Vector Calculus Questions and Answers

1 2 Suppose 3 3 is the terminal point on the unit circle determined by t Find the termina point determined by each of the following a t b t B2 2 N c t d t n

Use the given vector valued function to find each of the following r t t i 2t 3 j 3t 5 k a t b 7 t c t d r t t e t x 7 t Which choice below best describes the third derivative of the given vector valued function O The third derivative is a vector The third derivative is a scalar The third derivative is a real valued function of t The third derivative does not exist

Given that u 3 4 2 and v 1 2 3 find a The vector component of u along w b The vector component of u orthogonal to v W Which of the vector components that you found above is the projection of vector u onto vector v Select the choice part a O part b

Given the matrix A shown below what is the element a 1 2 53 3

Given the matrix A shown below what is the element a21 B 13 1 3

What is the order of the following matrix Provide your answer below 2 4 2 3

Subtract the following matrices Give your answer as a matrix using the matrix tool Provide your answer below 0 4 4 4 0 3

P 67 1 2 If possible find an invertible matrix P so that D P AP is a diagonal matrix If it is not possible enter the identity matrix for P and the matrix A D You must enter a number in every answer blank for the answer evaluator to work properly D A

12 Set up but do not Evaluate Use sperical cordinates to set up the integral for the volumenof the solid bounded above by the sphere x y z 1 and below by the cone Z x y

Angle Degree Radian 0 10 15 20 30 40 45 50 60 70 75 80 90 Length of sides in decimeters Adjacent Opposite Hyp Measure Measure Measure Approx Approx Cosine Sine Adj Opp Hyp Hyp Exact Exact Cosine Sine Adj Opp Hyp Hyp XX XX X X XX XX XX X X X X

0 0 10 15 20 Degree Radian 30 40 45 50 60 Angle 70 75 One Decimeter 1 2 3 4 5 6 7 8 9 80 30 90 15 45 45 60 75 1 0 Length of sides in decimeters Adjacent Opposite Hyp Measure Measure Measure 40 Data Sheet for Measuring Triangles Approx Cosine Adj Hyp 20 10 50 Exact Exact Approx Sine Cosine Sine Opp Adj Opp Hyp Hyp Hyp XX XX XX XX XX XX XX XX C2821 OPEN Clubel Village Original Peoples Education Network Inc 501000 NOO Edualun qefit organization Th Be the home of health Fember and Semberyanbok 70 80

Determine the correct partial fraction decomposition for 9x 14x 2 5x 4 x2 x 2

Find t t and B t vectors of the following vector functions a F t sint 2 cost sint b r t e sint e cost e at t 0 10

110 101 101 101 Find and parameterize the set of all least squares solutions to this system 6 5 points Consider the inconsistent system A16 where A and b 3 6 5 4

6z 6z 140 z 5 3z 5 Determine the correct partial fraction decomposition for the expression above Provide your answer below

9 r t F t X G t where nswer 0 F t sin ti cos tj G t sin tj cos tk

7 r t F t G t where Answer TO F t cos ti sin tj tk G t cos ti sin tj

O Correct Consider a triangle ABC like the one below Suppose that c 66 a 31 and C 124 The figure is not drawn to scale Solve the triangle Carry your intermediate computations to at least four decimal places and round your answers to the nearest tenth If no such triangle exists enter No solution If there is more than one solution use the button labeled or B A 22 9 B 33 1 b 43 5 or X No solution S

Suppose you have just poured a cup of freshly brewed coffee with temperature 94 C in a room where the temperature is 20 C a When do you think the coffee cools most quickly What happens to the rate of cooling as time goes by Explain The coffee cools most quickly as soon as it is removed from the heat source The rate of cooling decreases towards 0 coffee approaches room temperature b Newton s law of cooling states that the rate of cooling of an object is proportional to the temperature difference between the object and its surroundings provided that this difference is not too large Write a differential equation that expresses Newton s law of cooling for this particular situation Use t as the independent variable y as the dependent variable R as the room temperature and k as a proportionality constant dy k dt X What is the initial condition y 0 94 In view of your answer to part a do you think this differential equation is an appropriate model for cooling The answer and the model support each other because as y approaches R dy dt approaches 0 since the so the model seems appropriate

not be able to come back to the skipped part Tutorial Exercise Express the number as a ratio of integers 0 82 0 82828282

a What is the difference between a sequence and a series A sequence is the sum of a list of numbers whereas a series is an ordered list of numbers A sequence is an unordered list of numbers whereas a series is the sum of a list of numbers A sequence is an ordered list of numbers whereas a series is the sum of a list of numbers A sequence is the sum of a list of numbers whereas a series is an unordered list of numbers A sequence is an ordered list of numbers whereas a series is an unordered list of numbers b What is a convergent series What is a divergent series A series is convergent if the sequence of Select is a Select sequence A series Need Help Read It Sub Select partial sums Select if it is not converger

Evaluating a Function In Exercises 11 and 12 evaluate the vector valued function at each given value of t 11 r t i t 1 j a r 1 Answer b r 0 Answer c r s 1 Answer d r 2 At r 2 Answer

Finding the Domain In Exercises 3 4 5 6 7 8 92 and 10 find the domain of the vector valued function 3 r t i j3tk Answer 4 r t 4 t i t j 6tk 5 r t ln ti e j tk Answer 6 r t sin ti 4 cos tj tk 7 r t F t G t where Answer F t cos ti sin tj tk G t cos ti sin tj 8 r t F t G t where F t ln ti 5tj 3t k G t i 4tj 3t k 9 r t F t x G t where F t sin ti cos ti G t sin ti cos tk

