Calculus Questions

Use the second derivative test for local extrema to indicate any local extrema or saddle points of the following function f x y x y 3xy

Use the method of Lagrange multipliers to minimize 2 f x y x 4xy y subject to the constraint x y 16 You may assume a minimum exists

Upload solutions to Problem 1 here Find the following for the function a fx 1 2 b fyy x y c fxy x y f x y 3x y 5x 26 9

Which of the following is the total length of the curve of intersection of the surfaces x y 2 4 and y 22 4 a 2 3 b 4 3 c 2 d 4 e 8TT

Sketch the region of integration for integral the integral using spherical coordinates 0 9 x z dz dy dx Hence evaluate 10 marks

15 points Solve for the variable in each equation a 4 sin 0 10 b 2logg x log3 5 1

Let F xyz xy x yz eff cut Use Stokes Theorem to evaluate curlFdS where S consists of the top and the four sides but not the bottom of the cube with o corner at 5 5 5 and the diagonal corner at 1 1 1 Hint Use the fact that if S and S share the same boundary curve C that curl F dS curl F d S S F F dr

Given that 6f 12g 18h 72 4f 3g 42h 248 and 2f g 22h 132 solve the system of equations using Gaussian elimination Select the correct answer below 2 4 6 O 2 6 4 O 4 2 6 infinitely many solutions

Given matrix A shown below find 6A Give your answer as a matrix using the matrix tool Provide your answer below 4 A 4 3 1 4 5

Given matrix A shown below find 9A Give your answer as a matrix using the matrix tool A 0 5 3 3 1

A world famous prix fixe menu includes the following choices Beverage choose one wine soft drink coffee tea milk juice or cappuccino First course no choice baguette bruschetta whole roasted garlic and butter Second course choose one quesadilla onion soup salad Caesar salad lettuce wedge spinach salad or romaine salad Third course choose one rib eye steak swordfish rack of lamb whitefish grilled chicken southern fried chicken lobster or vegetaria plate if you choose steak then choose one of the following fries mashed potatoes creamed corn spinach broccoli or grilled onions Fourth course choose one chocolate cake cinnamon tart apple tart pecan tart creme brulee mango sorbet or chocolate tart How many different choices are possible meals

2 8 6 4 2 0 2020 StrongMind Created using GeoGebra 6 g x Of x and g x have the same rate of change O f x does not have a rate of change for the interval ala increases at a faster rate 8 Which statement is true about f x and g x and their rates of change for the interval 0 3 GeoGebra Graphing Calculator

Tarot cards are used for telling fortunes and in a reading the arrangement of the cards is as important as the cards themselves How ma different readings are possible if three cards are selected from a set of seven Tarot cards readings

Some automobiles have four button locks To gain entry push four buttons in a particular order repeats allowed to unlock the door How ma different four digit codes are possible codes

QUESTION 14 3 points The goal in this question is to estimate the value In 1 05 with the help of the Taylor polynomial of order 3 a Find the Taylor polynomial near r 0 for f x ln 1 r So find f 0 f x f 0 f z f 0 f r and f 0 and arrange them in the polynomial b Find the value of r such that f x ln 1 05 desimal places

Let g x f t dt The graph of y f t is given below 5 4 3 2 1 1 2 3 4 5 3 4 5 6 7 8 9 10 Determine the open interval s on which g is decreasing Enter the solution using interval notation Enter DNE if g is decreasing over no interval Determine the open interval s on which g is increasing Enter the solution using interval notation Enter DNE if g is increasing over no interval Determine the location x value of any local minimum of g over 0 10 OH Og has no relative minimum Determine the location x value of any local maximum of g over 0 10 0x og has no relative maximum

QUESTION 15 7 points Find the integrals Must show all work including all steps for and Sub a 2r 3 dr b Szln 2 c r dr 1 r dr 4

11 ft 23 ft Find the angle of elevation of the sun from the ground to the nearest degree degrees

Solve the equations on the interval 0 2 1 3 tan 0 2

ON 4 2 points A population of punishers is described by the DTDS 1 1 density after t weeks FIND the general Solution of the DTDS if initially to

Given A ABC find the value of the variable If necessary round your answer to the nearest hundredth Figure is not to scale A 25 76 O Sine O Cosine O Tangent B X C To answer this question which of the following will you need to use to solve for x

Classify the problem as a permutation a combination or neither How many three digit numbers can be formed using the numerals 3 4 5 6 7 without repetition permutation O combination O neither Answer the given question

