Vector Calculus Questions and Answers

1 Find T N B K at t ln 2 if r 2t e 4pt s 2 Draw the curve C of r 2pt s 3 On C draw T N and the circle of curvature at t In 2 2p

Perform a first derivative test on the function f x x 5x 2 5 5 a Locate the critical points of the given function b Use the first derivative test to locate the local maximum and minimum values c Identify the absolute maximum and minimum values of the function on the given interval when they exist a Locate the critical points of the given function Select the correct choice below and if necessary fill in the answer box to complete your choice A The critical point s is are at x Simplify your answer Use a comma to separate answers as needed OB The function does not have a critical point b Locate the minimum value Select the correct choice below and if necessary fill in the answer box within your choice A There is a local minimum at x Simplify your answer B There is no local minimum OA 52 Locate the maximum value Select the correct choice below and if necessary fill the answer box within your choice There is a local maximum at x Simplify your answer OB There is no local maximum 5 3 2 4

Find the area of the sector shown in the figure 5x 6 OU The area is Simplify your answer Type an exact answer in terms of Use integers or fractions for any numbers in the expression

Convert the radian measure to degrees 2 radian s

Directions Consider the following function f x f x 3x 10 Part 1 Difference Quotient If h 0 then the difference quotient can be simplified into the form A Bh Ch That is f x h f x h A Bh Ch where the coefficients A B and C can be functions of a or just constant terms Find these coefficients A 0 B 3 C 0 Note It s possible for one or more of these coefficients to be 0 Part 2 Derivative Use the simplified expression from Part 1 to then calculate the derivative f x f x lim h 0 f x h f x h

or wages less than the maximum taxable wage base Social Security contributions including those for Medicare by employees are 7 65 of the employee s wages T a Find an equation that expresses the relationship between the wages earned x and the Social Security taxes paid y by an employee who earns less than the maximum taxable wage base y b For each additional dollar that an employee earns by how much is his or her Social Security contribution increased Assume that the employee s wages remain less than the maximum taxable wage bas c What Social Security contributions will an employee who earns 55 000 which is less than the maximum taxable wage base be required to make

Given the equation y 8x 9 answer the following questions www a If x increases by 1 unit what is the corresponding change in y units b If x decreases by 5 units what is the corresponding change in y units

Find the slope of the line shown in the figure If an answer is undefined enter UNDEFINED 3 2 1 y 3 26 1 25 35 L 1 X 2 3

A firework is launched into the air from ground level with an initial velocity of 128 ft s If acceleration due to gravity is 16 ft s2 what is the maximum height reached by the firework h t at vt ho 256 ft 512 ft 448 ft 1 024 ft

Let u 15i 3j 3k v 5i j k w j 5k Which vectors if any are a perpendicular b Parallel a Which vectors are perpendicular Select the correct choice below and fill in the answer box es within your choice OA The vectors OB Vector is perpendicular to vectors OC None of the vectors are perpendicular are perpendicular Use a comma to separate answers as needed Use a comma to separate answers as needed

represents due north What is the component form of the velocity of the airplane Round the final answer to one decimal place as needed WAP Find the component form of the velocity of the airplane assuming that the positive x axis represents due east and the positive y axis

Suppose that a box is being towed up an inclined plane as shown in the figure Find the force w needed to make the component of the force parallel to the inclined plane equal to 2 1 lb The force w is about lb Round to the nearest hundredth as needed W 33

kr Consider an angle in standard position with a measure of radians for some integer constant k 6 Which of the following statements is true A If k is odd then the angle in degrees is a multiple of 60 B If k is odd then the angle in degrees cannot be a multiple of 45 C If k is even then the angle in degrees is a multiple of 60 D If k is even then the angle in degrees is a multiple of 90

8 Which of the following angles when graphed in standard position does NOT have a terminal side in the second quadrant A 200 B D 27T 5 8TT 11 4 100

An angle measure of x radians corresponds to an angle measure of 210 Find the value of x

3 At t 0 a train approaching a station begins decelerating from a speed of 160 mi hr according to the acceleration function a t 1280 1 4t 3 mi hr2 where t 0 How far does the train travel betwee t 0 and t 0 2 Between t 0 2 and t 0 5 Between t 0 and t 0 2 the train travels 49 78 mi Round to three decimal places as needed

Find the values of x in the given set of equations using the i Row transformatio and ii Gaussian elimination 3x 3x 1 2x 2 x 3 4 2x 1 x 2 2x 3 10 x 1 3x 2 4x 3 5 14mks

Given the system of inequalities x 2y 10 3x 4y 0 x 0 V 0 A GRAPH THE SYSTEM ON A SHEET OF PAPER be sure to shade the appropriate region 15 points B List all of the vertices 10 points C Is the graph bounded or unbounded 10 pointe

