Review problems for the final exam Calculus III Fall 2003

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1 Review problems for the final exam alculus III Fall Perform the operations indicated with F (t) = 2t ı 5 j + t 2 k, G(t) = (1 t) ı + 1 t k, H(t) = sin(t) ı + e t j a) F (t) G(t) b) F (t) [ H(t) G(t)] c) 2e t F (t) + tg(t) + 10H(t) 2. etermine the position r(t) as a function of time t for a particle with velocity v(t) = d r = 3 ı + 2 j position at time t = 1 given by r(1) = 2 ı + 5 j. 3. etermine the velocity, acceleration speed at time t = π 6 of a particle whose position is given by the vector function r(t) = sin(t) ı + 2 cos(t) j + 3t k. 4. Find the position vector R(t) the velocity vector V (t) given the acceleration vector A(t) = 3t 2 ı 3t j + 5 k the initial position R(0) = 3 ı + 2 j k the initial velocity V (0) = ı j 2 k. 5. For the following curves find the unit tangent vector T (t) the principal unit normal vector N(t). a) R(t) = sin(t) ı cos(t) j + t k b) R(t) = ln(t) ı + t 2 k 6. Find the length of the curve R(t) = t ı + 2t j + 3 k over [0, 2]. 7. Find the length of the curve R(t) = cos 3 (t) ı + cos 2 (t) k over [0, π 2 ]. 8. A car weighing 2,700 lb makes a turn on a flat road while traveling at 56 ft/s. If the radius of the turn is 70ft, how much frictional force is required to keep the car from skidding? 9. A car weighing 2,700 lb moves along the elliptic path 900x y 2 = 1, where x y are measured in miles. If the car travels at the constant speed of 45 mi/h, how much frictional force is required to keep it from skidding as it turns the corner at ( 1, 0)? What about the corner 30 (0, 1 20 )? 10. Find the critical points classify each point as a relative maximum, a relative minimum or a saddle point. a) f(x, y) = x 2 + y 2 6xy + 9x + 5y + 2 b) f(x, y) = 2x 3 + y 3 + 3x 2 3y 12x Find the equations of the tangent plane the normal line to the following surfaces at the given point: a) z = x 2 + y 2 at P 0 (3, 1, 10). b) x 2 + y 2 + z 2 = 3 at P 0 (1, 1, 1). 1

2 2 12. Evaluate the derivative of f(x, y, z) = z 3 x 2 y at the point P(1, 6, 2) in the direction of the vector v = 3 ı + 4 j + 12 k. 13. Find the direction from P 0 in which the given function f increases most rapidly compute ( the magnitude of greatest rate of increase. y ) a) f(x, y, z) = z ln P 0 (1, e, 1) z b) f(x, y, z) = 2x 2 + 3y 2 + z 2 at P 0 (2, 3, 1) 14. Use implicit differentiation to find z z x y where 4x2 y+y 2 z z 2 = Given that z = xy + y, use the chain rule to determine z z θ at r r = 2 θ = π, where x = r cos(θ) y = r sin(θ) Let z = u e u2 v 2 where u = 2x 2 + 3y 2 amd v = 3x 2 2y 2. Use the chain rule to find z z x y. 17. Evaluate the following double integrals: a) cos(x 2 )da where is the region bounded by y = x, y = x, x = 0 x = π 2 b) 4xdA where is the region bounded by y = 4 x 2, y = 3x x = etermine the volume of the solid that lies under the paraboloid z = x 2 + y 2 above the disk x 2 + y Evaluate xyzdv where is the tetrahedron with vertices (0, 0, 0), (2, 0, 0), (0, 1, 0) (0, 0, 1) (i.e., the tetrahedron bounded by the coordinate planes the plane x + 2y + 2z 2 = 0). 20. Evaluate z 2 dx dy dz x 2 + y 2 + z2 where is the solid sphere x 2 + y 2 + z Evaluate z(x 2 + y 2 ) 1 2 dx dy dz where is the solid bounded above by the plane z = 2 below by z = 2(x 2 + y 2 ). 22. Find the divergence curl of the following vector fields: a) F (x, y, z) = xyz ı + y j + x k at (1, 2, 3) b) F (x, y, z) = e x sin(y) ı + e x cos(y) j + k at (1, 3, 2). 23. Evaluate [(x 2 +y 2 ) dx+2xy dy] where is the quarte circle x 2 +y 2 = 1 from (1, 0) to (0, 1).

