Limits & Continuity Questions and Answers

Charles plots the point with the polar coordinates How does he plot this point Drag a value or phrase into each box to correctly complete the statements Charles first determines which line the angle of rotation places the point on This line tells Charles that the point must lie in 5 3 will lie quadrant I if r is positive or The radius of 5 tells Charles that the point lies on the fifth circle of the polar plane The value of r is negative Therefore the point the angle of rotation quadrant II quadrant III if r is negative quadrant IV in the opposite quadrant of in the same quadrant as

Let R be the finite region enclosed by the curve y 62 5 4x and the z axis Let V be the volume obtained by rotating R about the axis Express the volume as an integral of the form V f z dr a b b THE Evaluate this integral to find the volume in question Give your answer in terms of

Suppose an object moves along a linear path according to the velocity function v t e where v t is in meters per seconc If the object s position at t 1 is s 2 what is the function in terms of t that represents the position at any time t 02 e dt 02 e du 02 2 2e du 0 2 edu

Let 5 x 2 Use a graphing utility to confirm the inequality graphically Then complete the table to confirm the inequality numerically Round your answers to five decimal places S4 In x 1 55 0 0 X Sn In x 1 Sn 1 0 2 0 4 0 6 0 8 1 0

At a point which is 92 feet from a building the angle of elevation of the top of the building is 50 and the angle of elevation to the top of a flagpole mounted on top of the building is 58 How tall is the flagpole

45 50 Rectangular Equations to Polar Equations Convert the equation to polar form 45 x y 47 y x 49 x 4 46 x y 9 48 y 5 50 x y 1

a Rate of change b Average Rate of Change c Secant

3 Limits from a Graph The graphs of f and g are given Use them to evaluate each limit if it exists If the limit does not exist explain why a lim f x g x c lim f x g x 3 0 e limx f x y f x b lim f x g x f x 1 g x f limV3 f x d lim ya x g x X

5 18 Limits at Infinity Find the limit 6 I 5 lim 7 lim 2x 1 St 1 4x 1 2 3x 9 lim 6 lim 8 lim 3 2 3x 4x 5 10 lim I x 2 x 1

19 32 Finding Limits Evaluate the limit if it exists x x 6 x thi X 2 4x 3x 4 19 lim 1 2 21 lim x x 6 x 2 7 9 1 3 21 71 3 23 lim 20 lim 22 lim 16 3 2 X 2 24 lim

5 10 Estimating Limits Numerically Complete the table of values to five decimal places and use the table to estimate the value of the limit Vx 2 5 lim x f x x 2 2x x 6 6 lim x f x 7 lim X x 3 9 3 99 3 999 4 001 4 01 4 1 f x 1 9 0 9 1 99 1 999 0 99 0 999 2 001 2 01 2 1 1 001 1 01 1 1

Tutorial Exercise Find the interval of convergence of the power series Be sure to include a check for convergence at the endpoints of the interval 9x 1 Step 1 n 17 1 Recall the Ratio Test which states that ifa is a series with nonzero terms and lim n 81 For any fixed value of x such that x 0 let a 9x n 1 n 1 Then lim 818 an 1 an lim 318 lim 81B n 1 9x n 2 n 9x 1 n 1 n 1 9x n 2 n 1 a n 1 1 then a converges absolutely If lim n 318 a n P a n n 1 1 or lim 318 a n 1 a n then a diverges n

11 14 Using a Calculator Use a calculator to evaluate the expression Round your answer to five decimal places 11 a sin 22 b cot 23 12 a cos 37 b csc 48 13 a sec 13 b tan 51

Which of the following sets of parametric equations represents x 3 y 5 4 Ox 3 2cos 0 y 5 2sin e Ox 3 4cos 0 y 5 4sin e Ox 3 2cos e y 5 2sin e Ox 3 4cos e y 5 4sin Jhy

On an EKG strip a nurse determines that a patient s heart is beating 5 times every 4 seconds Calculate the time for each heart beat in seconds Calculate the patient s heart rate in beats min Hint Divide 60 by the time per beat

DETAILS Find the interval of convergence of the power series Be sure to include a check for convergence at the endpoints of the interval If the answer is an interval enter your answer using interval notation If the answer is a finite set of values enter your answers as a comma separated list of value 2 nix n 1 2n

Consider the equation 2x 7y 5 Solve the equation for x in terms of y

Evaluate the determinant of the matrix A 47 491 38 2

Sketch out the graph of the following functions Show steps and make a table of the graphs Include the labels for y intercepts axis asymptotes scales functions and explain the transformation from g x to h x C O g x 2x X h x 3 2 4

Question Write the following system of equations as a single matrix equation 9x 8y 5z 1 3x 6y 3 3y2z 12

You know that 12 I St 9 t g 7 1 g t dt 33 g 12 3 Evaluate 12 S 7 g x 4 g x dx Upload your handwritten work in the box below Make sure to show the entire process of evaluating this integral Report EXACT answer

Analyze the equation make a theta roe table and graph 0 4 cos 20

Given the following matrices if possible determine 34 If not state Not Possible Answer A Selecting a checkbox will replace the entered answer value s with the checkbox value If the checkbox is not selected the entered answer is used

2 Find the limits in the following problems a b lim 1 0 49 7 0 02105 000 x lim H 5 2 25 x 5 4 989 98 4 999 99 19 5 5 01 10 01 5 02 10 02 3 a lim 72 00 0 0 12x5 3x4 2x 5 24x5 4x 6 22 b lim 8118 x 9 9 x 2 96 81 4 Find the horizontal and vertical asymptotes of the following as well as any other points of discontinuity c functions 9r 6 5x 3x 2x 5 24x 4x 6 u f x 2x 10x 12

f x 4x 3 f 3x 2 x 3 X A 11 B 8 C 4 D 4 E 6

each month to reduce the loan What is the total cost of this loan that is what is the sum of all the monthly interest payments Cost of loan Problem 10 1 point The government has decided on a tax rebate program to stimulate the economy Suppose that you receive 700 and you spend 70 of this and each person who receives what you spend in turn spend 70 of what they receive and this process continues indefinitely What is the total amount spent to the nearest dollar that results from this government action Total amount spent

