Vector Calculus Questions and Answers

The function below has at least one rational zero Use this fact to find all zeros of the function f x 6x 31x 4x 5 If there is more than one zero separate them with commas Write exact values not decimal approximations 0 0 0 6

Which equation is impossible to solve for x A C sin x cot x 0 B D tan x 1 1 1 sec x 0

4 Rationalize the denominator Tip multiply by a form of 1 Case 1 denominator is a monomial multiply the numerator and denominator by the nth root of factors that produce nth powers of all the factors in the radicand of the denominator Case 2 denominator is a binomial multiply the numerator and the denominator of the fraction by the conjugate of the denominator 5 Simplify each expression a 2 V7 VII Examples 1 a b 515 2 2 3 5 3 5 3 5 3 5 3 2 10 3 2 10 9 5 4 b 2 9 3 10 5

Simplifying radicals When simplifying radical expressions factor the radicand and look for factors with powers that match the index 2 Simplify the following a 50 b 28 Example 45 9 5 3 5 3 5 24 23 3 2 3 FE c V 81

An object oscillates as it moves along the x axis Its displacement varies with time according to the equation x 4 sin rt F where t time in seconds and x displacement in meters What is the displacement between t 0 and t 1 second meters

4 True or false 2 3 2 3 5 Explain

4 15 points For A 12 26 3 and a vector x 3 1 5 points Find the row space and nullspace of A 2 5 points Find the projection x of x onto the row space of A and the projection x of x onto the nullspace of A 3 5 points Explain why x x xn

12 11 10 9 8 7 6 V 5 4 3 u 2 1 1

Problem 6 Sketch a graph of 2x 7 1 x 7 What is the domain of f x What is the range of f x f x if x 4 if 4 x 1 if x 1

Write the sum using sigma notation 9 13 17 21 A A 65 B where n 1

The polar equation r 0 3 6 sin is graphed below 15 14 13 12 11 10 9 10 9 8 7 6 5 87 7 65 1 2 3 4 5 Q As this lima on is traced out from 0 0 to 0 27 what values of 0 mark the starting and ending po of the inner loop Give your answers in radians separated by a comma Separate your answers by a comma

You pick 7 digits 0 9 at random without replacement and write them in the order picked What is the probability that you have written the first 7 digits of your phone number Assume there are no repeats of digits in your phone number Give your answer as a fraction

3 Jamie s parents gave him some money at the start of the school the year He spends 15 each month on paper pens and other supplies The graph shows his money decreasing as he spends from month to month 8 marks y 100 80 60 40 20 2 4 6 8 10 x

Problem 1 late where C is the curve given by 2xyzds c t 12t 5t 84t 0 t 1

Listen a Determine the magnitude of a vector perpendicular to both a 6 0 1 2 1T 2A and 1 0 2 5 Describe a strategy from this course that could be used to prove that the vector found in part a is perpendicular to both vectors 2C

Listen Using concepts from this course explain the process you would use to type of intersection that occurs between a line and a plane 2T 2C O

4 Listen Convert the following from parametric form to scalar form 1K 1T x t 2 y 5t 4

Listen The diagram below shows a square based right pyramid State a sh to each expression 27 a AB BD b AB AD A E

Find a particular solution to the nonhomogeneous equation below given that f t e 3t is a solution to the corresponding homogeneous equation ty 3t 1 y 3y 27t 2 e 18t

s are increasing at a rate of f x 8 52x 87 24 where x 0 corresponds to the year 2000 Total expenditures were 1581 82 billion in 20 a Find a function that gives the total expenditures x years after 2000 b What will total expenditures be in 2016 a What is the function for the total expenditures F x Simplify your answer Use integers or decimals for any numbers in the expression billion b In 2016 total expenditures will be Type an integer or a decimal

TEACHER A rectangular poster is to contain 242 square inches of print The margins at the top and bottom of the poster are to be 2 inches and the margins on the left and right are to be 1 inch What should the dimensions of the poster be so that the lea amount of poster is used in smaller value in larger value

5 Given that h x x find ho h x h o h x Simplify your answer

A storage unit is in the shape of a rectangular prism has a volume of 5440 cubic feet The length of the storage is 4 feet more than three times the width w and the height is 6 feet more than the width Find the equation in terms of w that could be used to find the dimensions of the storage unit in feet Your answer should be in the form of a polynomial equals a constant Sorry that s incorrect Try again

At the beginning of a walk Roberto and Juanita are 6 6 miles apart If they leave at the same time and walk in the same direction Roberto overtakes Juanita in 11 hours If they walk towards each other they meet in 1 hour What are their speeds Roberto s speed is Juanita s speed is mph mph SCCED

