Limits & Continuity Questions and Answers

Plot a shape as your original figure Object 20pts THE SHAPE SHOULD HAVE AT LEAST 6 POINTS o Dilate the figure using three different scale factors of your choice 2 enlargements 10 pts 1 reductions 10 pts each o Label each object and image 10pts o Write coordinates for each shape 20 pts Also create a horizontal vertical stretch 20 pts of your ORIGINAL figure 10 pts each Color each dilation a different color 5 pts

erpetual inventory using LIFO Beginning inventory purchases and sales for Item 88 HX are as follows 94 units 31 75 units July 1 Inventory July 8 Sale July 15 Purchase 104 units 33 July 27 Sale 87 units Assuming a perpetual inventory system and using the last in first out LIFO method determine a the cost of goods sold on July 27 and b the inventory on July a Cost of goods sold on July 27 b Inventory on July 31

Find the average rate of change of the following functions over the given interval a f x 3x 7 over the interval 0 3 b g x 2 over the interval x x h

Solve the given equation Enter your answers as a comma separated list Let k be any integer Round terms to three decimal places where appropriate If there is no solution enter NO SOLUT tan 8 9 2 cos 0 1 0 8

Solve the given equation Enter your answers as a comma separated list Let k be any integer Round terms to three decimal places where appropriate If there is no solution enter NO SOLUTIC 3 sin 8 7 sin 8 2 0 A

Find the vertical horizontal and oblique asymptotes if any for the following rational function R x X 2 Find the vertical asymptotes Select the correct choice below and if necessary fill in the answer box es to complete your choice A The function has one vertical asymptote x 2 Type an equation Use integers or fractions for any numbers in the equation OB The function has two vertical asymptotes The leftmost asymptote is and the rightmost asymptote is Type equations Use integers or fractions for any numbers in the equations OC The function has no vertical asymptote Find the horizontal asymptotes Select the correct choice below and if necessary fill in the answer box es to complete your choice OA The function has one horizontal asymptote Type an equation Use integers or fractions for any numbers in the equation and the bottom asymptote is OB The function has two horizontal asymptotes The top asymptote is Type equations Use integers or fractions for any numbers in the equations OC The function has no horizontal asymptote

Find the vertical horizontal and oblique asymptotes if any of the given rational function x 8 X 7x 10 H X 2 Find the vertical asymptotes Select the correct choice below and if necessary fill in the answer box es to complete your choice OA The function has one vertical asymptote Type an equation Use integers or fractions for any numbers in the equation OB The function has two vertical asymptotes The leftmost asymptote is and the rightmost asymptote is Type equations Use integers or fractions for any numbers in the equations OC The function has no vertical asymptote

For the continuous distribution below what does the shaded area represent 74 OP x 7 OP x 0 OP 6 x 7 0 1 2 3 4 5 6 7 8 9 9 10 X

Compute the directional derivative of the function g x y sin x x y at the point P 3 2 in the direct 8 15 17 197 Be sure to use a unit vector for the direction vector The directional derivative is IMOK

The function t f s represents the number of school busses t needed to transport s students to school Interpret the meaning of the statement f 33 650

Let m f t represent the total money earned while working at a company for t years Explain what information is given by the function f Be sure to include both the input and the output in your explanation

A 2 If f x 3 5x then f 1 A 8 B 2 C 2 D 8

If tanh x 3 sinh x cosh x 1 coth x 41 9 sech x 40 41 csch x 40 9 9 40 find the values of the other hyperbolic functions at x X X x X X

Find the numerical value of each expression Round your answers to five decimal plac a tanh 0 b tanh 1

Select the correct choices that complete the sentence below If two functions f and g are inverses then fog x and g x f x

y r x b graph f and on the same axes and c give the domain and the range of f and if the function is not one to one say so 1 x x 5 a Write an equation for the inverse function in the form y ro Select the correct choice below and if necessary fill in any answer boxes to complete your choice A The function f x is one to one and r x Simplify your answer B The function is not one to one

If a function f has an inverse and f 3 6 then 6 If a function t has an inverse and 1 3 6 then f 6

b lim tanh x 8118 12 0 17 Points Use the definitions of the hyperbolic functions to find each of the following limits If the limit is infinite enter co or co as appropriate If the limit does not otherwise exist enter DNE a lim tanh x X18 c lim sinh x X18 d lim sinh x X 00 e lim sech x X 00 DETAILS SCALC lim coth x X18 195 1

