Vector Calculus Questions and Answers

On the axes below draw the graph of the function f x x 3 Provide your answer below 5 10 5 0 5 4 10

Set the point Provide your answer below 10 1 3 21 3 13 10 0 TT 11 3 51 3

What trigonometric function would you use based upon the given information in the triangle shown below Select the correct answer below O O sin B sin B cos B b a 100 cos B tan B 2 B a

The distance from the Sun to Earth is approximately 149 600 000 km The distance form the Sun to Mercury is approximately 57 900 000 km The elongation angle a is the angle formed between the line of sight from Earth to the Sun and the line of sight from Earth to Mercury Suppose that the elongation angle for Mercury is 20 Use this information to find the two possible distances between Earth and Mercury Mercury Mercury CX Sun

Match each of the given functions with the BEST descriptor It is the only function we saw that had multiple vertical asymptotes It s domain includes all real numbers it always has a horizontal asymptote and may be increasing or decreasing It s the inverse of an exponential function It has no asymptote and normally it is increasing but slows down A It is the most basic of functions and is also a polynomial This function has two horizontal asymptotes A Linear B Rational C Root Radica D Logarithmic E Exponential F Logistic

Write a polynomial in standard form that represents the area of the dhaded region x 3 X 9 X 1 X 4

Graph of the function f x log x You can only graph integer points for example you cannot plot the point 1 There is one point that can be moved above and below the x axis as needed Do not move either point off of the graph as you will not be able to click on the point again Provide your answer below 3 0 2 3 A 5 6

A ball is thrown eastward into the air from the origin The initial velocity is 45i 75k measured in feet per second The spin on the ball gives an acceleration of 5 ft s 2 northward Where does the ball land and with what speed

2 Proof the following cos2x 1 a tanx sin2x 1 cos2x 1 cos2x tan x cos2x 1 sin2x cos2x 1 sin2x d 4 cos x cos2x 5 6sin x b

Draw the function f x by dragging the blue points to the locations on the x axis that correspond to the zeros of the function Use the red point as a slider to select the number of zeros that your graph will show You may click on a point te verify its coordinates Provide your answer below 10 O zeros 4 5 40 8 30 20 10 0 10 20 A f x 4 x 2 x 1 x 3 5 10

7 a Find the arc length parametrization of the curve that is the intersection of the elliptic cylinder y 1 and the plane z 2y 7 5 Use s as the arc length parameter with s 0 corresponding to the point 0 1 9 oriented counter clockwise as seen from above Also if the density is given by 8 x y z z set up an expression for its mass

4 point bins 96 to 99 92 to 95 etc to make a frequency table for the set of exam scores shown below Include columns for relative frequency and cumulative frequency 78 81 82 97 98 97 85 92 88 78 93 92 84 83 99 77 88 96 83 92 Complete the frequency table below Scores Frequency 96 to 99 92 to 95 88 to 91 84 to 87 80 to 83 76 to 79 Total Relative Frequency 7 Cumulative Frequency ww

Part 1 of 2 Points 0 of 4 A boat on the ocean is 2 mi from the nearest point on a straight shoreline that point is 12 mi from a restaurant on the shore A woman plans to row the boat straight to a point on the shore and then walk along the shore to the restaurant Complete parts a and b below COLLE 2 mi To minimize the total travel time the boat should land miles from the restaurant Type an exact answer using radicals as needed 12 mi Save a If she walks at 3 mi hr and rows at 2 mi hr at which point on the shore should she land to minimize the total travel time

Part 2 of 2 a Squares with sides of length x are cut out of each corner of a rectangular piece of cardboard measuring 7 ft by 5 ft The resulting piece of cardboard is then folded into a box without a lid Find the volume of the largest box that can be formed in this way Points 0 of 1 b Suppose that in part a the original piece of cardboard is a square with sides of length s Find the volume of the largest box that can be formed in this way C a The maximum volume of the box is approximately 15 02 ft Round to the nearest hundredth as needed b The maximum volume of the box is Type an expression using s as the variable Save

The electric current i in mA in the circuit i 7 4 1 e 0 42t where t is the time in s Evaluate i for t 9 0 m i mA Round to four decimal places as needed

1 point Consider the second order differential equation with initial conditions Without solving it rewrite the differential equation as an equivalent set of first order equations In your answer use the single letter u to represent the function u and the single letter v to represent the velocity function u Do not use u t or u t to represent these functions Expressions like sin t that represent other functions are OK u U Now write the first order system using matrices d dt u V u 6 5u 3u 1 5 sin 31 u 1 6 u 1 4 5 The initial value of the vector valued solution for this system is u 1 v 1 BIO E

