Vector Calculus Questions and Answers

Simplify the following expression 8 9a1810 81 819a1483 31 Show your work here Hint To add an exponent a type exponent or press A

4 6 pts The population of a city is described by the function P t 40 600 1 021 where t is the number of years after 2010 In what year will the population reach 52 500 Give exact and approximate answers

2 6 pts Solve the equation log 5 x log 2 x 2

Find the absolute maximum and minimum values of the function over the indicated interval and indicate the x values at which they occur f x 9x 1 A 0 4 B 5 6 A Find the first derivative of f f x 9 Simplify your answer The absolute maximum value is at x Use a comma to separate answers as needed

Find the degree of this polynomial 8 3x1 816x 8 140

Express in terms of x using the inverse sine inverse tangent and inverse secant functions 1 0 sin Simplify your answer including any radicals Use integers or fractions for any numbers in the expression Points 0 of 1

Match the graphs below with the following descriptions by indicating the graph corresponding to each description in the answer blank to the left of the description Note that because this is a matching question it does not show which parts are incorrect year 1 The concentration of tree pollen in the air over the course of a 2 The mass of Carbon 14 in a historical specimin 3 The cost of a product in an economy suffering from a constant inflation rate that persists over a very long time 4 The distance from the finish line of a ru a race and then running at a constant speed waiting for the start of 5 The number of bacteria on a nutrient petri dish after a group of bacteria is initially introduced A E

O Points 0 of 1 a radioactive isotope used in medicine to test the function of the thyroid gland Its half life is 13 1 hours A 575 microaurie 0 dose of iodine 123 is admini Answer parts a through d patient The quantity Q left in the body after t hours is given by the equation Q 575 a Determine the time it takes for the level of iodine 123 to drop to 25 Ci It takes hr for the level of iodine 123 to drop to 25 Ci Type an integer or decimal rounded to two decimal places as needed 13 1 points

Consider the differential equation B y 1 Be sure not to round your calculations at each step Now use four steps with initial condition y 0 2 A Use linear approximation Euler s method with two steps to estimate y when x 1 dy da B Magnitude of error in Euler with 2 steps 52 y 1 Be sure not to round your calculations at each step B What is the solution to this differential equation with the given initial condition Hint this is asking for a function that has a derivative of 5x What function do you have to start with to get that derivative Try using your knowledge derivatives to guess the solution C What is the magnitude of the error in the two Euler approximations you found Magnitude of error in Euler with 4 steps Note that in this example the right hand side of the differential equation is a function of the independent variable instead of the dependent variabl Nonetheless you can apply Euler s method

The radioactive isotope carbon 14 is present in small quantities in all life forms and it is constantly replenished until the organism dies after which it decays to stable carbon 12 at a rate proportional to the amount of carbon 14 present with a half life of 5699 years Suppose C t is the amount of carbon 14 present at time t a Find the value of the constant k in the differential equation C kC k b In 1988 three teams of scientists found that the Shroud of Turin which was reputed to be the burial cloth of Jesus contained about 91 percent of the amount of carbon 14 contained in freshly made cloth of the same material 1 How old is the Shroud of Turin according to this data Age years 1 The New York Times October 18 1988

Use the symbols provided below to fill in the phase line for the differential equation x x 3 x 1 To place a symbol on the line Click a symbol button then click a point on the line To remove a symbol from the line Click the delete X button then click the symbol on the line decreasing steady state slope undefined increasing delete X

For what positive value of k does the function y sin kt satisfy the differential equation y 169y 0 For what negative value of k does the function y cos kt satisfy the differential equation y 169y 0

Consider the slope field shown a For the solution that satisfies y 0 0 sketch the solution curve and estimate the following and v 1 y 1 b For the solution that satisfies y 0 1 sketch the solution curve and estimate the following y 0 5 y 1 c For the solution that satisfies y 0 1 sketch the solution curve and estimate the following and and y 1 17 1111 LLLL 1 1 1 1 1 1 111 11 LLL 17 1111

Use the curve sketching procedure to analyze the function a y x x 20x 1 x 1 x Determine the horizontal asymptote x 2x 6 2 x b y a y b y x 3x x 7 x5 3x 8 2x x 3x 4 c f x

Find the intercepts of the function a f x x 2x 15 x 10 x 5 Determine if the function is even odd or neither b y a f x 1 x x b y x x 7 c f x 3 X X State the domain of the function x 3x 2 X 4 a y b y 4x 7x 13x 100 c f x 2x 10 Determine the vertical asymptote s x 5x 7 x 3x 28 a y b f x x 4x 4x 6 x 2x 5x 6

For each function determine i the critical values ii the intervals of increasing or decreasing iii the maximum and minimum points a f x 4x 12x 7 b f x x 9x 24x 10 x 2 x 2x 15 c y 1 Find the intervals of concavity for the function and state the points of inflectio a f x x 2x x 2 x x 4 b f x

K Use the definition of the derivative to evaluate the limit 7 h 343 lim Im 1 0 7 h h Question 24 7 3 343