Determine whether the series n 1 6 5 10 OD Select the correct answer below and if necessary fill in the answer box within your choice OA The series diverges because it is the sum of two geometric series at least one with r21 OB The series converges because it is the sum of two geometric series each with r 1 The sum of the series is Simplify your answer OC The series diverges because lim n18 converges or diverges If it converges find its sum 6 57 10 The series converges because lim Simplify your answer 6 5 81x 10 0 or fails to exist 0 The sum of the series is

2 If f x x 1 and g x 3 4x then gof 1

Find the exact average value of f x x 4 NOTE Give an exact answer 1 x 1 The average value of f x on 0 x 4 is on the interval x 0 to

Use the Fundamental Theorem to calculate the definite integral below NOTE Give an exact answer So 2x x 1 dx

If z x y with x 0 d 2 and d 5 find when x dt 8 and y 15 Answer dz dt

QUESTION 22 The function f is one to one Find its inverse f x x 6 x 0 O f x x 6 x 6 Of x x 6 x 0 Of x x 6 x 6 f x x 6 x 0

Find the vertical asymptotes of the rational function f x x2 16 x2 5x 4 x 1 x 4 X 1 X 4 x 1 x 4 x 1

Given f x 4x 5 g x 5x 1 20x 9 20x 24 20x 1 20x 4 Find f g x

Find the x and y intercepts of f f x x 1 x 4 x 1 Ox intercepts Ox intercepts Ox intercepts Ox intercepts 1 1 4 y intercept 4 1 1 4 y intercept 4 1 1 4 y intercept 4 1 1 4 y intercept 4

Find the domain of the function f x In 4x x 145x 0 44x44 0 x 4 Ox 4

Evaluate the expression using the values given in the table g f 1 X 1 71012 5 2 1 3 14 f x 2 10 1 14 g x 1 87 10 800 1 3 2 13 10 2 7 8 X

2 A Given the following information about the first and second derivatives of a function of that is continuous on indicate all possible critical points the locations of the local max min and the inflection points and then sketch a possible graph of f 0f 0 f 0f 0 f 0f 0 f 0f 0 f 0 f 0 f 0f c I

5 Determine the difference qerotient x h f h where h o for each of the following following function for t

o f x e f x h f x h x n write the numerator as a factor of e th ex h f x h f x e e e h h

1 For the function F x 5 a graph the rational function using transformations b use the final graph to find the X domain and range and c use the final graph to list any vertical horizontal or oblique asymptotes What is the range of the given function Select the correct choice below and if necessary fill in the answer box to complete your choice O A The range of the given function is yly Type an integer or a simplified fraction B The range of the given function is yly Type an integer or a simplified fraction Use a comma to separate answers as needed OC The range of the given function is yly Type an integer or a simplified fraction O D The range of the given function is the set of all real numbers

4x 5X 9 x 7 00 8x 3x 8x b lim

11 lin NEX 4x 5x 9 8x 3x x

ASK YOUR TEACHER Ornithologists have determined that some species of birds tend to avoid flights over large bodies of water during daylight hours It is believed that more energy is required to fly over water than land because air generally rises over land and falls over water during the day island 5 km W L B 9 km 1 144 D nest A bird with these tendencies is released from an island that is 5 km from the nearest point B on a straight shoreline flies to a point C on the shoreline and then flies along the shoreline to its nesting area D Points B and D are 9 km apart Assume that the bird instinctively chooses a path that will minimize its energy expenditure Round your answers to two decimal places a In general if it takes 1 2 times as much energy to fly over water as land to what point C should the bird fly in order to minimize the total energy expended in returning to its nesting area 1 5 km from B MY NOTES W L b Let W and L denote the energy in joules per kilometer flown over water and land respectively Assuming the bird s energy expenditure is minimized determine a function for the ratio in terms of x the distance from B to C 25 c What should the value of Wbe in order for the bird to fly directly to its nesting area D L X PRACTICE ANOTHER d If the ornithologists observe that birds of a certain species reach the shore at a point 3 km from B how many times more energy does it take a bird to fly over water than land 1 144

