Vector Calculus Questions and Answers

Use the laws of exponents to simplify the algebraic expression Your answer should not involve parentheses or negative exponents 2x 5y 2x3 5y 3 3 Simplify your answer Use positive exponents only

6 Select the expression that is equivalent to see cot 25 2 B S X X C 25 1 Cot 25 D E S 25

Which of the following are true for the vector valued function r t cos t i sin t j Select all that apply The graph of r t is a line parallel to the z axis Or t 1 for all t Olim r t j The graph of r t is a circle in the xy plane lim r t 1 0 0 1 0 r t cos t sin t 0

Let f x X X 7 and g x 12 X 12 x Express f x g x as a rational function f x g x for x Use a comma to separate answers as needed

Let 1 3 2 02 X 5 0 04 Then i x 0 ii x is any real number there is no value for x iv x 8 V None of the options

Let x y be any two vectors in space Which of the following are properties of the cross product x Xy Select all that apply x x y is orthogonal to the plane determined by x and y x x x y 0 and y x x y 0 xxy y x x x x y is the area of the paralellogram determined by x and y xxy y x x xxx y xy 0

Suppose a ib is a complex number Then a ib is equal to i b 3ab 3a b a ii ib 3iab 3ia b a iii ib 3iab 3ia b a iv ib 3iab 3ia b a V ib 3iab 3ia b a vi None of the options

An equation of the line tangent to the graph of f x 4x 8x 3 4 at the point where x 1is OA y 1 3 37497 x 1 B y 1 C X 1 D y 5 x 1 E y 81 384 x 1

Systolic blood pressure levels above 120 mm Hg are considered to be high For the 100 systolic blood pressure levels listed in the accompanying data set the mean is 123 62000 mm Hg and standard deviation is 14 30602 mm Hg Assume that a simple random sample has been selected Use a 0 05 significance level to test the claim that the sample is from a population with a me greater than 120 mm Hg Click the icon to view the data set of systolic blood pressure levels Identify the null and alternative hypotheses Ho H V Type integers or decimals Do not round Systolic Blood Pressure Levels 100 106 116 116 130 158 114 122 132 126 128 134 104 126 130 114 132 136 112 126 134 130 136 158 106 x

M Which of the following represents the graph of f lx OA None of these OB n M O C O D

Luvusta 5 v hennet n muuttuja x Muuttuja x kerrotaan luvulla 13 ja tulo lis t n luku 9

The x and y coordinates of a moving particle are given by the parametric equations below Find the magnitude and direction of the velocity for the specific value of t Sketch the curve and show the velocity and its components x 3t y 1 t t 4 Find the magnitude of the velocity of the particle for the specific value of t The magnitude is approximately Type an integer or decimal rounded to two decimal places as needed

and Dot Products Consider the vectors u 4 0 10 and v 2 1 9 Compute u 2 1 9 Compute v 2 29 Compute u v 86

P 4 5 8 R 9 3 8 Find the point Q such that R is the midpoint of Po Q

Use the geometric definition of the cross product and the properties of the cross product to make the following calculations a k 3 K b i j ix 3 X c 2kx k 3 d i 3 3 LED

v 1 6 2 w 8 10 1 Find the cosine of the angle between v and w cos 8

1 point Find the cross product i e j e k 6i e j e k

Let a 5 3 9 and b 9 7 8 be vectors Compute the cross product a x b 7 axb

Write the expression as a single logarithm W log3y 4log3z log3 w 3 log

A 1 0 L buffer solution is 0 060 M HC2H302 and 0 250 M LIC2H302 Which of the followi actions will destroy the buffer O adding 0 060 moles of NaOH O All of the above will destroy the buffer O adding 0 060 moles of HC2H302 O adding 0 060 moles of LiC2H302 O adding 0 060 moles of HCI

5 Let g be the function given by g x x 4x 6x 4x k A On what intervals is g increasing Justify your answer I where B On what intervals is g concave upward Justify your answer IS d

3 Below is the graph of the velocity of an object moving in the positive direction along the x axis Answer the following questions using only information from the graph A 2 3 4 6 t 11 11 21 7 A Find a time t when the object changes directions If this never happens say This never happens Explain your answer B Find the time when the object is furthest to the right Explain your answer I C What is the object s maximum acceleration Explain how you found this