Find the correct partial fraction decomposition for 9y 11y 3 5y 4 y 1

Differentiation of Vector Valued Functions In Exercises 3 4 5 6 7 8 9 and 10 find r t to and r to for the given value of to Then sketch the curve represented by the vector valued function and sketch the vectors r to and r to to 3 3 r t 1 t2 i tj Answer 4 r t 1 t i t j 5 r t cos ti sin tj Answer to 1 to 6 r t 3 sin ti 4 cos tj 7 r t e e t to 0 to

fferentiation of Vector Valued Functions In Exercises 3 4 5 6 7 8 2 and 10 find r t and r to for the given value of to Then sketch the curve represented by the vector valued function and etch the vectors r to and r to 3 r t 1 i tj to

Finding a Unit Vector In Exercises 15 16 17 and 18 find a unit vector that is orthogonal to both u and v 15 u 4 3 1 v 2 5 3 Answer 16 u 8 6 4 v 10 12 2 17 u 3i 2j 5k v i j 4k

Area In Exercises 19 20 21 and 22 find the area of the parallelogram that has the given vectors adjacent sides Use a computer algebra system or a graphing utility to verify your result 19 u j v j k Answer 20 u i j k v j k 21 u 3 2 1 v 1 2 3

Finding Dot Products In Exercises 3 4 5 6 7 8 9 and 10 find a u v b u u c v d u v v and e u 3v 3 u 3 4 v 1 5 Answer 4 u 4 10 v 2 3 5 u 6 4 v 3 2 Answer 6 u 7 1 v 4 1 7 u 2 3 4 v 0 6 5 Answer 8 u 5 0 5 v 1 2 1 9 u 2i j k

43 u 9i 2j 4k v 4j 4k Answer a 0 3 3 b 9 1 1

Find the exact values of the center vertices and foci of the following equation 2 3 v 2 1 16 Choose the one best and most complete match for each Foci Vertices Center 1 3 2 2 2 3 3 3 2 4 1 2 and 7 2 5 0 2 and 6 2 6 3 1 and 3 5 7 3 2 and 3 6 3 2 7 3 7 2 8 9

Find the curl of the vector field F F 5xy z i 3yz x j 2xz y k curl F i k www

Finding the Projection of u onto v In Exercises 37 38 39 40 41 42 43 and 44 a find the projection of u onto v and b find the vector component of u orthogonal to v 37 u 6 7 v 1 4 Answer 38 u 9 7 v 1 3 39 u 2i 3j v 5i j Answer 40 u 2i 3j v 3i 2j 41 u 0 3 3 v 1 1 1 Answer 42 u 8 2 0 v 2 1 1 43 u 9i 2j 4k v 4j 4k

Finding Dot Products In Exercises 3 4 5 6 7 8 2 and 10 find a u v b u u c v d u v v and e u 3v 3 u 3 4 v 1 5

Solve for x 2 sin x 2 cos x 0 2 X

Find the curl of the vector field F F xy z i yz x j xz y k curl F i i k

0 000081 0 81 1 10000 0 01 Step 2 We have written the given repeating decimal as follows 0 81 81 1 0 01 0 0001 Notice that each term of the sequence in the parentheses can 1 0 01 0 01 b 0 0001 0 01 where a 0 01 b 0 0001 cannot come back and c be written written as a power of 0 01

Graph the curve r 6 sin 40 4 2 y 2 2 4 y 2 6 X y 2 0 4 2 2 4 2 4 X

Find the image of vif T v v V V V3 V3 v 2 1 2

40 sin x 5 15 0 2 X Select 2 correct answer s 0 5n 6 11 6 27 3 500 4x 3 1 30 Search

21 xy 0 2 3 Answer Three units below the xy plane and below either Quadrant I or Quadrant

If cost is represented by C x 2 8x 100 000 and revenues are represented by R x 5 3x which equation could be used to find the break even point Select the correct answer below 8 1x 100 000 O 2 5x 100 000 O 5 3x 100 000 O 2 5x 100 000 O 8 1x 100 000

45 9x 9y 2 9z 6x 18y 1 0 Answer X W 2 3 y 1 z 1 3 3 1 0 Center Radius 1

Finding the Midpoint In Exercises 33 34 35 and 36 find the coordinates of the midpoint of the line segment joining the points 33 4 0 6 8 8 20 Answer 6 4 7 34 7 2 2 5 2 3 35 3 4 6 1 8 0 Answer

Of the following which is most likely to improve employee motivation and satisfaction Allowing employees to set their own schedules Providing additional job training Upgrading the physical workplace with new furnishings and d cor Allowing employees to work from home

Hinding the Distance Between Two Points in Space In Exercises 25 26 27 and 28 find the distance between the points 25 4 1 5 8 2 6 Answer 3 2 26 1 1 1 3 5 3 27 0 2 4 3 2 8

75 Resultant Force Forces with magnitudes of 500 pounds and 200 pounds act on a machine part at angles of 30 and 45 respectively with the x axis see figure Find the direction and magnitude of the resultant force Answer 10 7 584 6 lb 30 45 200 lb 500 lb X

Since x 88 is always positive we can use various rules for operating with logarithms to write In x in x Inlyl In 88 C 88 In

A tire shop counted 80 invoices at the end of the month They charge 325 for a set of new tires and 50 for a set of used tires The total dollar amount of the invoices was 6 750 Let x represent the number of sets of new tires and y represent the number of sets of used tires Which system of equations represents the situation Select the correct answer below C C O O x y 325x x y 325x x y 325x x y 325x x y 325x 80 50y 6 750 80 50y 6 750 80 50y 6 750 80 50y 6 750 80 50y 6 750 B

Find the general solution of the following system of equations Select the correct answer below x 7 O x 7 O x 4x 7 O x 4x 11 O x 4x 7 O x 4 7 4x y 12x 3y 21