For what values of x does 0 none of these 0z 1 orz 2 Ox 1 0 1 x 2 0x 2

If log x 5 3 then a 08 O none of these O 13 0 1032 5 O log2

Solve the equation for a 0 1 4 Othere are no real solutions 0 2 0 2 2 2 0 2 4 2 x 4x s

Choose the function g x whose graph is obtained by transforming the the graph of f x as follows the graph of f x x shifted 3 units to the right and 2 units up O g x x 2 3 O g x x 3 2 O g x x 2 3 O none of these O g x x 3 2

What is the radian measure for an angle whose degree measure is 240 O 47 3 6 3 OT 3 0 277 3

The graphs of the functions f and g in the figure are obtained by vertical and horizontal shifts and scalings of y x Find formulas for f and g Verify your answers with a graphing utility f x Type an expression using x as the variable y xl x g x 8 f x X

Expre 54 Do not evaluate

5 0 4 Points DETAILS 50 feet The cup on the 9th hole of a golf course is located dead center in the middle of a circular green which is 40 feet in radius Your ball is located as in the picture below The ball follows a straight line path and exits the green at the right most edge Assume the ball travels 9 ft sec Introduce coordinates so that the cup is the origin of an xy coordinate system Provide numerical answers below with two decimal places of accuracy ball green PREVIOUS ANSWERS 40 feet 9 smallest x coordinate 40 20 Lav cup ball path rough a The x coordinate of the position where the ball enters the green will be 40 The ball will exit the green exactly 8 89 c Suppose that L is a line tangent to the boundary of the golf green and parallel to the path of the ball Let Q be the point where the line is tangent to the circle Notice that there are two possible positions for Q Find the possible x coordinates of Q MY NOTES X seconds after it is hit X

A tank holds 3 000 gallons of water which drains from the bottom of the tank in half an hour The values in the table show the volume V of water remaining in the tank in gallons after t minutes t min 5 v gal 2 058 1 320 765 336 81 0 a If P is the point 15 765 on the graph of V find the slopes of the secant lines PQ when Q is the point on the graph with t 5 10 20 25 and 30 Round your answers to one decimal place 5 2 058 10 1 320 20 336 25 81 30 0 Slope 10 15 20 25 30 b Estimate the slope of the tangent line at P by averaging the slopes of the two adjacent secant lines corresponding to the two points closest to P Round your answer to one decimal place

3 Convert the following equation from Cartesian coordinates to Polar coordinates x 4 y 5 41 a r 10r sin 8 8rcos 0 b 7 Brsin 8 10r cos 0 41 8rsin 10rcos 80 10r sin 80 Br cos 0 0 c d e r 10rsin 8 8rcos 8 41 4 Let f x y x 2 Find f x y cos 3x 2y 2xsin 3x 2y x y x y cos 3x 2y 2x sin 3x 2y x x 3 x y cos 3x 2y 2x sin x 2y x y 3 a a b c d e 3 x y cos 3x 2y 2x sin 3x 2y x y x y cos 3x 2y sin x 2 x x2 5 Given f x y 3x y3 2xy Find the critical points and classify them a 0 0 is a saddle point is a local maximum point b 0 0 is a local maximum point is a local minimum point c 0 0 is a saddle point is a local minimum point d 0 0 is a local maximum point is a saddle point e 0 0 is a local minimum point is a saddle point 6 Which of the following integral do we get if we interchange the order of integration Sony dydx f x y dxdy b fff x y dxdy c fff x y dxdy d ff x y dxdy e fff x y dxdy S SVEY

4 K 2 D C 7 4 A B X Analitik d zlemde verilen ABCD karesinde C 7 4 noktas veriliyor K ve D noktas d do rusu zerinde oldu una g re K noktas n n apsisi a a dakilerden hangisidir A 7 B 6 C 5 D 4 E 3

The point 1 1 can be represented in polar coordinate as

Use the inverse matrix equation X A 1 B to find the solution to 4x 7y 2 x 2y 5 4 2 thn 4 7 A 1 2 Hint If A

Perform the indicated operation 13 1 4 5 3 3 8 2 6

Given that A B and A 13 y 22 11 B Solve for X 2x 5 8 28 7 and 4 32 7

Shari has an electronic entry system on her house To enter one must input a six character code where the first four entries are letters of the alphabet A through Z and the last two are single digit numbers 0 through 9 If repeated letters numbers are not allowed how many different codes are possible

