Vector Calculus Questions and Answers

1 point If r t cos 5t i sin 5t j 6tk computer t and r t dt with C a constant vector i i j j

Use cos t and sin t with positive coefficients to parametrize the intersection of the surfaces x y2 25 and z 2x r t 3 3

1 point Find the cross product cx d where c 3i j k and d 4i 4j 2k cxd Entoring Vectors htmal

Find an equation of a plane containing the three points 5 2 5 6 3 6 6 4 8 in which the coefficient of x is 9 0

1 point Find the vector and parametric equations for the line through the point P 4 3 3 and orthogonal to the plane 2x 3y 4z 0 Vector Form r 3 1 Parametric form parameter t and passing through P when t 0 x x t y y t z z t 4

1 point Determine whether the lines and 4 11 3 4 2 17 2 Point of intersection x 11 5 y 10 z 24 10 4 intersect are skew or are parallel If they intersect determine the point of intersection if not leave the remaining answer blanks empty The lines L 7

Consider the line which passes through the point P 5 3 2 and which is parallel to the line x 1 1t y 2 1t z 3 3t Find the point of intersection of this new line with each of the coordinate planes xy plane xz plane yz plane 5 3

1 point Rewrite the vector equation r t 3t i 1 5t j 4 3t k as the corresponding parametric equations for the line x t y t z t

The profit Pix that Rivka s Delivery Service makes from delivering packages between local businesses is Pix where x is the number of 0 10 price increases for each delivery Use the graph below to answer the question profit 40 number of price increases

Textbook Videos Given the information below estimate the total distance travelled during these 6 seconds using a left endpoint approximation time sec velocity ft sec 0 1 2 3 4 5 6 20 36 37 23 6 2 15

The radius of a cone is increasing at a constant rate of 2 meters per minute and the volume is increasing at a rate of 333 cubic meters per minute At the instant when the radius of the cone is 8 meters and the volume is 170 cubic meters what is the rate of change of the height The volume of a cone can be found with the equation V r h Round your answer to three decimal places

Using a compass and a straightedge Dx XY Using a compass and a straight edge D x 4 XY X Y What can be concluded about XY and its image

Please do in 20 minutes Determine whether B 1 3t 2 t 5t 1 2t is a basis for P

f o t is an increasing function of t What can you say about the angle between the gradient Vf and the velocity vector o

Table for I by I sandstone tile Edge length of laid tile in feet x 1 2 3 4 5 Area covered sq ft x2 1 4 9 16 25 Number of samistone tiles y 1 4 9 16 25 Table for Edge length of laid tile in feet x 1 2 3 4 5 1 by natural wood tile Area covered sq ft 1 4 9 16 25 Number of natural wood tiles y 4 16 36 64 100 2 Compare the data in the tables for the sandstone tile and the natural wood tile a How are the data values numbers of tiles needed different for the same x value edge length of laid tile b According to this pattern how many natural wood tiles will Carly need for a 10 foot by 10 foot floor c Write a function rule that represents the relationship between the edge length in feet and the number of natural wood tiles

arty and Rodrigo are considering three different types of tile for their floors First Choice Type Sandstone Size 1 foot by 1 foot Second Choice Type Natural Wood Size foot by foot 1 The grid shown represents a 10 foot by 10 foot room Make a sketch and complete the table to show the relationship between the edge length of the area to be tiled and the number of 1 2 by 2 foot wood tiles needed to tile it Third Choice Type Granite Size 2 feet by 2 feet Edge length of laid tile in feet Area covered sq ft foot Number of natural wood tiles

A solid region E lies within the cylinder x z 1 above the plane y z 3 and below the cone y 5 x z a Draw the diagram using mathematical application GeoGebra etc 3m b Draw the projection of diagram using mathematical application GeoGebra etc 3m

Design component Number of options Primary color 8 Secondary color 8 Sole color 8 Lace color 5

Evaluate the following as true or false Because Arectangle L x W W x L it doesn t matter if we use the X easy method or the Y easy method Select one a false h true

1 4 X 5 S 6 7 A certain test is designed to measure the satisfaction of an individual with his her relationship Suppose that the scores on this test are approximately normally distributed with a mean of 55 and a standard deviation of 9 An individual with a score of 45 or less is considered dissatisfied with his her relationship According to this criterion what proportion of people in relationships are dissatisfied Round your answer to at least four decimal places 11 8

1 Prove Hall s Marriage Theorem and provide an original example where Hall s condition holds and and an original does not hold respectively 2 For the following graph find all shortest paths from Node 1 to Node 5 using labelling method with depth first breadth first and Dijkstra schemes Explain how dynamic programming can be used for this task Explain in general the advantages and this advantages of these four methods 1 2 counterexample 1 2 3 1 3 0 4 0 5

estion 6 10 marks Given the vector function F 2 v 0 v S1 Compute the surface integral 2 and S be the surface of the unit cube F n dA using the theorem of divergence

and intercepts of f b Find all intervals on which f is increasing decreasing and extremum points

Thas the coordinates 6 3 state the bordinates of T after the following composition of transformations R180 Tyx T x 4 y 3 O