G Use the method of elimination to solve the following system of equations If the system is dependent express the solution set in terms of one of the variables Leave al fractional answers in fraction form Answer Selecting an option will display any text boxes needed to complete your answer Only One Solution 3 7 2 2x 3y 14 4x 6y 28 2 Inconsistent System O Dependent System ER 2 Keypa Keyboard Short

e Consider the function e x 5x 6 f R 1 R f x x 1 Find a rule for a function g R R so that g x f x for all x 1 and g x continuous at x 1 Justify your choice carefully

Using the appropriate theorems explain why the function is continuous at every number in its domain x 9 2x 9 F x is a polynomial so it is continuous at every number in its domain F x is a rational function so it is continuous at every number in its domain F x is a composition of functions that are continuous for all real numbers so it is continuous at every number in its domain F x is not continuous at every number in its domain none of these Find the domain Enter your answer using interval notation F x

The graphs of f and g are given Use them to evaluate each limit if it exists If an answer does not exist enter DNE y f x b lim a lim f x g x lim f x g x c lim f x g x 0 ya d lim f x 1 y y g x 0 1 1

Use the graph of g in the figure to do the following A Find the values of x in 3 3 at which g is not continuous B Find the values of x in 3 3 at which g is not differentiable C Sketch a graph of g 4 2 10 8 6 4 2 Ay J 10

Let f x C cx 2x x CX cx if x 5 if x 5 For what value of the constant c is the function f continuous on

We are asked to find the following limit h 5 25 h lim h 0 h We will first attempt to do so using direct substitution Let f h h 5 2 25 We know that as f is a rational function if 0 is in the domain off th the given limit is equal to f 0 The function Select defined at h 0 therefore we cannot evaluate the limit by this direct substitution Therefore we will next attempt to use the quotient law which we recall is as follows lim g x h 0 lim 9 x h 0 k x lim k x h 0 We note that the quotient law requires that the limit in the denominator is not equal to 0 so we must check that as follows lim h h 0 So this means that the quotient law Select

Given that lim f x 4 lim g x 4 find each limit if it exists If an answer does not exist enter DNE a lim f x 2g x X 2 b lim g x X 2 c lim f x X 2 5f x x 2 g x d lim g x x 2 h x e lim f lim g x h x X 2 f x lim h x 0 X 2

For the function to be continuous at x 2 we must have a f x sin x 2 x 2 2 4 x 2 a x 2 x 2 and f 2

Suppose f x and g x are differentiable at x 0 and f x and g x are continuous at x 0 If f 0 g 0 0 f 0 6 and g 0 2 evaluate the limit f e t 1 lim x 0 g sin 6x Type your solution in the space provided Explain in English how you solved the problem Provide sufficient details of your mathematical calculations

Suppose f x and g x are two functions satisfying the following properties f 1 3 lim f x 2 lim f x 1 x 1 x 1 g x g 1 1 lim g x 2 lim g x x 1 Determine whether lim 2f x 3f x x exists or not If it does exist what is it equal to If it does not exist why doesn t it exist g x x the details For example if you find a limit include the answer to that limit etc Type your answer an

4 15 cos x y if sin x 54 5 2 cos x sin y 5 5 11 and cos y 7 11

Find the limit if it exists lim x 0 x 5 O 5 0 0 O Does not exist 5

23 Tara recorded this data to show how far her pet turtle traveled over an 8 hour wo period of time Based on this informa tion how far did the turtle travel after 6 hours 8 Time h 2 4 6 8 uch A 105 m B 125 m C 135 m D 150 m Distance m 45 90 180 1992

2 A boat is sailing at a bearing of 140 with a speed of 26 mph It hits a current of 8 mph at a bearing of 195 Find the resultant a magnitude and b direction as a true bearing of the boat

Consider A ABC where a 8 c 11 and m2C 33 What are the measures of the missing angles and missing side Enter your answer by filling in the boxes Enter all values rounded to the nearest ten m A 23 33 m B 16 8 b 123 7

7 Sketch the curve on polar graph paper a r 0 b r sin 20 c 1 cose d r 1 cos 20 r

6 find a Cartesion equation for r 6 0050 Graph it X rcosa r rsing LanG Y

Use your knowledge of trig identities not a calculator to find each trig value 3 9 1 2 9 If sin 0 51 find csc

Let f x be given by the following graph 1 5 1 0 5 1 8 5 0 5 1 0 5 1 1 5 2 T Show Transcribed Text what s the f 1 where are the turning point of f x

Find the extreme values of the function f x y 2x 3y on the circle x y 4 Use Lagrange multipliers findf Ag Answerin ve Her

11 000 The function N t 1 999e t weeks after the initial outbreak Step 1 of 2 How many people were ill initially Round to the nearest person models the number of people in a small town who have caught the flu AnswerHow to enter your answer opens in new window 2 Points people Keypad Keyboard Shortcuts

Use Stoke s theorem to evaluate Find curl F x y z 7 0 0 1 2y Answer format a b c If curl F curl F dS for the vector field F x y z yi 2 x k for the surface S generated by the intersection of y z 3 and the cylinder x y 1 S X

Find the sum of the first n terms using the formula a 1 1 2 4 8 16 25 terms Round your answer to the nearest hundredth