Vector Calculus Questions and Answers

11 Using the points 0 1 3 4 d A 3 2 B 5 2 OC 4 Pl D 5

Find three mutually orthogonal unit vectors in R3 besides i j and k There are multiple ways to do this and an infinite number of answers For this problem we choose a first vector u randomly choose all but one component of a second vector v randomly an choose the first component of a third vector w randomly The other components x y and z are chosen so that u v and w are mutually orthogonal Then unit vectors are found based on u v W Start with u 2 0 2 v x 1 1 and w 1 y z The unit vector based on u is Type exact answers using radicals as needed

1 point Determine whether each of the following vector fields F is path independent conservative or not If it is path independent enter a potential function for it that is a function f x y so that Vf F If it is path dependent enter NONE a If F x y x 2 i 3y 5 then x 2 2 2x 3y 2 2 5y f x y b if F x y 4x 2y i 2x 3y then f x y 2x 2 2xy 3y 2 2 c If F x y 5y 5x then f x y none d If F x y 4x y 2xy then f x y 4x 3 3 xy 2 4x 4y 47 4 In x 4 then e If F x y x 4 f x y 4yln x 4 c 1 if F x y 6y sin xy i 6x sin xy then f x y NONE

Q8 15 marks The following f t is a periodic function of period 2 defined over the period 0 t 2 f t 5 cos t when 0 t 0 when t 2 Express f t as a Fourier series expansion

1 point Compute the curl of the vector field F 2y i sin x cos z k curl

A car drives east from point A at 30 kph Another car B starting from B at the same time drives S 30 W toward A at 60 kph B is 30 km away from A 5 How fast in kph is the distance between the two cars changing after one hour a 76 94kph b 78 94kph c 75 94kph d 77 94kph

1 point Find F dr for F 4yl sin y on the curve counterclockwise around the unit circle C starting at the point 1 JcF dr

man stands on a bathroom scale in a onless elevator When the elevator begins to e the scale briefly reads only 0 71 of her ar weight Express your answer to two significant figures and include the appropriate units a x H Value Q X 10n Units ELITA

Find the limit of the trigonometric function lim sin X 10 TTX 12

Find the domain of the function g x 8x 2 X 9 The domain is

Calculate Tu Tu and n u v for the parametrized surface at the given point Then find the equation of the tangent plane to the surface at that point u v 2u v u 4v 7u u 7 V 7 T Tu The tangent plane 9z To 3 n u v

Determine whether the vector field is conservative and if so find the general potential function F cos z 2y 7 x sin z 4 c Note if the vector field is not conservative write DNE

1 point Find the divergence of each of the following vector fields at all points where they are defined a div 2exyi 5 cos xy j 3e dx y 5 k X x 3 z 5 c div 2x sin xz i 3j sin xz k Z y i x 3 2 1 5 x y 2 1 5 b div k

1 point Let F 4xyz 2 sin x 2x z 2x y Find a function f so that F Vf and 0 0 0 0

Let F e5y sin x cos7 x Calculate curl F curl F

1 point Show that the vector field F x y z 7y cos 8x 8x sin 7y 0 is not a gradient vector field by computing its curl How does this show what you intended curl F VXF

1 point Find the line integral of F x y z x i y j zk along the line segment C from the origin to the point 4 2 8 F dr

1 point Find F dr for F 4yi sin y j on the curve counterclockwise around the unit circle C starting at the point 1 0 JcF dr

Compute and sketch the vector assigned to the points P 6 6 and Q 6 5 by the vector field F x x F P F Q

1 point Evaluate in spherical coordinates the triple integral of f p 0 cos over the region 0 0 2 0 x 6 1 p 4 integral

12 cos 0 4 7 tan 0 0 Find sin e

Let u 1 5 4 and v 6 5 8 Evaluate u v and 3u v u v 0 0 0 Simplify your answers

Find two vectors parallel to v of the given length V v 9 12 0 length 30 The vector in the direction of v is D

19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 and 36 Sketch the region enclosed by the given curves and find its area

