Limits & Continuity Questions and Answers

Sketch the graph of a function that satisfies all of the given conditions f 0 f 2 f 4 0 f x 0 if x 0 or 2 x 4 f x 0 if 0 x 2 or x 4 f x 0 if 1 x 3 f x 0 if x 1 ora 3 2 2 6 2 4

The data in the table below represents the percentage of patients who have survived after diagnosis of advanced stage breast cancer at 6 month intervals of time Time after Diagnosis years 0 5 1 1 5 2 2 5 3 3 5 Percentage Surviving 95 8 83 4 74 0 58 7 46 6 40 9 32 2 a Using a graphing utility build an exponential model from the data y 120 5681 0 6918 Type integers or decimals rounded to four decimal places as needed b Express the function found in part a in the form A t Aet where t x A t Type integers or decimals rounded to four decimal places as needed

Atte Sketch the region enclosed by the given curves x 9y2 x 2 7y 1 0 y 0 5 0 5 1 0 1 0 0 0 5 0 5 8 1 0 M 1 0 0 5 0 5 y 1 0 1 0 0 5 webassign net JS Secti y 8 MacBook Air 6 8 0 5 1 0 X

nd the domain and range of the function 6 1 g z 2 1 A D 1 00 R 0 00 C D 1 R 0 00 B D 1 0 R 0 D D 0 R 0 00

The function f x x5 52x 2x 147x 98 has the graph given to the right Use the graph to factor the polynomial What is the factored form of the polynomial f x Simplify your answer Type your answer in factored form Factor completely 10 6000 12000 10 O G

Find the partial sum without using a graphing utility 100 50 En En n 51 n 1

Find the sum of the infinite geometric series if possible If not possible enter IMPOSSIBLE n 0

Solve the logarithmic equation for x Enter your answers as a comma separated list log x log x 48 2 X

Determine whether the sequence converges or diverges If it converges find the limit If the sequence diverges enter DIVERGES 9n 20 lim an 11 an

8 Consider the function f x 3x 5x 4 7x a Find a term that when added to f would change the end behavior of f as but not as x b Find a term that when added to f would change the end behavior of f on both sides as x and as c Find a term that when added to f would NOT change the end behavior of f

2 It is known for a function f that as f x and as a o f x What must be true about the leading coefficient of f and the degree of f 3 Consider the graph of the polynomial function f shown a Describe the end behavior of f b Is it possible for f to have a degree of 5 Give a reason for your answer y 30 25 20 15 10 5 3 2 1 6 Go 0 10 15 20 25 30 2 3 4 5 6 X

roblem 6 Find an integral which represents the volume of the solid obtained by rotating the region enclosed by the curves y x and y 8x about the line x 8 Problem 6 64 A 50 D G A 8 8 dy B 50 7 z 8 8 5 C a 8 2 8 2 dy 64 64 a 8 2 8 y 8 5 8 2 8 8 dv 64 Just Save Problem 6 Your Answer 8 E a 8 8 5 f r 8 8 5 8 dy F dy Submit Problem 6 for Grading Attempt 1 Attempt 2 Attempt 3 G A 4

Problem 6 Find an integral which represents the volume of the solid obtained by rotating the region enclosed by the curves y x and y 8x about the line x 8 64 4 7 8 5 8 2 64 D n 8 2 8 64 0 64 x 8 8 C z 8 8 2 TU 8 2 8 5 1 a 8 2 8 G 8 8 2 H 8 5 8 4 Problem 6 Select you

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In Silicon Valley a number of computer firms were found to be contaminating underground water supplies with toxic chemicals stored in leaking underground containers A water quality control agency ordered the companies to take immediate corrective action and contribute to a monetary pool for the testing and cleanup o the underground contamination Suppose that the required monetary pool in millions of dollars is given by the following function where x is the percentage expressed as a decimal fraction of the total contaminant removed Complete parts A through E 3x P X 1 X 0 x 1 w C Find any horizontal and vertical asymptote Find the vertical asymptote s if any Select the correct choice below and if necessary fill in the answer box to complete your choice CEILE A The vertical asymptote s is are x Simplify your answer Use a comma to separate answers as needed OB There are no vertical asymptotes Find the horizontal asymptote s if any Select the correct choice below and if necessary fill in the answer box to complete your choice OA The horizontal asymptote s is are y Simplify your answer Use a comma to separate answers as needed OB There are no horizontal asymptotes D Find the x and y intercepts Find the x intercept s Select the correct choice below and if necessary fill in the answer box to complete your choice A The x intercept s is are x Simplify your answer Use a comma to separate answers as needed B There are no x intercepts

Use the graph of y f x assuming f x 0 i x bord and f x 0x f to answer parts A through F Q Q AV VA A Identity intervals on which the graph of t is concave upward Choose the correct answer below OA The graph is never concave upward OB a c c e g h OC d e g h OD b c c e g h OF a b d e g h OE a b e g O G a c c d e g OH e g B Identify intervals on which the graph of f is concave downward Choose the correct answer below OA a c c e g h OC b c c e g h OE d e g h OG b ck c dk e a OB a c c d e g OD a b e g OF e g OH The araph is never concave downward