A cardiac monitor is used to measure the heart rate of a patient after surgery It compiles the number of heartbeats after t minutes When the data in the table are graphed the slope of the tangent line represents the heart rate in beats per minute The monitor estimates this value by calculating the slope of a secant line Use the data to estimate the patient s heart rate after 42 minutes using the secant line between the points with the given values of t Round your answers to the nearest hundredth a t 36 and t 42 b t 38 and t 42 c t 40 and t 42 d t 42 and t 44 What are your conclusions t min 36 38 40 42 44 Heartbeats 2530 2661 2806 2948 3080 a b c d

9 A heater of temperature 60 C is put at the middle of a straight corridor in a house It is given that the room temperature in the house without the heater is 6 C Suppose the air temperature is decreased by 2 C for every 10 cm away from the heater a What is the air temperature 2 2 m away from the heater 3 marks b Jessica thinks that 24 C is the most comfortable temperature How far should she stay from the heater 2 marks

2 f 2x cos 3x ln x dx

8 Using vectors explain how to determine the volume of the following parallelepiped Determine its volume B 6 3 5 A 4 0 0 O C 2 3 5 y D 6 5 3

5 A father is pulling his son on a sled at an angle of 20 with the horizontal with a force of 25 lb He pulls the sled in a straight path of 50 ft How much work was done by the man pulling the sled to the nearest unit 20 25 lb

3 The area of a parallelogram is 104 m and the lengths of its sides are 9 m and 13 m Determine the measure of the acute interior angle of the parallelogram Round to the nearest degree

A 0 6 2 0 1 3 0 A B a Find A21 b Simplify and find 2AAT 5ABT 3ABT 2 Solve these systems 2x1 2x2 4x3 6 2x13x2 x3 19 3x2 4x3 13 3x1x24x3 1 3 Find production level which satisfies this consumption matrix and demand vec 2x1 x2 2x3 8 21 3x3 1 3x12x2 11 0 1 0 25 0 25 0 3 0 5 2 0 0 3 0 2 0 25 50 3 6 1 0 B X1 S 10 20 30

1 If A7x5 is a coefficient matrix how many unknown variables system of linear equation has 5 2 When it is possible to say that det A is equal 0 without any calculation If three rows are equal If all elements in one column are equal to 0 If we have diagonal matrix 3 What are the properties for bounden function f x f x A f x B bounded function always has limit not all bounded functions have limit when z 7 12 35 4 If lim f x 3 and lim 1 1 removable breakpoint first type breakpoint 6 second type breakpoint function is continuous at point ro f x what type of breakpoint to is 5 Which point s are critical points Points where function is equal zero or does not exists Points where function s derivative is equal zero or does not exists Points where function s second order derivative is equal zero or does not exists Extreme points Inflection points

An object moves along the y axis marked in feet so that its position at time x in seconds is given by the function f x x 15x 72x a Find the instantaneous velocity function v f x b Find the velocity when x 1 c Find the time s when v 0 a v b The velocity of the object at x 1 is ft s Simplify your answer c The velocity is zero at x seconds Simplify your answer Use a comma to separate answers as needed

y 2 2 21 Choose the correct graph below OA B 3 5 XXXI do 3 OD

Graph the equation Then describe the transformation from the parent function f x x 1 y xl G Now describe the transformation from the parent function f x xl Choose the correct answer below OA The parent function f x x is reflected across the x axis OB The parent function f x x is reflected across the y axis OC The parent function f x x is vertically stretched and reflected across the x axis OD The parent function f x xl is vertically compressed and reflected across the x axis G

The question asks us to sketch the function y cos 3t First note that the function is a cosine function and will have that basic shape From the text the general form of a non shifted cosine function is y a cos bt The text also tells us that a is the amplitude that is the maximum height above zero that the function attains Note that the amplitude must be a positive number Looking at our function the amplitude is 1 1 Step 2 2x Next we need the period From the text the period of the function y a cos bt is b Looking at our function and simplifying the period is of complete cycles the function goes through between 0 and 2 More intuitively the number b gives the number

Tutorial Exercise Sketch the graph of the function showing the amplitude and period of the function below y cos 3t Step 1 The question asks us to sketch the function y cos 3t First note that the function is a cosine function and will ha that basic shape E From the text the general form of a non shifted cosine function is y a cos bt The text also tells us that a is the amplitude that is the maximum height above zero that the function attains Note that the amplitude must be a un number Looking at our function the amplitude is

Find the absolute minimum and absolute maximum of f x x 4x 5 on the interval 1 5

Express the given quantity as a function f x of one variable x The product of two numbers x and y whose sum is 24 Hul

Use Laplace transforms to solve the following differential equations 4 y 5y 4y et sinht with y 0 1 y 0 0 Show the full explanation for this problem