Calculus AB Assignment Applications of the Derivative I A Find an equation that relates the total length of the elastic x to the distance y that the button has moved

3 If a right triangle has legs 6 and 8 its hypotenuse is 10 The triangle will be inscribed within a circle with area 25T The hypotenuse will be the diameter of the circle B Use a tangent line approximation to estimate the area of the circumscribed circle Now consider the sphere that just contains the triangle so the hypotenuse is the diameter of the sphere Use a tangent line approximation to estimate the volume of this sphere

Which of the following statements are always true for any pair of nonzero vectors v and w Select all that apply v w is a vector orthogonal to both v and w O v w is a number If is the angle between v and w then cos 0 V W v w

Which of the following equations are true for any two nonzero orthogonal vectors u and v in space Select all that apply u v u v Oux v 0 0 ux v u v u v 0

Which of the following statements apply to any unit vector u in space Select all that apply O cu c for every number c u u 1 Ou 1 u 0

Two equivalent vectors always have the the same select all that apply O direction terminal point O magnitude initial point

Use this graph of g to find the solutions to g x 1 10 8 9 6 4 2 104 8 6 4 2 0 2 4 6 8 10 y 2 4 6 8 10 X

3 Suppose you take a car trip traveling east along a very long highway starting at time 1 0 Let x t be the number of gallons of gasoline used during the first hours and let s t be the distance traveled in that time Because you re using very cheap gasoline that s not good for the engine your fuel efficiency measured in miles gallon decreases as fuel is consumed The graph of fuel efficiency as a function of gasoline used is shown in the graph below ds dx 25 201 151 101 5 5 I 10x x gallons 15 20

Suppose your friend gives you the following problem There is some function p x that represents profit p in dollars as a function of the number of units sold x Your friend doesn t know what the function p x is explicitly but does know that p x 2 for all x Your friend also knows that p 200 3175 24 Now your friend has enough information to approximate p 203 using tangent line approximation but your friend really wants to know how accurate that approximation will be The question to you then is What can you tell your friend about the accuracy of the tangent line approximation for p 203 given the above information about the function p You may explain yourself mathematically or by presenting graphs or just by discussing in complete sentences

2 You measure a cubic container and find it to be about 10 cm on each side From this you conclude that it holds 1000 cc However your measurement is accurate only to within 10 1 cm So you can be sure that the side is between 9 9 and 10 1 cm A What are x f x and a in this problem B Use the tangent line approximation to estimate the error in your value of 1000 for the volume I C Give the percent error in your measurement of the length of the side

The function y 55 0 3 graphed below S Which of the following Riemann sums yields the exact area under the curve on the interval 0 6 O A R 1 f wk Ack where the subdivisions are taken to be at 0 2 3 4 6 and right hand endpoints are used and R E 1f w Axk where the subdivisions are taken to be at 0 2 3 4 6 and midpoints are used O B R f wk Axk where the subdivisions are taken to be at 0 2 3 4 6 and midpoints are used O c R f wk A k where the subdivisions are taken to be at 0 2 3 4 6 and right hand endpoints are used O D R f w A k where 6 equal subdivisions and right hand endpoints are used O E R f wk Ak where 6 equal subdivisions and left hand endpoints are used

The ellipse x 3y2 13 below has two points when x 1 m 2 7 O T 28 1 Find the slope of the ellipse at both these points I 2 Thoroughly explain the method you used to find the slope in part 1 What are the strength and weaknesses of the method you used 3 Explain whether your method will work to find the slope of x2 xy y2 3 at points 1 2 ar 1 1 If it succeeds please show your work as you find these two slopes If your method fai that s fine just explain thoroughly why it failed

1 Below is a chart that gives some information regarding a twice differentiable function fx The n a in the chart means not applicable x 4 x 4 4 x 0 x 0 0 x 4 x 4 4 x n a n a 1 n a 5 n a negative 0 positive 0 negative 0 positive n a f x x Concavity 3 n a n a Fill in the last row of the chart the four empty spaces with the proper concavity either concave up or concave down and then sketch a graph that satisfies all the conditions in the chart

Find a power series solution of the form y x 0 Cnx of the given differential equation x y y 0 First compute y x Use cn for c in your answer Then x y x y Co C1 Cn n 1 Use c0 for co c1 for c cn for c and cnm1 for C 1 in your answer Requiring that the terms of this series for x y y vanish gives X7 1 Cn 1 for n 2 100 x 2 The series solution converges to the familiar function y f x x