3 24. Evaluate [x 2 y dx + (x 2 y 2 ) dy] where is the arc of parabola y = x 2 from (0, 0) to (2, 4). 25. how that the following integrals are independent of path evaluate them using the fundamental theorem of line integrals: a) (xy 2 ı + x 2 y j) dr, where is any path from (4, 1) to (0, 0). b) 2x sin(y) dx+(x 2 cos(y) 3y 2 ) dy, where is any path from ( 1, 1) to (5, 1). 26. ompute the line integral of the vector field F (x, y) = (5y + x 3 cos(x)) ı sin(y 2 ) j where is the boundary of the triangle with vertices (0, 0), (1, 2), (1, 0) traversed counterclockwise. 27. Evaluate (3y e x ) dx + (7x + y 4 ) dy, where is the boundary of the rectangle 1 x 4, 2 y 6 traversed counterclockwise. 28. Evaluate F N d where F = x ı + y j + 2z k is the surface of the cube bounded by the planes x = 0, x = 1, y = 0, y = 1, z = 0 z = Evaluate F d r, where F (x, y, z) = y 2 ı + x j + z 2 k is the boundary of the triangular region on the plane 3x + 4y + z = 8 in the first octant, traversed counterclockwise. 30. Evaluate the line integral of the vector field F (x, y, z) = 3y ı + 2z j 5x k along the curve of intersection of the xy-plane with the hemisphere z = 1 x 2 y 2 traversed counterclockwise as viewed from above. 31. Evaluate F N d where F = x ı + y 2 j + xyz k where is the part of the paraboloid z = 4 x 2 y 2 with z 0. Use the upward unit normal vector. 3

4 4 Formulas that you should know 1. Polar coordinates: x = r cos(θ), y = r sin(θ), r = x 2 + y 2, θ = tan 1 ( y x ). 2. Integration in polar coordinates f da, where da = rdrdθ. 3. ylindrical coordinates: x = r cos(θ), y = r sin(θ), z = z, x 2 + y 2 = r 2, θ = tan 1 (y/x). Integration in cylindrical coordinates: f(r, θ, z) dv, where dv = r dz dr dθ. 4. pherical coordinates: x = ρ cos(θ) sin(φ), y = ρ sin(θ) sin(φ), z = ρ cos(φ), x 2 +y 2 +z 2 = ρ 2. Integration in spherical coordinates: f(ρ, θ, φ) dv, where dv = ρ 2 sin(φ) dρ dθ dφ. 5. cos 2 (θ) = cos(2θ) 6. sin 2 (θ) = cos(2θ) 7. irectional derivative: u (f(p 0 )) = f 0 u, where u is a unit vector. 8. Operations with vector fields in parametric form such as dot product, cross product, magnitude, etc. 9. Arc length: b a R (t), where R (t) sts for the length of R (t). 10. econd Partial Test for determining relative extrema (using = f xx f yy fxy) curl( F ı j k ) = x y z u v w, where F = u ı + v j + w k. 12. div( F ) = u x + v y + w z 13. f = f f f x ı + y j + z k 14. urface Integral: g(x, y, z) d = g(x, y, f(x, y)) fx 2 + fy da xy. R xy Note that the flux integral of a vector field F across the surface is given by the surface integral F N d, where N is an outward normal vector to. 15. Fundamental Theorem of line integral: R F d R = f(q) f(p ), where is any curve joining the points P Q f = F. 16. Equations of plane, line. 17. Tangent plane normal line to a surface (formulas on page 746 of your text book). Note that when the surface is given by z = f(x, y) the equation of the tangent plane is the one given on page 720. R

5 5 18. Formulas for the dot product cross product. 19. Implicit differentiation, chain rule. The following will be provided in the test: ( N Green s theorem: (M dx + N dy) = x M ) da. y tokes Theorem: F dr = ( F ) N d, where N is a unit normal vector to the surface. ivergence Theorem: F N d = div F dv Tangential normal components of the acceleration: ( ) A T = d2 s ds 2 2, A N = κ (note that ds in A N equals the speed). Tangential normal components of the force: ( ) F T = m d2 s ds 2 2, F N = mκ, where κ is the curvature, the mass m = W, W denotes the weight g is the acceleration of gravity (g g 32ft/s 2 ).