Vector Calculus Questions and Answers

Find the angle between vectors u 2 3 and v 2i 5j

3 Josh is swimming in a lake at a speed of 4 feet per second in the direction of N 24 W Find the magnitude of the a horizontal component and b vertical component

7 If F x y z P x y z i Q x y z j R x y z k is a vector field on R and the partial derivatives of P x y z Q x y z and R x y z all exist then the curl of F is the vector field on R defined by OP y and the divergence of F is the function of three variables defined by curl F R 5 y Q R 302 OP 32 i ap Q R r y z Solve the following problems div F x y z j Q 00 k a A vector field F x y z is irrotational if curl F is equal to zero Find constants a b c so that F x 2y az i br 3y z j 4x cy 2z k is irrotational b Show that F 6ry z i 3x z j 3xz y k is irrotational c A vector field F x y z is soleinoidal if its divergence is zero Is F 3y z2 i 4x z j 3r y2 k soleinoidal d For which values of a the vector field F a x y i ay xz j xy 6z k is soleinoidal

Find the angle between the given vectors to the nearest tenth of a degree 1 3 110 7 108 4 112 5 O 71 6 2 3

Find a b a 4i 8j b 2i 3j O 32 O 16 8 24 Ka

Use the dot product to find Ivl V 5 1 O 26 O 4 O 2 6 O 6

Consider the following equation of an ellipse x 4 v 1 36 16 Step 3 of 3 Find the coordinates of the two foci of this ellipse Round your answer to two decimal places Answer Enter the coordinates to plot points on the graph 1 1514 Enable Zoom P

Given the following vectors calculate 10u 6v nswer 3 6 v 8 3 U

To find the equation of a line in 3D what do you need Check Answer Save Answer format a b c Check Answer Save Find the point associated with x y 2 3 on the surface generated by x y z where z f x y 2 3 22 1 0 16 Step By Step Example Answer format a b c Check Answer Save Step By Step Example Live Help Step By Step Example For fixed y 3 the tangent vector direction vector is given by the derivative of x y f x y with r Live Help D Live Help What is the equation of the tangent line to the trace of y 3 2 1 3 22 167 Slope of the line in traditi A direction vector A vector perpendicular to A point on the line CINE Answer format a bt c dt e ft where a b c d eandf are constants and is the parameter

Let F x y z 4x yzi 10j 2ryk and a smooth curve C given by the line segment from 1 2 0 to 2 0 1 Find ind 7 47 Find the parametric equation in parameter for the line segment starting from point 1 2 0 to 2 0 1 and call it 7 1 X Answer format a b c

Use Stoke s theorem to evaluate Find curl F x 3 Answer format a b c curl F 5 for the vector field 7 x y z yi zj x k for the surface S generated by the intersection of y z 3 and the cylinder 7 1

Fitzwilliam is buying a new car The sticker price for the base model is 10 000 but the options he wants will cost him an extra 15 He then talks the dealer into giving him 10 off Fitzwilliam has also decided to put down a 25 down payment on the car Find the that Fitzwilliam needed Fitzwilliam has three options for financing for the balance on the car the dealership the bank or his parents o Through the dealership he can get a 60 month 5 APR monthly installment loan o The bank will offer Fitzwilliam a 5 5 APR monthly installment loan for 48 months o His parents would loan him the money and he would need to pay it all back after 15 months with 20 simple interest

find a Cartesion equation for r 6 0050 Graph it

Given F yi xj 0k and parametric surface 7 u v ui v u v k F a5 F ds F dS S S for 0 u 2 and 0 y 1 find

Use Stoke s theorem to evaluate Find curl F x y z x y 1 Answer format a b c curt F curl d5 for the vector field F x y z yi j rk for the surface S generated by the intersection of y z 3 and the cylinder x y 1 x

Use Stoke s theorem to evaluate vector field F x y z yi rk for the surface S generated by the intersection of y z 3 and the cylinder r y 1 Find curl F x y z y z curl F d S for the X

arameterize the flow line of the vector field F 1 2 3 that passes through me point 2 2 3

Lxp xy ds where C is the first quadrant are of the unit circle x y

If you began earning 45 000 per year with a 1 8 raise every year what would your gross earnings be after three years Round your answer to the nearest cent and do not include a dollar sign in your answer

5 On a dice the sum of the dots on opposite faces is always 7 Draw dots on the three could fold up to make a dice sinag to show good birlw 5 empty

x y iven the equation 1 determine the foci the vertices and the asymptotes of the assor 9 36 yperbola 10 pts

11 Let 2 3 a b Determine w 2 and give a geometric interpretation of w in terms of v 3 pts Determine wll 2 pts

Convert the polar equation r 8 tan 0 to rectangular coordinates 5 pts

A wagon is pulled along a flat sidewalk by exerting a force of 25 pounds on the handle at an angle of 45 respect to the sidewalk How much work is done in moving the wagon 75 feet 6 pts

Given that the angle 0 is in Quadrant II and tan 0 3 determine the values of the other 5 trigonometric functions 2 pts each sin 0 cos 8 cot 0 sec 0 csc 8