Assume f x is continuous on 0 00 Use the given information to sketch the graph off 1 0 3 6 7 1 2 0 f x X f x 4 22 ND 0 2 ND 0 2 Choose the correct graph below OA 3 0 0 OB 6 20 S CAFK O C Q 5 OD

sinx1 lim 4 x 0 x 2x 6x 2x 6x How should the given limit be evaluated Select the correct choice below and if necessary fill in the answer box to complete your choice OA Use l Hopital s Rule exactly once to rewrite the limit as lim Xx 0 B Use direct substitution C lim x 0 BURD which is convenient and applicable Use l Hopital s Rule more than once to rewrite the limit in its final form as lim x 0 OD Multiply the expression by a unit fraction to obtain lim x 0 Evaluate the limit e sinx 1 eme Type an exact answer e sinx 12x 12x 12

Prove that lim 1 X 8 lim X 8 818 In 1 a x 1 x Apply l H pital s Rule by finding the derivative of the numerator and the denominator separately 2 lim ax 1 a x lim 2 X 8 X X lim a x48 Evaluate the limit e for a 0 lim Simplify the expression before finding the limit ax 1 a x X X 1 818 a 2 a 2 A

Recall the following definition Let 0 be an angle in standard position The reference angle of 0 is the acute angle formed by the terminal side of and the horizontal axis Below express the reference angle for in the same units degrees or radians as 0 You can enter arithmetic expressions like 210 180 or 3 5 pi The reference angle of 100 is The reference angle of 350 is The reference angle of 4 is 0

For the given values of an angle a find the terminal point P x y on the unit circle a If a then x b If a then x c If a then x d If a then e If a 7 then x f If a 37 then x x and Y and y and y and y and y and y GI

Evaluate the following limit lim sinx cotx X 0 lim sinx cotx X 0 Simplify your answer

e sinx 1 lim 4 3 2 x 0 x 2x 6x Use l H pital s Rule when it is convenient and applicable How should the given limit be evaluated Select the correct choice below and if necessary fill in the answer box to complete your choice OA Use l Hopital s Rule exactly once to rewrite the limit as lim X 0 B Use direct substitution OC Use l Hopital s Rule more than once to rewrite the limit in its final form as lim x 0 Com OD Multiply the expression by a unit fraction to obtain lim x 0

3 Prove that lim 1 x 00 X 00 OD L lim x In X 00 mix OC L lim x In 1 X 8 lim ea for a 0 lim X10 lim x 00 lim X 8 A X In 1 a x Apply l H pital s Rule by finding the derivative of the numerator and the denominator separately In 1 a x 1 x x x a X 1 In 1 ax

Stan lim 7 sin x x 0 em Choose the correct answer below and if necessary fill in any answer boxes to complete your choice OA L lim 6 tan x In 7 sin x is finite and therefore lim 7 sin x Stan x 0 X 0 Type exact answers in simplified form OB can be written as the expression The limit does not exist because it has the indeterminate form 0 which cannot be written in the form 010 or O D The limit does not exist because it has the indeterminate form 1 which cannot be written in the form 10 The evaluated limit is Colo 88 or O C 0 00 The limit does not exist because it has the indeterminate form oo which cannot be written in the form so that H pital s rule can be applied or so that H pital s rule can be applied 80 88 so that Hopital s rule can be applied

limit methods to determine which of the two given functions grows faster or state that they have comparable growth rates 0 05x xe Use the given functions to create an expression whose limit can be used to determine which function grows faster lim 818

OA lim x b OC lim OC x X 00 b lim b Apply L H pital s Rule Choose the correct answer below lim lim Evaluate x 00 b lim x b X x p X x p lim x xb P X x p x p P 1 x Inb p b lim 1 lim x 00 Inb p b X lim x 00 Inb p b x p x p x p P J Simplify your answer TIM B lim lim x 00 b D lim OB OD x x 00 b lim x b xxb x p lim x wb x p lim x p x b x p lim 300 Inb p b a lim a p Clear all

Evaluate the following limit or explain why it does not exist lim 1 2x X 0 Select the correct choice below and if necessary fill in the answer box to complete your choice lim 1 2x X 0 Type an exact answer OB The limit does not exist because it is not defined as x 0 OC The limit does not exist because l H pital s Rule cannot be applied OD The limit does not exist because the limit approaches co as x 0 Occas OA

Use analytical methods to evaluate the following limit sin rx lim x 4 x 4 sin xx lim x 4 X 4 Simplify your answer

Use analytical methods to evaluate the following limit lim x 00 lim x48 6 x 4 4x 4x Simplify your answer

Evaluate the following limit lim 1 ax for positive constants a and b x 0 b lim 1 ax x 0 Type an exact answer

