Vector Calculus Questions and Answers

Use the formula for the general term the nth term of a geometric sequence to find ag with the given first term a 6 and common ratio r 2 OA 1 536 OB 1 532 OC 22 OD 3 072

Approximate the number using a calculator Round your answer to three decimal places 2 6 OA 6 000 OB 5 462 OC 4 899 OD 32 000

Find the zero of the function shown below f x 2 2x 3x 4 8 x 4 O A OB 12 5 5 C 4 OD 4

5 The surface area of a cube is equal to the sum of the areas of each of its six square faces The formula to find the surface area of a cube is A 6x where x is the side length of one of the square faces Jack has two cubes that he needs to paint One of them has a side length of 2 inches and one has a side length of 1 5 inches How many square inches does he need to paint

Find the value of the summation 3 2k

Find the period in degrees Then sketch the graph using degrees 54 y tan 0 55 y sin 0 60 120 180 240 300 360 90

For the rectangular coordinates 2 2 2 2 a plot the point and b give two pairs of polar coordinates for the point where 0 0 360 K a Choose the correct plot of 2 2 2 2 below OA Ay B Ay COL O C b Give two pairs of polar coordinates for the point where 0 0 360 Type an ordered pair Simplify your answer including any radicals Use integers or fractions for any numbers in the expression Type your answer in degrees Do not include the degree sym Use a comma to separate answers as needed

For the point a Choose the graph that represents the point 1 1 37 OA x 2 in polar coordinates a plot the point b give two other pairs of polar coordinates for the point and c give the rectangular coordinates of the point 4 3x 2 Q B x 2 3x 2 0 Q Q OC x 2 3x 2 0 Q b What are the coordinates of the point for which r 0 and 2x 0 4x 11x 4 Type an ordered pair Simplify your answer Type an exact answer using as needed Use integers or fractions for any numbers in the expression What are the coordinates of the point for which r 0 and 0 0 2x Type an ordered pair Simplify your answer Type an exact answer using as needed Use integers or fractions for any numbers in the expression OD R 2 3x 2 Q

5 Determine the value of x such that the points A 3 5 B x 7 C 8 10 and D 0 2 form the parallelogram ABCD Hint The order of the letters matters

Watch the video that describes plotting points with polar coordinates Click here to watch the video Plot the point 3 240 Then find the rectangular coordinates of the given point In rectangular coordinates 3 240 is Type an ordered pair Simplify your answer including any radicals Use integers or fractions for any numbers in the expression Choose the graph OA 180 180 90 270 90 270 0 Q Q OB below 180 180 90 270 90 270 0

4 Given u and 7 draw the following vectors a u b 20 V V c u v d u v V

calculator a Choose the graph that represents the point A polar coo1 hates plot the point Give two other pairs of polar coordinates for the point Do not use a R 2 3x 2 2 3x 2 0 OC x 2 3x 2 b What are the coordinates of the point for which r 0 and 2 0 4x O D x 2 3x 2 Type an ordered pair Simplify your answer Type an exact answer using it as needed Use integers or fractions for any numbers in the expression

For the point 1 and c give the rectangular coordinates of the point 3x 4 x 2 3x a Choose the graph that represents the point 1 4 OA 3x 2 in polar coordinates a plot the point b give two other pairs of polar coordinates for the po B x 2 3x 2 Q O C x 2 3x 2 Q G O D x 2 3x 2 Q L b What are the coordinates of the point for which r 0 and 2n 0 4x Type an ordered pair Simplify your answer Type an exact answer using as needed Use integers or fractions for any numbers in the expression

11 In the following diagram OA a OB b and OC c Points P Q R and S are the mid points o OA AB BC and CO respectively Determine the following vectors in terms of a b and c a P A S b B R a PQ b QR

7 Given a 6cm b 3cm and the angle between these vectors when arranged tail to tail is 35 determine 2a b geometrically as well as the angle formed between 2a b and vector a

3 cos vcos 3v sin vsin 3v

7 cos 3vcos 5v sin 3vsin 5v

8 sin 3vcos v cos 3vsin v

4 cos 6xcos 2x sin 6xsin 2x

6 cos 50cos 20 sin 50sin 20

24 4x 20x 5 x 5

Given u and 7 draw the following vectors 4 a u b 20 V V c u v 15 v d u v V v

2 For the vectors 3 12 0 and b 5 5 7 determine the vector 2a 5b 3 For the point P 2 4 5 plot this point in R and list the coordinates of the points that form a rectangular prism with this point and the origin Y 2

12 Vector a joins the x and y intercepts tail at the x intercept and head at the y intercept of the line 2x 3y 6 Determine a as a quadrant bearing

Application 3 An airplane is on a heading of 030 at an airspeed of 800 km h There is a wind of 75 km h from 315 Using trigonometry find the resultant velocity as a true bearing of the airplane Include a diagram