QUESTION 8 2 points Use 2 iterations of Newton s Method with the initial condition To 1 4 to estimate the solution of f x e r 0 with a precision of 3 decimal places

e graph shows the function f x 19 18 14 12 10 0 1 CO 6 4 20 2 5 15 59 2020 StrongMind Created using GeoGebra 4 9 3 5 2 2 3 1 1 73 2 4 6 8 10 12 What is the average rate of change for the function f x over the interval 0 3 Enter your answer as a number like this 42 E

Calculate the average rate of change of f x over the interval 1 x 3 I f x 4 2 877 3 2 270 2 0 423 1 2 728 O 5 546 What is the average rate of change for the function f x over the interval 1 x 3 00 0 2 524 0 2 524 2 728 0 423 O 1 364 1 2 3 4

Use the table to calculate the average rate of change of f x over the interval 0 4 x f x 1 2 4 3 2 1 0 1 2 3 4 4 8 16 32 64 128 256

cos x 2 1 QUESTION 3 2 points Find the derivative of f x In x 2 4

5 2 points Consider the function f r r 12r 1 defined on the closed be and its Global absolute maximum and find its Global absolute minimum

Evaluate the expression 6 n 91

QUESTION 7 2 points Consider the function y f x given Implicitly by the equation ry 2y 1 e 4 around the point 2 0 FIND The Derivative of y at the point 2 0

3pts Consider f 1 1 x 3 mly The Definition of the DERIVATIVE to find f 1 e equation of the tangent line to the graph of f at the

QUESTION 6 2 points Let F be an antiderivataive of f x 2 I F 4 e such that F 0 3 Then find

QUESTION 10 3pts Consider defined by f x Find the value of k such that f is continuous 1 9 if r 3 3 ek 1 if I 3

In how many ways can seven books be arranged on a bookshelf ways OF

Use the to Give Homogeneous Substitution method solve the following differential Equation a general xy y x F box solution

5 Use the Method of Coefficient to the Undetermined general to the following differential equation Give the comple following non homogeneous genenal y y 6 cos x Solution Solution

Of differential equations following homogeneous Using the characterist equation Give the general Show of your Wonk all y qy 0 3y 10y 0 44 54 the solution

QUESTION 2 2 points Information on functions f x g r f r and g r for some r is given in the table Let h x gof x I f x f x g x g x Find h 1 and h 1 1 27 3 3 2 312 QUESTION 3 2 points Find the derivative of f x ln 3 5 2 3 cos x 22 11 x 2 4e 4 3 1 1 4

1 2 Suppose 3 3 is the terminal point on the unit circle determined by t Find the termina point determined by each of the following a t b t B2 2 N c t d t n

3 Use the integration factor Method to fir the general solution of y xy 4x

2 Use the Following Give the YE separation Separable Particular x ex technique to solve th equation differential solution y 0 3

fine Set up and solve a Definite integral to the particular solution of the following equation must Use Propert differential integral with all the a Definite notation AISO Bounded integral 0 6 must use dy 4x 6x ex dx

Given matrices A and B shown below find 6 A 9 B B 2 5 1 0 3 3 3 2 4 0 1 1 0 Give your answer as a matrix using the matrix tool A 3 b n b 3 L 5 T

Given the function f x x 3 x 3 a Find f x 1 1 b Graph f and f in the same rectangular coordinate system c Use interval notation to give the domain and the range of f and f a Find f x Be sure to include any domain restrictions f x 0 x 0

Given matrix A shown below find 1A Give your answer as a matrix using the matrix tool 2 3

7 Find the term that does not contain x in the expansion of 10 4x 2x

ADVANCED ANALYSIS Assume that the consumption schedule for a private open economy is such that consumption C 50 0 75Y Assume further that planned investment g government spending G and net exports Xn are independent of the level of real GDP and constant at g 30 G 0 and X 10 Recall also that in equilibrium the real output produced Y is equal to aggregate expenditures Y C Ig G Xn Instructions Round your answers to the nearest whole number a Calculate the equilibrium level of income or real GDP for this economy b What happens to equilibrium Y if lg changes to 10 What does this outcome reveal about the size of the multiplier Multiplier

Add the following matrices 1 3 0 1 5 Give your answer as a matrix using the matrix tool 2 1 5 4 3 5 3 3 1

Find the radius of convergence and interval of convergence of the series x 6 n 1 R I n 0

Find the price p per unit that produces the maximum profit P if C x 25 16x and p 24 2x OA 16 OB 20 O C 22 O D 12