Bonus A bridge is built in the shape of a semielliptical arch The bridge has a span of 60 feet and a maximum height of 20 feet Find the height of the arch a at distances of 10 and 20 feet from the center 10 pts

x 19 Determine the partial fraction decomposition of x 4 x 1 x 3 8 pts

Convert the point 2 4 to polar coordinates where angles are given in radians Round answer to 1 decimal place 2 pts

Many states are carefully considering steps that would help them collect sales taxes on items purchased through the Internet How many randomly selected sales transactions must be surveyed to determine the percentage that transpired over the Internet Assume that we want to be 99 confident that the sample percentage is within two percentage points of the true population percentage for all sales transactions Round up to the nearest integer www

Solve the following elementary exponential equation Answer 32 4 4x 1

Compute the directional derivative of the function g x y sin x 3y at the point P 1 3 in the direction of 13 Be sure to use a vector for the direction vector Show all steps

5 points ind the domain of the vector valued function 97 3447 Show all steps Express your answer in interval notation

Find two vectors orthogonal to the plane formed by a 4 5 1 and 2 0 3 Show all steps

bints ulate the work done pulling a sled 20 m along horizontal ground with a constant force of 10 N at an angle of 55 above the horizontal Show all step

id the center vertices and foci of the hyperbola 2 16 223 25 center vertices foci x y x x x y x y x y 10 Find the asymptotes of the hyperbola Enter your answers as a comma separated list of equations 10 Sketch its graph using the asymptotes as an aid Use graphing utility to verify your graph y y 10 10 10 10 10 10 10 y smaller x value 10 larger x value LH 5 smaller x value 10 larger x value 10 10 5 y 10 10 10

Determine the equation of the ellipse with foci 2 18 and 2 6 and vertices 2 19 and 2 7

6 3 0 A 1 2 911 B X TO 5 2 7 Determine the dimensions of AB

4 Find the power series representation for the function and determine its radius of convergence x 1 3x f x

Find a polar equation for the hyperbola with its focus at the pole and vertices 18 0 90 Hint These functions are plotted in polar coordinate system 0 and 7 refers to angles

Write the polynomial f x that meets the given conditions Answers may vary 2 4 3 Degree 3 polynomial with integer coefficients and zeros of f x 8 and 5

1 2 3 PL K 111 LLL Evaluate the following integrals using the Cauchy integral formula a 2 dz m n 0 e z 1 mdz f f 2 1 1 5 4 2 1 Log dz b dz n20 g h sin z dz z cosh z c d 21 123 z 1 dz Solution We rearragnde the Cauchy integral formula Let D be bounded domain with piecewise smooth boundary D and let f be an analytic function defined in a domain which contains D Then for any zo E D and positive integer m we have D f z z dz dz m c The nominator has one zero at z 0 inside z 1 thus by by Cauchy s theorem sin z 1 1 z dz z 1 z 2 4 e 2 1 222 22 4 e dz dz 1 1 z z 4 e 2 i 2 i dm m1 dzm f 2 i dz sin 2 0 0 g The nominator has one double zero at z 0 inside z 1 thus by by Cauchy s theorem 2 20 h The nominator has one zero at z 0 and one zero at z 2 inside z 1 2 thus by by Cauchy s theorem 2 i d 1 2 2 1 dz 1 1 2 i 1 1 i i

How many radians is 288 Give the exact answer in simplest fo radians

A Buiding contractor is buiding a play ground and Wishes to put down rubber mulch The mulch will go in a rectangular pit 27 feet long 15 feet wide and 6 inches deep If the muich cost 10 per cubic feet what will Cost of the mulch bez

in a 10 A relative frequency distribution is computel by frequency

1 What digit will be in the one place C Precuen d

The area of a regular 200 dode cagon with a side 2 A lengths is given by A 38 what is the approximate area tv of a regular dode cagon with a Side length of S 5 inches Bound your answer to the

A date set has the following five number Summary min 4 Q1 15 median 20 Q3 25 max 28 which of the following provides an estimate of the standard deviation of the date Set Round your answer to the nearest to tenth A 54 8 7 4 4 0 14 8 5 0

In AHIJ j 690 cm i 900 cm and I 15 Find all possible values of ZJ to the nearest degree

5 Parallelogram PQRS with vertices P 3 4 Q 0 3 R 1 8 and S 4 9 a translation along the vector 7 0 b 90 counterclockwise rotation about 3 2

Preview Activity 4 1 1 Suppose that a person is taking a walk along a ong straight path and walks at a constant rate of 3 miles per hour a On the left hand axes provided in Figure 4 1 1 sketch a labeled graph of the velocity function v t 3 8 4 mph 1 2 hrs 8 41 miles 1 2 hrs Figure 4 1 1 At left axes for plotting y v t at right for plotting y s t Note that while the scale on the two sets of axes is the same the units on the right hand axes differ from those on the left The right hand axes will be used in question d b How far did the person travel during the two hours How is this distance related to the area of a certain region under the graph of y v t c Find an algebraic formula s t for the position of the person at time t assuming that s 0 0 Explain your thinking d On the right hand axes provided in Figure 4 1 1 sketch a labeled graph of the position function y s t e For what values of this the position functions increasing Explain why this is the case using relevant information about the velocity function V