ule to calculate the partial derivatives. Express your answer in terms of the independent variables. (a Find F y where F(u,v = euv, u = x 2, and v = xy. (b Find f θ where f (x,y,z = xy z2, x = r cos(θ, y = cos 2 (θ, and z = r. (c Find df where F(x,y,z = xy + z 2, x = r + s, y = 3r, z = s 2, r = t 2 1, and s = 2t 1. dt 2. Two spacecraft are following paths in space given by r 1 = sin(t,t,t 2 and r 2 = cos(t,1 t,t 3. If the temperature for points in space are given by T (x,y,z = x 2 y(1 z, use the Chain ule to determine the rate of change of the difference in the temperatures the two spacecraft experience at time t = π. 3. Calculate x of the function defined implicitly by the equation sin(xyz + y2 z + xy = 1. Optimization in Multiple Variables: Section 11.7 and 11.8 Fermat s Theorem: If f (x,y has a local extrema at (a,b, and f x and f y exist, then f x (a,b = 0 = f y (a,b; or simply f (a,b = 0. For this reason, the points (a,b where f (a,b = 0 are called critical points. Second erivative Test: Suppose (a,b is a critical point of f and f xx, f xy, f yy are continuous at (a,b. (a,b = f xx(a,b f xy (a,b f yx (a,b f yy (a,b = f xx(a,b f yy (a,b [ f xy (a,b] 2 (I If (a,b > 0 and f xx (a,b > 0, then (a,b is a local minimum point. 2

3 (II If (a,b > 0 and f xx (a,b < 0, then (a,b is a local maximum point. (III If (a,b < 0, then (a,b is a saddle point. Extreme Value Theorem: If z = f (x,y is continuous on a closed and bounded set in 2, then f (x,y attains an absolute maximum and an absolute minimum. To find the absolute extrema, we take the following steps: (I Find all critical points (a,b and their values f (a,b. (II Find the extrema of f on the boundary of. (III The largest of the values from (I and (II are the absolute maximum, and the smallest is the absolute minimum. An alternative step (II is to use Lagrange Multipliers: To find the extrema of z = f (x,y subject to the constraint g(x, y = k: (a Find all values a, b, and λ such that g(a,b = k and f (a,b = λ g(a,b. (b Find values of f at all (a,b from (a. The largest of these is the absolute maximum value and the smallest is the absolute minimum value. 1. Find the critical points of the function and analyze them using the Second erivative Test: (a f (x,y = x 4 4xy + 2y 2 (b g(x,y = e x+y xe 2y 2. Find the global extrema of f (x,y = 2xy x y on the domain y 4 and y x Find three positive numbers that sum to 150 with the largest possible product of the three. 4. A box with a volume of 8m 3 is to be constructed with a gold-plated top, silver-plated bottom, and copper-plated sides. If gold plate costs $120 per square meter, silver plate costs $40 per square meter, and copper plate costs $10 per square meter, find the dimensions that will minimize the cost of the materials for the box. 5. Use Lagrange multipliers to find the dimensions of a cylindrical can with a bottom but no top, of fixed volume V = 100cm 3, with minimum surface area. ouble and Triple Integrals: Section Fubini s Theorem: If f (x, y is continuous on a rectangle = [a,b] [c,d] then or x=b ( y=d y=d ( x=b f (x,ydy dx f (x,ydx dy 3

4 Type I egion: = {(x,y a x b, g 1 (x y g 2 (x} Type II egion: = {(x,y c y d, h 1 (y x h 2 (y} Limits for the outer integral are [a,b] while the limits for the inner integral, for each fixed x, g 1 (x y g 2 (x. x=b ( y=g2 (x y=g 1 (x f (x,ydy dx Limits for the outer integral are [c,d] while the limits for the inner integral, for each fixed y, h 1 (y x h 2 (y. y=d ( x=h2 (y x=h 1 (y f (x,ydx dy An Application: Suppose a lamina occupies a region of the xy-plane and its density (in units of mass per unit area at a point (x,y in is given by ρ(x,y, where ρ is a continuous function on. The mass m = ρ(x,yda. The moment about an axis, is( the measurement of the tendency a region has to rotate about that axis. The My center of balance is the point m, M x where m M x = yρ(x,yda M y = xρ(x,yda Surface Area: At the point r(a,b on the parametric surface r(u,v = f (u,v,g(u,v,h(u,v, the normal vector to the tangent plane is n = r u (a,b r v (a,b. The area of the surface parametrized by r(u,v with domain is calculated r u (u,v r v (u,v da 1. Evaluate the following double integrals: y (a da where = [1,2] [1,2]. x + y2 (b sin(x + yda where = [ 0, π [ ] 3] 0, π 6. (c e 2y sin(3xda where = [ 0, π [ ] 6] 0, 1 2. (d xyda where is the region bounded by x = 0, x = 2, y = 0, and y = 2x x 2. (e cos(xyda where is the region bounded by y = 1, y = 2, x = 0 and x = 1 y. (f Use polar coordinates to calculate sin(x 2 +y 2 da where is the region between the circles x 2 + y 2 = π 2 and x2 + y 2 = π. 4

5 2. Change the order of integration for each double integral: 1 1 y 2 (a f (x,ydxdy 1 y 2 (b x f (x,ydydx. 3. Let T (u,v = (x(u,v,y(u,v be a transformation where x = 2u and y = u+v. Calculate the Jacobian of the transformation T and the transformation T 1. etermine the image under T of the following sets: (a The u and v-axes. (b The rectangle = [0,5] [0,7]. (c The line segment joining (1,2 and (5,3. (d The triangle with vertices (0,1, (1,0, and (1,1. 4. Let be the region consisting of points (x,y where 1 xy r and 1 y x 4. Evaluate the integral (x2 + y 2 da by changing the variables using the transformation x = uv 1 and y = uv. 5. epresent f (x,yda by viewing as a collection of Type I and II regions. 6. Find the center of mass of the lamina with density f (x,y = xy over the region bounded by y 2 = x+4 and x = escribe the domain of integration for the integral 2 4 z 2 5 x 2 z z 2 1 f (x,y,zdydxdz 8. Calculate the integral of f (x,y,z = e z over the tetrahedron with points (0,0,0, (4,0,0, (0,4,0, and (0, 0, Let be the region bounded by z = 1 y 2, y = x 2, and z = 0. Calculate the volume of using a triple integral in the order dzdydx. using a triple integral in the order dydzdx. 5