3 Find an equation of the line tangent to the graph of f x 2 at 7 7 X The equation of the tangent line is y 31x

Let f x 6 tan x 2e What is f 7 h

6 Sketch the graphs of the following equations and identify the surface a x 16y 42 b 4y x 2

green 16 Find the equation of the line parallel to x 1 and passing through 4 5

Compound Inequalitites Solve each compound inequality and gra 3x 4 34 and 7 x 14

1 point Find the volume of the solid bounded by the planes x 0 y 0 z 0 and x y z

1 point Consider the following integral Sketch its region of integration in the xy plane 1 180x y dx dy a Which graph shows the region of integration in the xy plane b Evaluate the integral A Click on a graph to enlarge it B

19 sinx 2sin x cosx 0 for 0 x 2n 58

1 point We have a rectangular piece of cardboard that is 33 centimeters by 44 centimeters We form a box by cutting squares of sidelength x from the corners then fold up the four flaps What are the possible values for x the side of the small square Assume that the answer is a closed interval help intervals Suppose we remove squares of sidelength 5 centimeters from the rectangle What is the volume of the box we form help numbers Find a formula which gives the volume as a function of X V x help formulas

Question 3 1 point 4 Listen Which of the following is a vector quantity Energy Velocity Mass Speed Question 4 1 point E4Listen The diagram shows a cube where AB AD Vand AE w

Estimate the angle to the nearest one half radian radians

Find the length of the arc on a circle of radius r intercepted by a central angle 8 Round your answer to two decimal places Radius r Central Angle 8 15 inches in 300

A research study is conducted to examine the relationship between reading ability and math ability in a sample of public school children kindergarten grade 12 They believe that students with higher scores on a reading test will also have higher scores on a math test Which of the following is likely a confounding variable

O Find the equation of a plane that Passes through 1 4 1 and has normal vector line x 4 t y 4t that is parallel to the 2 t a

Factor completely If the polynomial cannot be factored say it is prime 9x 18x 8 9x 2 x 4 3x 2 3x 4 prime 3x 2 3x 4

cal values should be exa Let lim f x L Write the formal definition of what this means HIP

Consider the parallelepiped with adjacent edges u 3i 2j k v i j k wi 3j 2k Determine the angle between u and the plane containing the face determined by v

Two sprinters of equal mass leave the starting blocks with the following horizontal and vertical force vectors Runner Horizontal Vertical 150N 300N 200N 250N Runner 1 Runner 2 Newton s Second Law states that force is equal to mass times acceleration F ma a Draw a diagram for each runner showing the horizontal vertical and resultant forces b Which runner has the greatest acceleration out of the blocks c How does your answer to b relate to the diagrams in a

Consider a first fundamental form of the form ds a Calculate your Christoffel symbols b Calculate its Gaussian curvature dr f r de

Find the position vector of a particle that has the acceleration a t ti e t j e t k with initial velocity and position v 0 k and r 0 j k

Increasing 6 5 4 3 2 1 5 3 2 1 3 4 A C 3 3 3 3 1 2 3 5 6 x B 3 3 D 3 3

Identify the theorem or postulate that is related to the measures of the angles in the pair and find the unknown angle measur m ACD 5z 25 m BDC 25z 35 B Alt Int zs Thm m ACD 45 mzBDC 135 Same Side Int zs Thm mcACD 135 mzBDC 135 O Alt Int cs Thm m ACD 135 mzBDC 45 Same Side Int 25 Thm m ACD 45 mzBDC 135

Given the value domain by R 2 x 4 y x 2

Find the arclength of the curve r t 7t 2 7 t Int for 1 t 3 L

1 point Given that the acceleration vector is a t 1 cos t 1 sin t 2t the initial velocity is v 0 1 0 1 and the initial position vector is r 0 1 1 1 compute A The velocity vector v t B The position vector r t i i Note Your answers must be functions of t e g 5 cos 2t j j

1 point For the curve given by r t 8t 7t 1 8t Find the derivative r t Find the second derivative r t Find the curvature at t 1 K 1 2