Vector Calculus Questions and Answers

The vector w ai bj is perpendicular to the line ax by c and parallel to the line bx ay c It is also true that the acute angle between intersecting lines that do not cross at right angles is the same as the angle determined by vectors that are either normal to the lines or para to the lines Use this information to find the acute angle between the lines below 2x 5y 2 7x 3y 9 The angle is radians Type an exact answer using x as needed SCIE

Give a geometric description of the set of points in space whose coordinates satisfy the given pair of equations 9 y 8 Choose the correct description OA The circle with center 0 8 0 and radius 3 parallel to the xz plane OB The circle with center 3 0 3 and radius 8 parallel to the xy plane OC The line through 3 8 3 parallel to the y axis OD The line that passes through the points 3 0 3 and 0 8 0

Let u 6 1 and v 5 5 Find the component form and magnitude length of the vector 2u 2v The component form is 2u 2v Simplify your answers

3 FIND THE REDUCED ROW ECHELON FORM OF THE BELOW MATRIX USING AND DEMONSTRATING OTHERWISE YOUR SOLUTION ALL ELEMENTARY ROW OPERATIONS WILL BE INVALID 4 0 3 7 6 9 9 9 7 5 10 19 1 2 4 1

Using the graph of f complete the following a y b 4 3 2 1 1 3 2 1 2 12 1 Determine whether dy dt is positive or negative given that dx dt is negative O positive O negative 1 3 ii Determine whether dx dt is positive or negative given that dy dt is positive Opositive negative 1 14 2 X 3 X 1 Determine whether dy dt is positive or negative given that dx dt is negative Opositive negative i Determine whether dx dt is positive or negative given that dy dt is positive

S s 4 s 5 Show Transcribed Text Find the Inverse Laplace transform of the above function

Consider the equation and the given point f x x 4 x 8 1 27 a Find an equation of the tangent line to the graph of f at the given point y b Use a graphing utility to graph the function and its tangent line at the point Use the tangent feature of the graphing utility to confirm your results y y 2 10 20 30 40 50 2 6 X 6 2 50 40 30 20 10 2 X 6 O 2 y 10 20 30 40 50 2 X O 2 y 50 40 30 20 10 2

Find the equation for the plane through Po 7 5 2 perpendicular to the following line x 7 t y 5 5t z 2t o t Choose the correct answer below OA x 5y 2z 9 OB 7x 5y 2z 14 OC 7x 5y 2z 9 OD x 5y 2z 14

Find a parametric equation for the line through the point 9 0 1 and perpendicular to the plane 4x 3y 2z 11 Find a parametric equation for the line Choose the correct answer below OA x 9 4t y 3 z 2 t OB x 9 4t y 3t z 1 2t OC x 4 9t y 3 z 2 t OD x 4 9t y 3t z 2 t

Find the angle between the vectors u 5i 6j v 4i j 2k radian s The angle between the vectors is 0 Round to the nearest hundredth as needed

Find the following for the vectors v 2i 3j 6k and u 7i 24j a v u vl Jul b the cosine of the angle between v and u c the scalar component of u in the direction of v d the vector proj u a v u Type an exact answer using radicals as needed

Find the center and radius of the sphere x 2 y 2 2 2 2 The center of the sphere is Type an exact answer using radicals as needed www

Determine the equation for the sphere whose center and radius are given Center Radius 5 3 4 18 Choose the correct equation of the sphere O A 2 2 x 5 y 3 2 4 18 2 2 OB x 5 y 3 z 4 18 OC x 5 y 3 z 4 18 2 OD x 5 y 3 z 4 18

Let C be the boundary of the region enclosed by the parabolas y x and x y Then 2 3y 2 cos r dr 6x 3e v dy

rewrite the rational expression so it has the given denominator D 1 x 2 D x 2 4

a Find an expression for f w in terms of unit step functions u w b The inverse Fourier transform of f w is where F x and G x Use I for the imaginary unit i in Mobius TO Ay IMH FO F 1 1 0 F w 16 ww i 2w 4w w 4 w 4 F x sin 4x G x cos 4x

Find an equation of the parabola with focus 7 4 and directrix x 11 0 00 0 0

Find an equation of the circle that has center 2 1 and passes through 1 5 0 8 Questi

6 Find an equation of the line that passes through the point 12 4 and has the slope m that undefined

Find direction numbers for the line of intersection of the planes x y z 6 and x z 0 Enter your answers as a comma separated list

a Find parametric equations for the line through 4 5 6 that is perpendicular to the plane x y 3z 2 Use the parameter t x t y t z t b In what points does this line intersect the coordinate planes xy plane x t y t z t x t y t z t x t y t z t yz plane xz plane

Consider the figure below Find lux vl u 6 Ointo the screen 150 v 9 Determine whether u x v is directed into the screen or out of the out of the screen

Let v 2j and let u be a vector with length 9 that starts at the origin and rotates in the xy plane Find the maximum and minimum values of the length of the vector u x v maximum minimum 4 In what direction does u x v point When u is pointed in the positive x direction u x v is pointed in the positive z direction When u is pointed in the negative x direction u x v is poin