1 cOSX 1 COSX fadesi a a dakilerden hangisine e equal

3 cos2x 2sin x 1 ifadesinin de eri ka t r

I 1 sin cos 1 ifadesinin sonucu ka t r

The tables to the right give price demand and price supply data for the sale of soybeans at a grain market where x is the number of bushels of soybeans in thousands of bushels and p is the price per bushel in dollars Use quadratic regression to model the price demand data and linear regression to model the price supply data Complete parts A and B below A Find the equilibrium quantity and equilibrium price The equilibrium quantity is thousand bushels Round to three decimal places as needed GIIB Price Demand X 0 10 20 30 40 Price Supply p S x 6 33 6 39 p D x X 6 60 0 6 51 10 6 45 20 6 43 30 6 40 40 6 44 6 51 6 54

Write the first five terms of the geometric sequence an 4an a 2 OA 8 32 128 512 2 048 OB 4 8 32 128 512 O C OD 2 2 6 10 14 2 8 32 128 512

Write the first five terms of the arithmetic sequence a 2 d 2 A 2 4 6 8 10 4 6 8 10 12 0 2 4 6 8 OD 2 3 4 5 6 OB O C

5 Solve the equation Do not use a calculator Give an exact value 5 points 1 4 ln 2z 1 5

7 Find the Cartesian coordinates of the polar point 2 3

dy Is there a value of n which makes y x a solution to the equation 3x y If so what is the exact value dx Enter NA if there is no value for n n

Use the definition of continuity to determine the value of the constant c for which the function fis continuous on where f y cy 8 if y 5 c y 8 if y 5

A factory selling cell phones has a marginal cost function C x 0 01x 3x 229 where a represents the number of cell phones and a marginal revenue function given by R x 429 2x Find the area between the graphs of these curves and x 0 What does this area represent O total revenue for 200 cell phones sold O total profit for more than 200 cell phones sold total profit for 200 cell phones sold O total cost for 200 cell phones sold

Graph the curves r 2 4 cos 0 and r 2 4 sin 0 Identify the correct sketch of r 2 4 cos 0 O A 8 8 OA 8 Identify the correct sketch of r 2 4 sin 0 8 Q 8 U ON OB B 8 8 A 001 Q ON AST OC O C 84 18 T Di Q Q This test 25 point s possible This question 4 point s possib OD OD B C 8 Q

a the z axis at 1 2 3 b the x axis at 2 6 1 c the y axis at 2 4 4 a Choose the correct equation of a plane perpendicular to the z axis the plane perpendicular to each of the following axes at the given points with a single equation or a pair of O A x 1 B y 2 C Z 3 b Choose the correct equation of a plane perpendicular to the x axis A Z 1 B x 2 O c y 6 c Choose the correct equation of a plane perpendicular to the y axis A y 4 B x 2 www

20 points We are given a vector field F x y z axy z Bx y where a are some nknown constants It is known that H identically and that 3 2k 2x x ind a and 3

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Prof Rudy Meangru Solve the following equ 1 sin 1 2

press 1 234234234234 as a ratio of intege

Find the volume V of the described solid S The base of S is the region enclosed by the parabola y 2 3x2 and the x axis Cross sections perpendicular to the y axis are squares V

Solve for 0 0 0 2x 2 sin 0 sin 6 31 21 5x 0 2 5 10

Given side a 24 side b 28 and side c 30 find the area of triangle AABC 0316 336 909 285

TABLE OF INDEFINITE INTEGRALS fcf x dx cff x dx fkdx kx C X 1 f xndx C n 1 dx In x C n 1 fe dx e C sin x dx cos x C sec x dx tan x C sec x tan x dx sec x tan x dx sec x C 1 x 1 sinh x dx cosh x C f x g x dx f f x dx g x dx dx tan x C ax a dx in a cos x dx sin x C fcsc x dx cot x C CSC x cot x dx csc C CSC X C 1 1 x S cosh x dx sinh x C enc dx sin x C 14 csc t 2e dt tai

Consider a 400 N weight suspended by two wires as shown in the figure Find the magnitudes and components of the force vectors F and F2 F N Round to one decimal place as needed F N Round to one decimal place as needed F Round to one decimal place as needed F F14 25 50 400 N

Find the volume of a parallelepiped with one of its eight vertices at A 0 0 0 and three adjacent vertices at B 2 1 0 C 0 5 1 and D 4 6 8 The volume of the parallelepiped is Simplify your answer

Suppose that a box is being towed up an inclined plane as shown in the figure Find the force w needed to make the component of the force parallel to the inclined plane equal to 2 2 lb The force w is about lb Round to the nearest hundredth as needed COO

Find the area of the parallelogram whose vertices are given below A 2 3 B 2 0 C 8 3 D 4 6 The area of parallelogram ABCD is square units