For the following system what is the correct corresponding matrix x z 6 y 2z 5 3x y 4

21 Which polar coordinate is equivalent to 3 5 a 3 11 b 3 117 c 3 57 3 17

10 Given vectors u and v find 4v 5u U 3 5 a 3 15 b 21 33 C 1 3 gib bro obutinopom erit nevig neriw ofbev ert to mol tnenogmoo erit OS V 1 2 d 11 17 90 2021 9 S81 2

Consider the following quadratic function f x 3x 18x 26 a Write the equation in the form f x a x h 2 k Then give the vertex of its graph Writing in the form specified f x 10 240 X Vertex b Graph the function To do this plot five points on the graph of the function the vertex two poin of the vertex Then click on the graph a function button 36 12 10 18 4 6 COL 00 12 X 9 0 20 E

6 Write the standard form equation of the ellipse Vertices 8 20 8 6 Dec 2521922 8 12 8 2 la 6 id 6718193 4pts 8 Write the standard form equation of the parabola Vertex 0 8 Focus 1 8

23 x y term containing yu A 25 200 x 6y 4 B 6 x 6y4 Z10x by 4 D 151 200x by 4

A taxi driver buys a new car for 50 000 The car will have a scrap value of 10 000 after 8 years Using the straight line depreciation method find the value of the car after 4 years A 10 000 B 13 000 C 15 000 D 25 000 E 30 000

2 Let R be the region enclosed by the graph of y 1 ln cos x the x axis and the lines x 3 and x 3 The closest integer approximation of the area of R is A 0 B 1 C 2 D 3 E 4

Verify the identity cos 2 0 2 sec 0 1 2 sec 0 Use the appropriate half angle formula and rewrite the left side of the identity Simplify your answer Rewrite the expression from the previous step by multiplying the numerator and denominator by sec 8 Multiply and distribute in the numerator Do not simplify The expression from the previous step then simplifies to A Even Odd Identity O B Quotient Identity point s possible OC Pythagorean Identity OD Reciprocal Identity sec 0 1 2 sec 0 using what

3 Let D C a b Co a b be D f f Show that D is not continuous in the sup metric

Write a general term an for the sequence Assume that n begins with 1 21 2 5 12 19 26 A an 7n 10 B an 7n 9 C an 7n 9 D an 9n 9 2

uestion 4 6 marks Use Gaussian Quadrature with 4 steps to approximate In 1 x dx to 6 decimal places

15 Which of the following has the best price per unit 1 pt Brand B 6 00 for 3 Brand C 8 00 for 3 Brand A 5 00 for 2 Brand C 7 00 for 7

Write the equation of the conic section shown below 10 6 4 3 2 1 10 9 8 7 6 y 5 4 3 1 2 13 4 1 2 3 8 9 10

Which equation choice could represent the graph shown below 10 9 87 6 5 4 3 2 1000 950 900 850 800 750 700 650 600 550 500 450 400 350 300 250 200 150 100 150 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 Of x x 7 x 8 850 900 950 1000 y 2 3 4 5 6 7 8 9 10 O f x x 7 x 8 Submit

Find the product of the following two matrices 3 0 0 1 1 2 1 Rows 2 Columns 2

What is the degree of the polynomial f x defined below f x 6x 3x 6

Find all holes of the following function Write your answer as a coordinate point in simplest form If no hole exists press the icon to indicate there is no solution 3x 108 x 2x 24

Consider the equation 4x 9y 32x 90y 197 0 a Rewrite the equation in standard form and identify the conic

Find all vertical asymptotes of the following function f x x 8x 20 2 3x 9x210

For which values of x is the expression undefined x 4x 81 12

Use a half angle formula to find the exact value of the following expression sin 165 sin 165 Simplify your answer including any radicals Use integers or fractions for any numbers in the expression Rationalize the denominator

Write the following expression as the sine cosine or tangent of a double angle Then find the exact value of the expression 2 sin 22 5 cos 22 5 Write the following expression as the sine cosine or tangent of a double angle Select the corect choice below and fill in the answer box to complete your choice Simplify your answer Type your answer in degrees Use integers or decimals for any numbers in the expression OA 2sim 22 5 cos 225 cos OB 2sin 22 5 cos 22 5 tan OC 2sin 22 5 cos 225 sin

In Exercises 59 to 64 find the equation of the graph that passes through the given points of the g Susqunuenos odi tudi enam ni 59 Equation of a Plane Find an equation of the plane th passes through the points whose coordinates are 1 0 2 1 5 and 1 1 1 Suggestion The equation of plane can be written as z ax by c

8 Convert from general form to standard form for 2x 2y 16x 8y 10 0 x 4 2 2 1 a 15 15 b x 2 y 4 1 15 15 C x 4 y 2 d 30 30 x 2 y 4 30 30 1 1 I

4 Use the law of sines or cosines to solve the triangle a B 116 a 7 6 c 4 b B 116 a 4 c 7 6 c B 139 a 5 9 c 9 2 d B 139 a 9 2 c 5 9 A B 43 10 2