If a 2 0 2 b 3 2 2 and c 0 2 4 show that a b c a x b x c a x bx c 4 4 4 a x b x c x

Find tan 1 if csc u 6 882 and u is in Quadrant III 2 tan 2 Your answer should be accurate to 4 decimal places

Using the appropriate theorem evaluate the surface integration f curl F ds given F x y z y cos z i y z j xk Where surface S is the portion of the paraboloid z 2 x y that lies above the xy plane The curve direction is counterclockwise when viewed from the top

9 Note You must not use vector components to solve this question You must use the Law of Sines and the Law of Cosines 6 marks The air velocity of a jet is 400 km h S 40 E A 70 km h wind is blowing from N20 W Determine the ground velocity of the jet Include a proper fully sketched diagram with the vectors arranged properly and show all your steps

Calculate the number of possibilities Planting a red rose bush a yellow rose bush a white rose bush and a pink rose bush in a row in a planter

S 2 4 4 4 4 3 Real axis Part A Identify the points in standard form and find the distance between them Part B Give the complex conjugate of z and explain how to find it geometrically Part C Find 2 Z1 geometrically and explain your steps

12 6pts Find all zeros of the polynomial P x x 3x 4x 12

9 10pts Use identities to find the exact value 3 8 a sin

The Leaning Tower of Pisa was completed in the year 1372 and originally stood perfectly perpendicular 186 5 feet high At a distance of 127 feet from the base of the tower Galileotis measures the angle of elevation to the top of the tower today to be 64 0 What is the leaning angle of the tower now i e what is angle RPQ Give your answer to 1 decimal place 186 5 ft R 64 127 ft Q

manda wants to save 7500 over the next three years to help pay for her first year of college How much would she have to deposit at the end of every 6 months in an account The Ccount pays 7 per annum compounded semi annually or the toolhar press ALT F10 PC or ALT FN F10 Mac

Rene buys a computer system for 80 down and 18 monthly payments of 55 each The first payment is due next month The interest rate is 15 per annum compounded monthly What is the selling price of the computer system For the toolbar press ALT F10 PC or ALT FN F10 Mac

QUESTION 4 Rachel needs 5900 per year for 4 years for university tuition She will start university in 1 year She can invest her money at 6 1 compounded annually How much will an ann cost For the toolbar press ALT F10 PC or ALT FN F10 Mac E EV A k T Q

2 points Shawn deposits 750 at the end of every three months in his account earning 4 2 annually compounded quarterly What will be the balance in the account after 4 years For the toolbar press ALT F10 PC or ALT FN F10 Mac

Without using a calculator find all solutions to the following equation in the interval 0 2 2sin u 9 sinu 5 0

et f x 3x 15x 1 xpress the function in the form f x a x h k by completing the squar dentify the vertex

0 2 4 6 8 10 12 Number of correct challenge questions Which of the following is a typical number of challenge questions one member answered correctly Choose 1

The graph of the function y f x is given below Find the domain of f x Select the correct answer below 0 5 3 2 3 21 10 0 1 2 3 01 2 4 5 6

is shown in blue Draw the graph of y f x shown in red by determining how a stretch or compression will transform the location of the indicated blue points Drag the movable red points to the desired coordinates You may click on a point to verify its coordinates Provide your answer below 10A 8 9 7 6 5 4 3 2 1 1 2 4 6 3 4 5 6 8 6 8 9 10

Siraph the piecewise function below by dragging the endpoints of the lines to the correct locations then select whether endpoints are included in the graph or not Provide your answer below 8 Included 8 Included f x 3 x if x 0 if x 20

1 Sketch a graph of a function f that satisfies the given values there are many correct answer f 2 0 f 2 0 lim f x 0 lim f x does not exist x 2 x 2 ty

Given the graph of y f x below solve for f x 2 If you have multiple answers enter your answer in the form x a b 6 5 4 3 2 1 1 4 10 N 4 5 6

8 Why is the derivative 1 2 x 1 equal to arcsin x 1 1 1 1 1 1 2 x 8 3 2 x 8 4 2 x 1 x cos x 2

If fis undefined at x c then the limit as of fas x approaches c does not exist True False Question 2 1 point Let f x 1 x2 Select all that apply fis undefined at x 0 The limit of f as x approaches O is infinity fhas a horizontal asymptote at y 1 fhas a vertical asymptote at y 0 fhas a vertical asymptote at x 0 The limit of fas x approaches 0 is negative infinity

Find k such that the line is tangent to the graph of the function Line Function f x k x y 4x 4 k

Find the equations of the two tangent lines to the graph of f that pass through the indicated point f x 8x x y 18 16 14 12 10 10 line with positive slope y line with negative slope y 4 17 Ho

What is the value of the first term for the geometric series described T an Sn a1 4 9 1024 7371 1024

Divide Show your complete solution 1 x 4x 12 by x 2 2 x 5x 6 by x 3 3 x x 20 by x 4 4 x 7x 10 by x 5 2 Ov V 19 bu