1 point This is the second part of a two part problem Show that y t is a solution to the system of linear homogeneous differential equations y y y3 y y 2y2 y3 et a Find the value of each term in the equation y 2y y2 y3 in terms of the variable 1 Enter the terms in the order given 2y y2 Y3 1 y2 2y3 b Find the value of each term in the equation y2 y y2 2y3 in terms of the variablet Enter the terms in the order given c Find the value of each term in the equation y y 2y2 y3 in terms of the variablet Enter the terms in the order given

1 point This is the second part of a two part problem Let 7 1 sin 6 cos 61 Compute the wronskian to determine whether the functions x 1 and x t are linearly independent Wronskian det x 1 6 sin 6t 6 cos 61

a On what interval s is increasing Enter your answer using interval notation 1 6 U 8 00 On what interval s is f decreasing Enter your answer using interval notation 0 1 U 6 8 b At what value s of x does f have a local maximum Enter your answers as a comma separated list x 6 At what value s of x does f have a local minimum Enter your answers as a comma separated list x 1 8 c On what interval s is f concave upward Enter your answer using interval notation

00 DETAILS SCALCET9 4 9 057 The graph of a function is shown Which graph is an antiderivative of 3 0 2 0 1 0 1 0 2 0 5 1 0 1 5 2 0 b 2 5 a

2 O No 4 6 Are the secant line and the tangent line parallel Yes Show My Work Required Wha 8 or reasoning did you use Your work counts towards your score

For the spinner below assume that the pointer can never lie on a border line Find the following probabilities Enter the probabilities as tractions a P D b P E c P F Need Help Read It Additional Materials F Tech Guide Excel E

Rewrite the complex number 6e4i in a bi form

An engineer is building experimental dome shaped living quarters 24 m high and modeled by the function z 24 2x 3y She wants to bolt a flat solar panel to the dome at the point 2 2 4 Find the vector of the direction in which the engineer should drill Give your answer using component form or standard basis vectors Express numbers in exact form Use symbolic notation and fractions where needed

Find the inverse of matrix A defined by Provide your answer below 2 4 1 Use the matrix tool to enter your answer Use fractions when necessary

Which of the following matrices are invertible Select all correct answers Select all that apply 3 2 4 5 2

Solve the following system of equations by using the inverse of a matrix 4x 6y 6z 54 9x Z 3 8x 2y 3z 21 Give your answer as an ordered triple x y z Use fractions when necessary

Is the following matrix invertible Select the correct answer below Yes the matrix is invertible No the matrix is not invertible 435 3 3 2 4 4 5

Given the following system of equations use the inverse of a matrix to solve for the values of x and y Give your answer as an ordered pair If no solution exists enter 2x 4y 4 x y 1

Consider a die with only four sides marked one two three and four a Write out a sample space for rolling a pair of these dice Die One c P three d P four e P five f P six 2 9 P seven 3 4 1 Assuming equally likely outcomes find the probability that the sum of the dice is the given number Enter the probabilities as fracti b P two 2 3 2 Die Two 4

Estimate the value of the sum of series S n 0 n methods taught in this class Do not refer to the function ex anywhere in your solution Yo 1 n to within an error of 0 0002 Use only

a Determine whether class 4n 5n 1 is convergent or divergent Use only methods taught in this

Given the polar equation r 4sin 30 0 Sketch the polar graph by first graphing the cartesian graph Show the exact correspondence between sections of the two graphs I e do it exactly the way that was shown In lecture

Find the Maclaurin series for the following function Find its radius of convergence R Use only methods taught in this class f x ln 1 4x

Show all work No Calculator is allowed 1 Write the equation in standard form identify the shape and center a and b ar graph it 7x 5y 42x 10y 23 0

scale from 0 to 50 in intervals of 5 Six plotted points are generally falling The approximate coordinates of the points are as follows 0 5 28 1 5 3 1 3 11

the function f is defined as f x 5x4 9x3 2 Use the power rule to find the derivative f x and evaluate that derivative at x 1 7 retain at least six decimal Use the forward difference and centered difference formulas to estimate f x at x 1 7 for three different values of the mesh sizes h 0 1 h 0 01 h 0 001 Calculate derivatives using two finite difference formulas h forward difference approximation centered difference approximation exact

Find the vector in the subspace W spanned by Answer which is closest to 23 3 4 2

Given matrices A and B below determine if they are inverses of each other 1 Select the correct answer below Yes A and B are inverses No A and B are not inverses A B 3 2

Determine each of the following for the giv function f x y x y 3y n x a f 1 2 5 fx x y fxy x y

Find the inverse of matrix A defined by 7 necessary Provide your answer below Use the matrix tool to enter your answer Use fractions whe

Given the matrix A 31 7 find det A

Given the matrix A 5 2 4 5 find det A

Problem 20 Use the method of Lagrange multiplies to minimize f x y z x 4y 22 subject to x 2y z 10

expression as a sum and or difference of logarithms with all variables to the first degree In 15w523 In 15w z Type an exact answer in simplified form Use integers or fractions for any numbers in the expression

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

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

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