Vector Calculus Questions and Answers

The integral Tea e e sin 80 de can be best matched by formula number from the Table of Int

Find an orthonormal basis for the solution space of the homogeneous system of linear equations x y z w 0 2x y z 2w 0 1 12

4 1 Find an expression for V v in terms of v 4 2 Find an expression for V 2 v 2 in terms of V 4 3 Use the result of 4 2 to deduce an expression for 2 whenever and V are orthogonal to each other

If the terminal side of angle 0 goes through the point Provide your answer below 5 5 2 5 5 on the unit circle then what is cos 0

If log x log x 6 3 then O x 3 only x 9 only Ox 9 or x 3 O x 16 5 only

True O False QUESTION 3 ee e b is true for all values of a and True

Greg and Charmaine are standing on a riverbank 100 meters apart at points A and B respectively See the figure below They want to know the distance from Charmaine to a house located across the river at point C Greg measures angle A angle BAC to be 54 and Charmaine measures angle B angle ABC to be 70 What is the distance from Charmaine to the house Round your answer to the nearest tenth of a meter 54 100 m 70 B meters X E

Use logarithmic differentiation to find dy dx y x x 44 x 0 dy dx 11

Evaluate the integral using the indicated trigonometric substitution Use C for the constant of integration 4x 25 O CHOICE C Sketch and label the associated right triangle CHOICE A X 0 dx 4x 25 0 5 X 5 X sec 8 2 5 X 4x 25 CHOICE B CHOICE D O 2x 5 4x 25 5 4x 25 2x

Find the average value of f on 0 16 1 0 4 100 8 12 AX

How many four digit passcodes are possible if the first digit can not be five and repetition is not allowed Write the answer as a whole number

Problem 54 Find the intervals on which f x is increasing the intervals on which f x is de creasing Then sketch the graph Add horizontal tangent lines f x x 3x 3x f x x 3x 3x

In a certain college 33 of the physics majors belong to ethnic minorities Find the probability of the event from a random sample of 7 students who are physics majors exactly five belong to an ethnic minority 0 037 0 147 0 274 O 0 085 0 015

3 If cos x 0 3 then cos x 4 If tan x 2 5 then tan x

The graph shows the number of trips taken annually on a certain state s mass transportation system Find the average rate of change of the number of trips from 1992 to 1995 from 1995 to 2000 and from 1992 to 1998 10 9 8 Annual Trips on a State s Mass Transportation System 8 4 million 7 1992 7 5 million 1994 1996 9 6 million 8 9 million 1998 2000 The average rate of change from 1992 to 1995 is Round to three decimal places as needed The average rate of change from 1995 to 2000 is Round to three decimal places as needed The average rate of change from 1992 to 1998 is Rould to three decimal places as needed million per year million per year million per year

Since we have verified that all three hypotheses of Rolle s theorem are satisfied we know that there is a number c in 1 3 such that f c 0 To find this number c we begin by finding f x f x 2x 4x 9 f x Therefore we have the following f c

Explain the difference between an absolute minimum and a local minimum O A function f has an absolute minimum at x c if f c is the smallest function value on the entire domain of f whereas f has a local minimum at c if f c is the smallest function value when x is near c O A function f has an absolute minimum at x c if f c is the smallest function value when x is near c whereas f has a local minimum at c if f c is the smallest function value on the entire domain of f O There is no difference O A function f has an absolute minimum at x c if f c is the largest function value on the entire domain of f whereas f has a local minimum at c if f c is the largest function value when x is near c O A function f has an absolute minimum at x c if f c is the largest function value when x is near c whereas f has a local minimum at c if f c is the largest function value on the entire domain of f

1 point Give a vector parametric equation for the line through the point 1 2 that is perpendicular to the line 3 5t 1 t L t

Find the magnitude of the complex number z 8 6i 2

Write log 8 using natural logarithms Note When entering natural log in your answer enter lowercase LN as In There is no natural log button on the Alta keyboard

Given the graph of the piece wise defined function below f x find the following 8 5 a 35 a Domain b Range 6 3 2 3 4 5 6 c Interval s of Increase enter DNE if the function does not increase d Interval s of Decrease enter DNE if the function does not decrease e Left hand end behavior As f Right hand end behavior As

Represent log k using the natural logarithm Note When entering natural log in your answer enter lowercase LN as In There is no natural log button on the Alta keyboard

Which of the following equations uses the inverse property of logarithms corm Select all correct answers Select all that apply logg 87 7 glog 7 7 glogg 7 7 logg 78 7 7log 8 8 glog 8 7

Question Use the Product Rule of Logarithms to write the completely expanded expression equivalent to logs 3x 6y Make sure to use parenthesis around your logarithm functions log x y Note If you are using log you need to type it in and then use the subscript button x on the keypad There is no log button

Find the arc length parameter along the curve from the point where t 0 by evaluating the integral s r t 5 3t i 8 2t j 3 7t k 1st 0 The arc length parameter is s t Type an exact answer using radicals as needed SIV T CT 0 The length of the indicated portion of the curve is Type an exact answer using radicals as needed dt Then find the length of the indicated portion of the curve