Use both the washer method and the shell method to find the volume of the solid that is generated when the region in the first quadrant bounded by y x y 25 and x 0 is revolved about the line x 6 Set up the integral that gives the volume of the solid as a single integral if possible using the disk washer method Select the correct choice below and fill in any answer boxes within your choice Type exact answers A B OA SO OB 0 25 6 y dy Set up the integral that gives the volume of the solid as a single integral if possible using the shell method Select the correct choice below and fill in any answer boxes within your choice GOODS 0 dx

x 4 y 5 Graph the given hyperbola 9 36 Graph by selecting the correct hyperbola clicking the center then clicking the point on the graph 10 9 8 7 654 3 to 10 9 8 7 65 5 732 2 3 6 8 2 6 7 8 9 10 1

K Determine whether the following statements are true and give an explanation or counterexample a When using the shell method the axis of the cylindrical shells is parallel to the axis of revolution b If a region is revolved about the y axis then the shell method must be used c If a region is revolved about the x axis then in principle it is possible to use the disk washer method and integrate with respect to x or the shell method and integrate with respect to y GEXEND CC True The rotation of the function about the axis creates a cylindrical shape parallel to the axis OD False The rotation of the function about the axis creates a disk shape perpendicular to the axis b Choose the correct answer below A True Any time a function is revolved about an axis the shell method must be used OB False The shell method can only be used when a region is revolved about the x axis OC True Then the axis of the cylindrical shells is parallel to the y ixis D False The disk washer method may be used by integration with respect to y c Choose the correct answer below OA True If the disks washers are perpendicular to the axis of revolution x must be used and if the shells are parallel to the axis of revolution y must be used OB False Use the disk washer method and integrate with respect to y or the shell method and integrate with respect to x C True Integration must always to with respect to the parallel variable be used when revolved about the x axis

A radioactive material decays according to the function A t Age grams of the radioactive material a What is the decay rate of the radioactive material b Graph the function using a graphing utility c How much radioactive material is left after 20 years d When will only 600 grams of the radioactive material be left e What is the half life of the radioactive material a The decay rate of the radioactive material is 2 44 Type an integer or a decimal b Graph the function Choose the correct graph below The graphing window is 0 1600 100 by 0 1 0 25 A e 0 0244t where Ao is the initial amount present and A is the amount present at time t in years Assume that a scientist has a sample of 8 OB OF O C OD OU

The population of a southern city follows the exponential law Use this information to answer parts a and b Point a If N is the population of the city and t is the time in years express N as a function of t N t Noekt Kt Type an expression using t as the variable and in terms of e b If the population doubled in size over 16 months and the current population is 60 000 what will the population be 5 years from now The population will be approximately 807259 people Do not round until the final answer Then round to the nearest whole number as needed

The size P of a certain insect population at time t in days obeys the function P t 700e0 04t a Determine the number of insects at t 0 days b What is the growth rate of the insect population c Graph the function using a graphing utility d What is the population after 10 days e When will the insect population reach 840 f When will the insect population double

A piece of charcoal is found to contain 39 of the element B that it originally had a When did the tree from which the charcoal came die Use 5500 years as the half life of the element B b Using a graphing utility graph the relation between the percentage of the element B remaining and time c Using INTERSECT determine the time that elapses until half of the the element B remains d Verify the answer found in part a a The tree died years ago Round to the nearest whole number as needed HEI

The half life of radium is 1690 years If 30 grams are present now how much will be pres grams Do not round until the final answer Then round to the nearest thousandth as needed

A radioactive material decays according to the function A t Ae 0 0244t where A is the initial amount present and A is the amount present at time t in years Assume that a scientist has a sample of 500 grams of the radioactive material a What is the decay rate of the radioactive material b Graph the function using a graphing utility c How much radioactive material is left after 40 years d When will only 300 grams of the radioactive material be left e What is the half life of the radioactive material

How long does it take for an investment to double in value if it is invested at 2 compounded quarterly Compounded continuously At 2 compounded quarterly the investment doubles in about 34 74 years Round to two decimal places as needed At 2 compounded continuously the investment doubles in about Round to two decimal places as needed years www

Points 0 of 4 After 20 years Trish will have 34898 19 Round to the nearest cent as needed After 20 years Sean will have Round to the nearest cent as needed Sa Trish invests 6 000 in her IRA in a bond trust that pays 9 interest compounded semiannually Sean invests 6 000 in his IRA in a certificate of deposit that pays 8 8 compounded continuously Who has m money after 20 years Trish or Sean

For each set of values calculate the missing variable using the ideal gas law n Set 1 Set 2 Set 3 Set 4 P 2 61 atm 302 kPa P3 782 torr V 1 69 L 2382 mL 3 0 0250 m V4 T 36 1 C T2 288 K 303 K n n1 3 23 mol 1 08 mol 5 26 mol mol T2

Suppose f x 0 for all x andf x dx 18 Let R be the region in the first quadrant bounded by the coordinates axes y f x 2 and x 3 Find the volume of the solid generated by 0 revolving R around the y axis Set up the integral that gives the volume of the solid Use increasing limits of integration Select the correct choice below and fill in the answer boxes to complete your choice Type exact answers OA f x dx GILD OB x dx Sav