Find the power series solution of the initial value problem Assume the form y x 0 Cnx n 0 y x n 1 Then compute n C x 1 0 Cn First compute y x 1 n 1 n n Cn 1 The recurrence relation implies Cn 0 n 1 y 4y 0 Then y 4y n 2 n 1 Requiring that the terms of this series for y 4y vanish gives the recurrence relation Cn 2 Cn for n 0 1 2 The initial conditions imply Co C for odd indices n Cn 2 0 Cn 1X Cn 1x 1 n 2 n 1 2m 1 y 0 1 Cn 1 4 for even indices n 2m The series 0 Cnx is familiar as the series for the elementary function in 0 y f x m 0 1 2 m 0 1 2 y 0 0 Cn x Cn 2x

3 Consider the function f x 1 x4 5x 4 Use graphical methods to discuss the differentiability of the function In particular find all values of x for which the function appears non differentiable and explain why Include at least one graph illustrating each point of non differentiability

Find a power series solution Cnx of the given differential equation Determine the radius of convergence of the resulting series and by comparing with a table of well known series identify the series solution in terms of familiar elementary functions y 4y y x 0 The radius of convergence is p The series gives the familiar function Use c0 for co in your answer Use c0 for co in your answer

x2 2x 1 2x3 3x 3x 1x4 x 9x 24 if 4 x Discuss the continuity and differentiability of the function at x 1 and x 4 Prove your assertions mathematically a graphic proof will NOT suffice 4 Consider the function f x W 5 Below is the graph of y g x 101 0 if x 1 if 1 x 4 Is it possible impossible or certain that the function is continuous atv 22 Explain

Find the amplitude phase shift and period of the function y 1 2 cos 2x 2 Give the exact values not decimal approximations Amplitude 1 Phase shift 3 X B

Khalil Problem 4 3 Consider the system x x 1 x x x x 1 x x Use a quadratic Lyapunov function to show that the origin is asymptotically sta

3 The int operator in your calculator is the greatest integer function It returns the greatest integer less than or equal to the operand and is written using square brackets x or x For example it would give back 82 for 82 976 In Smalltown AZ the students receive grade points as follows 4 0 for any grade 90 or higher 90 100 3 0 for any grade 80 or higher 80 90 2 0 for any grade 70 or higher 70 80 1 0 for any grade 60 or higher 60 70 Notice students NEVER fail And students never get 100 A Write a function that correctly returns grade points for gradesfrom 60 to 100 78 45 should give a 2 0 Is your function even odd or neither I

The number of times a new pop song has been downloaded t weeks after its initial release is given by f t a After how many weeks is the rate of change of the number of downloads maximized b What is the rate of change of the number of downloads at the time found in part a c How many times has the song been downloaded at the time found in part a a The rate of change of the number of downloads is maximized after weeks Do not round until the final answer Then round to two decimal places as needed 3 000 000 1 300 0 85

se Stoke s Theorem to calculate the circulation of the field F around the curve C in the indicated direction F 3y y z where C is the counterclockwise path around the ellipse 4 36 1 in the xy plane

1 point Use Stokes Theorem to find the circulation of F 4y 5zj 6xk around the triangle obtained by tracing out the 4 0 0 to 4 0 6 to 4 3 6 back to 4 0 0 F dr Circulation

Solve the IVP using Laplace Transform y y 8 t 2km y 0 0 y 0

The equation r t 3 cos t i 3 sin t j t k is the position of a particle in space at time t Find the particle s velocity and acceleration vectors Then write the particle s velocity at t 0 as a product of its speed and direction The velocity vector is v t i j k The acceleration vector is a t Oi j k The velocity vector at t 0 written as a product of the speed and direction is v 0 0 0 0j k Type exact answers using radicals as needed

Find the distance from the point 0 0 7 to the line x 3t y 2t and Z 2t The cross product used when calculating the distance is 0 0 7 x 3 2 2 14 21 0 The magnitude of the cross product is 637 Type an exact answer using radicals as needed 0 Type an exact answer using radicals as needed The distance is

Find the curve s unit tangent vector Also find the length of the indicated portion of the curve TU r t 3 cos t j 3 sin t k Osts Osts Find the curve s unit tangent vector T t j k Type exact answers using radicals as needed The length of the curve is Type an exact answer using as needed