C o X X C C X Choose the correct answer OA 1 OB X C C X OC x c X C OD 1

Raise the number to the given power and write the answer in rectangular form 3 cos 90 isin 90 6 3 cos 90 i sin 90

b How much of the sample in mg remains after 20 years After 20 years we have the following Round your final answer to two decimal places y 20 Step 6 50 2 31 50 t 30 2 3 c After how many years will only 1 mg remain 2 3 To find the time at which only 1 mg remains we must solve 1 y t 50 2 t 30 and so we get the following t 30 log 1 50 1 50 In 31 50 mg Step 7 Hence we conclude the following Round your final answer to one decimal place In 2

Consider the following quadratic equation 3x 19x 7 1 Step 1 of 2 Using the standard form ax bx c 0 of the given quadratic equation factor the left hand side of the equation into two linear factors

A Ferris wheel rotates 2852 per ride How far would a rider seated 8 feet from the center of the Ferris wheel travel during the ride Round your answer to the nearest thousandth

Consider the following g x x 2x 8 x 9 Exercise a State the domain of the function Click here to begin Exercise b Identify all intercepts leted sequentially If you skip a part of the question you will not receive any points for the skipped part and you will not be a Click here to begin Exercise c Find any vertical and horizontal asymptotes Click here to begin Exercise d Plot additional solution points as needed to sketch the graph of the rational function

maximize the value of the function z 2x 2y given the constrainst y 0 x 3 y 6 3x 4y

Find the number of subsets of the set 23 A xlx is a letter in the word ohio

List the members of the indicated set using set braces 25 An B

24 A xlx is a prime number between 30 and 40

Insert U or n in the blank to make the statement true 22 5 15 9 5 15 9

Find the x intercepts and y intercept of the following function f x x 4 x 1 x 8 Write your answer in coordinate pairs of the form x y Provide your answer below

lus Week 5 Local Behavior of Polynomial Functions Given the graph of the following degree 4 polynomial function find all of the zeros and their multiplicities Select the correct answer below 765432H 6 x 4 with multiplicity 2 and x 2 with multiplicity 2 x 4 with multiplicity 2 and x 2 with multiplicity 1 x 4 with multiplicity 1 and x 2 with multiplicity 1 x 4 with multiplicity 1 and x 2 with multiplicity 3 5 4 3 2 7 6 5 4 3 2 11 1 2 3 4 567 1 2 123 st 6

Decide whether or not the matrices are inverses of each other 6 3 11 0 5 2 2 1 5

the problem 5 There are three convenience stores in Gambier This week Store I sold 88 loaves of bread 48 quarts of milk and 16 dozens of eggs Store II sold 105 loaves of bread 72 quarts of milk and 21 dozens of eggs Store III sold 60 loaves of bread 40 quarts of milk and 50 dozens of eggs The sales for the three stores are shown in the matrix Store I Store II Store III Bread Milk 88 48 105 60 72 40 Eggs 16 21 50 a During the following week sales on these products at store I increased by 25 sales at store II increased by one third and sales at store III increased by 10

Introduce slack variables as necessary Maximize z X 5X and then write the initial simplex subject to tableau for the given linear programming problem Complete the initial simplex tableau X X2 1 2 6 2 S 1 0 0 0 with X 2x 14 6x x 10 3x 2x2 7 X 20 X 20 S 52 1 0 0 3 0 0 0 Z 0 0 10 7 0

Determine 2 the value of each variable in the matrix equation 2x 3 3y 4 7 3z 7 7 10 7 18

4 Use graphical methods to solve the following linear programming problem Minimize z 2x 3y subject to x ys14 3x 3y29 x20 y 20 Graph the feasible region using the graphing tool to the right at the comer point The minimum value of z is Simplify your answers Ay 16 14 12 10 8 6 4 2

Express as a product log z13 log z

2x 2 4 xy 4 3 x y2 1 Find lim 2x2 2 4 lim xy 4 3 x y 1 Type an exact answer

Find for the equation x y 2 using implicit differentiation 3 dy dy dx 11

and the equation of the tangent line of x y 13 at the point 3 2

First class postage with a private postal service costs 0 36 for all weights through 1 ounce plus 0 22 for each ounce or fraction of an ounce thereafter Each letter is required to carry one 0 36 stamp and as many 0 22 stamps as necessary Graph the function f that models the number of stamps y on a letter weighing x ounces over the interval 0 5 Choose the correct graph O A 5 4 3 2 1 0 0 9989 0 1 2 3 4 5 O B 54327 1 9 34 Q Q O C 5 012 8889

1 A spherical balloon is inflated with gas at the rate of 800 cubic centimeters per minute How fast is the radius of the balloon increasing at the instant the radius is 30 centimeters V 4 r 3 see text section 2 6

If possible find an absolute extremum of g x x 7 et on the interval 10 0 Select the correct choice below and if necessary fill in the answer box to complete your choice Simplify your answer Use a comma to separate answers as needed and no absolute minimum and an absolute minimum at x OA The function g x has an absolute maximum at x OB The function g x has an absolute maximum at x OC The function g x has no absolute maximum and an absolute minimum at x O D The function g x does not have an absolute maximum or an absolute minimum

46 2 tan x 3 shows the graph of the function