For f x a fog 14 11 Y 5 X 4 A and g x b gof 9 find the following composite functions and state the domain of each X c fof 9 4xing your UL Select the correct choice below and fill in any answer boxes within your choice c fof x 9 The domain of fog is x x 0 x x 2 Type an inequality Use integers or fractions for any numbers in the expression Use a comma to separate answers as needed OB The domain off og is all real numbers d gog 9x 36 b g of x 5 Simplify your answer Use integers or fractions for any numbers in the expression Select the correct choice below and fill in any answer boxes within your choice 5x 20 20 4x A The domain of g of is x x 4 Type an inequality Use integers or fractions for any numbers in the expression Use a comma to separate answers as needed OB The domain of g of is all real numbers Simplify your answer

Substitute m 2 and b 3 into the equation

The figure shows two right circular cones one upside down inside the other The two bases are parallel and the vertex of the smaller cone lies at the center of the larger cone s base What values of r and he will give the smaller cone the largest volume possible Note The volume of a right circular cone is V th H

K Find the inverse function on the given interval if specified and graph both f and f on the same set of axes Check by looking for the required symmetry in the graphs f x x 8 for x2 8 Find the inverse function f x x 8 for x 0 Graph f and f on the same set of axes Choose the correct graph below A OB HEZZES OC Ay

The function below has the form of f x log3 x c d 6 7 8 9 10 11 12 13 14 15 16

If two objects travel through space along two different curves it is often important to know whether they will collide The curves might intersect showing that objects will be in the same location at some time but we need to know whether the objects are in the same location at the same time Suppose the trajectories of two objects are given by r t t t t 7 t 1 and r s 1 38 1 9s 1 42s Find two points where the trajectories intersect Enter the point with the smaller x value first 888 1 Do the objects ever collide Point Point att at t and s and s

ti t j t k dt i j k

Find parametric equations for the tangent line at the point cos 2 sin 2 2 on the curve x cost y sint z t y t z t Hint Consider what value of t yields the given point

Find a vector function that represents the curve of intersection of the paraboloid z 3x 5y2 and the cylinder y 6x Use the variable t for the parameter Hint The answer prompts you to use x t r t t 00

Consider the vector function r t t t t5 Compute r t 000 T 1 0 r t r t x r t

For the given position vectors r t compute the tangent velocity vector r t for the given value of t A If r t cos 3t sin 3t Then I B If r t t t Then r 1 C If r t e ti e tj tk Then r 0 i j k

The function below describes the volume V of a three dimensional sphere in terms of its radius Find the inverse of this function that gives the radius in terms of V Assume that rand V are nonnegative Express your answer in the form r f V 4 3 v r Type an expression using V as the variable

Find all the inverses associated with the following function and state their domains 22x x 22 f x TWEE The inverse s is are f x Type an exact answer using radicals as needed Use a comma to separate answers as needed

K Find four intervals on which f is one to one making each interval as large as possible Four intervals on which f is one to one are Type your answer in interval notation Use a comma to separate answers as needed

K Give an example of a function that is one to one on the entire real number line Choose the correct answer below OA f x sin x 3 OB f x x x 2x O C f x x OD f x 3x 2

Given a 22 33 and 6 37 3 find a b a b 3

2 Here is a polygon A D B C a Draw the dilation of ABCD using center A and scale factor Label the dilation EFGH b Draw the dilation of ABCD with center D and scale factor Label the dilation IJKL

4 List the intervals for each section of the function and state whether it is increasing decreasing or constant Label them A E starting from left to right on the graph A B C D E Distance km 70 60 50 40 30 20 10 10 00 11 00 2 00 13 00 The Which interval has the fastest speed What is the speed on this interval 14 00 In a couple sentences write a story describing what this graph could represent Include a description for each interval

A water tank shown to the right is shaped like an inverted cone with height 4 m and base radius 1 m a If the tank is full how much work is required to pump the water to the level of the top of the tank and out of the tank Use 1000 kg m for the density of 2 water and 9 8 m s for the acceleration due to gravity b Is it true that it takes half as much work to pump all the water out of the tank when it is filled to half its depth as when it is full Explain a Draw a y axis in the vertical direction parallel to gravity and choose the bottom of the tank as the origin For 0 y 4 find the cross sectional area A y in terms of y 16 2 A y 1000x h 20 Simplify your answer Type an exact answer using as needed Use integers or fractions for any numbers in the expression 4 m

3 The function f x x 3 is one to one a Find the inverse of f and check the answer b Find the domain and the range of f and f c Graph f f1 and y x on the same coordinate axes and if necessary fill in the answer box to complete your choice A The domain is x x B The domain is x x C The domain is x x D The domain is the set of all real numbers Find the range of f if necessary fill in the Select the correct choice below and answer box to complete your choice 1 c Graph f f and y x on the same coordinate axes The graph of y x is shown using dashed line Choose the correct graph below OA O C B D E