How can it be shown that f x and g x have comparable growth rates f x g x Show that lim 848 OB Show that lim 818 O C O D f x g x Show that lim Show that lim f x lim g x 848 x x X 00 M where 0 M o M is nonzero and finite f x 9 x 00 grow at a comparable rate as x 00 0 Substitute the given functions f x and g x into the limit function lim 0

3 6 pts Solve the equation Give exact and approximate solutions In x 4 In 2x 5

When a plant or animal dies it stops acquiring carbon 14 from the atmosphere The amount y in grams of carbon 14 in the body of an organism after t years is y a 5 75730 where a is the initial amount What percent of the carbon 14 is released each year

Determine the domain of each of the following functions Report the solution in interval form Also identify the equation s of the vertical asymptote s of each function f z log 32 6 The domain of is f The function f has a vertical asymptote at h z In 12 2x The domain of his The function h has a vertical asymptote at g x log r 1 The domain of g is The function g has a vertical asymptote at 9 and

Use the definition of the derivative to evaluate the limit lim h 0 lim h 0 3 h 7 h 343 h 3 h 7 h 343 h 147 343 log 7

In the spreadsheet pictured an arrow indicates the contents of a cell are copied down its column n the spreadsheet below column A holds different values of h and column B computes 0 h f 0 for the function f x x Writing abs A1 in a spreadsheet will compute the absolute value of the number in cell A1 a What number shows up in cell A6 123456 A 1 A1 10 B abs A1 A1 o f 0 does not exist Select below the best explanation of a way you could suspect that from the spreadsheet The values in column B are not converging to one number The numbers in column B have different signs The values in column A are converging to 0 The values in column A are not converging to 0

Use the given graph of y f x to find the intervals on which f x 0 the intervals on which f x 0 and the values of x for which f x 0 Sketch a possible graph of y f x www OA X On what subinterval s is f x 0 Select the correct choice below and if necessary fill in the answer box to comp your choice Type your answer using interval notation Use a comma to separate answers as needed OB There are no such subintervals

Find the vertex focus and directrix for the following parabolas x 2 8 y 2 Vertex Focus

K Solve the following equation Simplify your answer Use a comma to separate answers as needed

the inverse of the function below on the given interval and write it in the form y f x b Verify the relationships f f x x and f f x x f x x 36 for x 0 A x x 36 36 Type integers or simplified fractions a f x x 36 b Verify the relationship f f x x Select the correct choice below and fill in the answer box es to complete your choice OB x 36 x tion 8 7 1 32 Part 2 of 3 X HW GOCOR P

Compute the derivative of the natural log of kx Begin by letting u kx and substituting u into the equation d dx d In kx In u dx Compute the derivative using the chain rule d dx Inu U dx CH U

a Find the inverse of the function below on the given interval and write it in the form y f x b Verify the relationships f f x x and f f x x f x x 36 for x 0 a f x

K 1 Use x e to explain why Inx for x 0 X d dx First explain why x must be greater than zero What is the range of the function f y OA 0 00 OB 0 00 O C 00 00 OD 1 00

K a Find the inverse of the function below on the given interval and write it in the form y f x b Verify the relationships f f x x and f f x x f x x 6 A OB OA 5 B 9 X 6 6 EX 5 x x 6 X Type integers or simplified fractions Verify the relationship f f x x Select the correct choice below and fill in the answer box es to complete your choice 5 9 9 f f x 6 x x 6 f1 f x x 6 HELEID X simplified fractions

The data in the table below represents the percentage of patients who have survived after diagnosis of advanced stage breast cancer at 6 month intervals of time Time after Diagnosis years 0 5 1 1 5 2 2 5 3 3 5 Percentage Surviving 95 2 82 6 74 0 58 7 46 4 40 2 33 8 a Using a graphing utility build an exponential model from the data y 0 0x Type integers or decimals rounded to four decimal places as needed

If the rate of inflation averages r per annum over n years the amount A that SP will purchase after n years is A P 1 r where r is expressed as a decimal If the inflation rate averages 3 1 how much will 2 000 purchase in 5 years In 5 years 2 000 will purchase Round to the nearest cent as needed

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 res y a Choose the correct answer below www OA False The shell method produces shells that are always perpendicular to the axis of revolution B True The shell method always produces shells parallel to the y axis C True The rotation of the function about the axis creates a cylindrical shape parallel to the axis D False The rotation of the function about the axis creates a disk shape perpendicular to the axis b Choose the correct answer below OA 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 axis O D False The disk washer method may be used by integration with respect to y Final chec

Suppose that g x 6 2 a What is g 1 When x 1 what is the point on the graph of g b If g x 38 what is x When g x 38 what is the point on the graph of g 13 6 a g 1 13 6 The point Type integers or simplified fractions b If g x 38 x The point Type integers or simplified fractions is on the graph of g is on the graph of g

Solve the following logarithmic equation log 3x 2 2 log 3x 15 Select the correct choice below and if necessary fill in the answer box to complete your choice A 1 The solution set is 27 Simplify your answer Use a comma to separate answers as needed Type an integer or a simp B The solution is the empty set