Find a vector of magnitude 3 in the direction of v 12i 5k The vector is i O Simplify your answer Use integers or fractions for any numbers in the expression

Find the volume of the parallelepiped box determined by u v and w u 2i 3j v 3i 2j 3k w i 2k The volume of the parallelepiped is units cubed Simplify your answer

Evaluate the following integral in spherical coordinates 2 3 3 csc S S S op 0 4 0 2 p sin p dp do de N 3 4 T 3

The graph of the function f x log x is stretched vertically by a factor of 2 reflected over the x axis reflected over the y axis and shifted up by 1 unit Find the equation of the function g x described above Note If you are using log you need to type it in and use the subscript button on the keyboard There is no log button

For each of the following multiplication tables determine whether it is the multiplication table group justify your answer and of a explain which of the group axioms hold I a b C a b a C d d I b d C a b C a C b d d b a d b C a b b bd f a I a J C a b a C a d a d b d f d f C b d C b a C a b d a c d d c

4 Let A be a 3 x 3 matrix with 3 columns a A2 A3 i e A a a a3 Suppose that A transforms 2 and 9 to 9 and C 1 and respectively Let B a a2 a3 the set of 2 the columns of A Find the B coordinate vector of 3 4

Find B if possible B 53 41 Select the correct choice below and if necessary fill in the answer box to complete your choice A B Simplify your answer OB The matrix B is not defined

Find a b c and d so that a b c d OA Select the correct choice below and if necessary fill in the answer boxes to complete your choice 11 88 B There is no solution 23 81 a b cd 10 7 8 10

Find 3 BA 4 AC if possible 1 03 A B A 2 1 1 0 0 2 C 0 01 21 2 1 3 Select the correct choice below and if necessary fill in the answer box to complete your choice A OA 3 BA 4 AC OB The matrix is not defined Simplify your answer

3 Describe in your own words what it means for the infinite series an to converge n 1

3 a log b Rewrite as an exponential equation log 6 1 1080 0 1 64 010 X G

Find the arc length of the image given below 103 26 ft Give your answer rounded to 2 places after the decimal

1 point Find an approximate value of each expression correct to at least five decimal places if it is defined otherwise input undefined a tan 1 53 1 10211 radians b sin 1 53 radians

r 3 5 0 P x y S 2 1 Using the image above find the x coordinate of point P Give your answer as a decimal rounded to 2 places after the decimal point

For the graph of y f x shown to the right find the absolute minimum and the absolute maximum over the interval 6 11 Select the correct choice below and if necessary fill in the answer boxes to complete your choice OA The absolute minimum is at x and x Round to the nearest integer as needed Use ascending order OB The absolute minimum is at x Round to the nearest integer as needed OC There is no absolute minimum Select the correct choice below and if necessary fill in the answer boxes to complete your choice OA The absolute maximum is at x and x Round to the nearest integer as needed Use ascending order B The absolute maximum is at x Round to the nearest integer as needed Af x 10 5 0 0 6 SE X 12

The graph of y f x is shown in red Graph y f x Choose the correct graph in blue on the right OA OC B OD M

Use the price demand equation below to find the revenue function Sketch a graph of the revenue function and indicate the regions of inelastic and elastic demand x f p 54 6 p R p Which graph below is the graph of the revenue function A 20 AR p 20 A 13 P 470 B AR p 59 10 Region A in the graph of the revenue function is Region B in the graph of the revenue function is B Q elastic 5 CHERCH neither elastic or inelastic OC AR p MA 650 D 85 AR P 650 W 11 85 K

O 0 0 0 0 Next question The amount of money A t in a savings account that pays 7 interest compounded quarterly for t years with an initial investment of P dollars is given by 4t A t P 1 0 07 4 If 18 000 is invested at 7 compounded quarterly how much will the investment be worth after 7 years How much money will there be in the account at the end of 7 years Round to the nearest cent as needed

Find the volume of the solid whose base is the semicircle y 4 x2 where 2 x 2 and the cross sections perpendiculam o the x axis are squares Use symbolic notation and fractions where needed

1 point For 0 0 2 find the values of the trigonometric functions based on the given one give your answers with THREE DECIMAL PLACES or as expressions e g you can enter 3 5 If cos 0 5 11 sin 0 sec 0 csc 0 tan 0 cot 0 then

Given the graph of a polynomial function 8 6 4 94 8 6 2 6 y Number of x intercepts 3 Number of turning points 3 4 6 X Least possible degree of the polynomial function 4

Find the zeros and multiplicity of the function f x x 4x x 4 List the zero from least to greatest Zero Zero F Zero have multiplicity of have multiplicity of have multiplicity of

Using the graph of a degree 6 polynomial below complete the table of values for either the x value of zeros or the multiplicity of zeros 5 Answer 7 to 8 1 5 3 2 1 1 Zeros ry 1 2 3 4 56 Multiplicity 3

Find the maximum rate of change of f at the given point and the direction in which it occurs f x y 8y x 16 7 maximum rate of change direction vector

Find the directional derivative of f at the given point in the direction indicated by the angle 0 Duf 0 1 f x y y cos xy 0 1 0 4