Analyze the polynomial function f x x x 5 using parts a through h below The last 2 FUNCTIS VI mupiicity z su te graphi vii Roches d Use a graphing utility to graph the function The graphs are shown in the viewing window Xmin A B OA Q OB aLA V The gitation the functions of pHCRY SO the graph UNI cusses the axis al x 20 Xmax 20 Xscl 2 Ymin e Approximate the turning points of the graph The turning point s of the graph is are 0 0 3 33 18 52 Type an ordered pair Round each coordinate to the nearest hundredth as needed Use a comma to separate answers as needed f Use the information in parts a through e to draw a complete graph of the function by hand Choose the correct graph below O C O C 30 Ymax 30 Yscl 3 Choose the correct graph below O D D is

9 For what angles x in 0 27 does the sin x have the same value as cos 7

Use l Hopital s Rule to evaluate the limit cos 3x 1 x2 17 lim x 0 A 18 lim x sin X 8 A 1 5x B 0 B 0 C 3 20 C 1 2 2 D D 5 17 18

Divide 1 m 7m 11 m 8 61

The graph of y f x is shown Assume f x is continuous 8 7 6 5 4 3 2 Find xf x 12 lim x4 f x x 13 Question Help Mu 1 8 7 6 5 4 3 2 1 2 3 4 5 6 7 1 8 1 2 3 4 5 6 7 q

Suppose f x and g x have the following values f 8 4 and f 8 6 g 8 1 and g 8 1 Assume all functions are continuous Find the following f x 3f x 4 lim

Suppose f 2 8 f 2 9 g 2 3 g 2 1 and that f x g x are continuous for all x f x 8 lim x2 g x 3 Question Help O Cannot Be Determined Video Message instructor

Match the function with its graph f x x cos x 4 2 4 2 Mw 2T 2 0 2 6 4 2 0 5 2 And 2 0 2 2 6 10 6 Describe the behavior of the function as x approaches zero As x 0 f x 2T 2 6 6 2 0 5 5 2 S 2T 2T

A company that produces snowboards which are seasonal projects forecasts monthly sales for one year to be S 77 00 42 25 cos 6 where S is the sales in thousands of units and t is the time in months with t 1 corresponding to January a Use a graphing utility to graph the sales function over the one year period S 120 100 O 80 60 40 20 120 100 80 60 40 20 2 8 10 10 12 12 O 120 100 80 O 60 40 20 S 120 8 100 80 60 40 20 b Use the graph in part a to determine the months of maximum and minimum sales minimum Select 10 10

Use the graph of the function to answer each question 2r sxs 2x f x cot x a Find all x intercepts of the graph of y f x Enter your answers as a comma separated list If an answer does not exist enter DNE b Find all y intercepts of the graph of y f x Enter your answers as a comma separated list If an answer does not exist enter DNE c Find the intervals on which the graph of y f x is increasing and the intervals on which the graph of y f x is decreasing Enter your answers using interval notation If an answer does not ex increasing decreasing d Find all relative extrema if any of the graph of y f x Enter your answers as a comma separated list of ordered pairs If an answer does not exist enter DNE relative minima relative maxima e Find all vertical asymptotes if any of the graph of y f x Enter your answers as a comma separated list If an answer does not exist enter DNE

Let f x arctan x The arctangent function and all of its derivatives are perfectly well behaved for every real value of x Use Maple to produce p x and p x when c 0 What is the interval of convergence Now repeat this problem using when c 4 What is the interval of convergence

4 a It is well known that 1 converges to e Test a few values of k and assemble evidence to support this fact Submit this evidence with your lab report b Investigate the behavior of two related sequences 1 2 and 1 Discuss how the sequences differ in their behavior as n gets large n Vn

Use the Limit Comparison Test to determine if the following series converges or diverges Hint Limit Comparison with n 5 3 2 n 1 n n 8 1 The series n 5 and Apply the Limit Comparison Test with an 2 bn Complete the sentence below n 1 n an n 1 n bn n 5 3 2 n 1 n n 8 because lim an bn n n an b and for all

Abaseball pitcher s earned run average ERA is A e i 9e i where e is the number of earned runs given up by the pitcher and i is the number of innings pitched Good pitchers have low ERAS Assume that e 20 and i 0 are real numbers a During his best season a certain pitcher pitched a total of 237 innings and gave up just 36 earned runs What was his ERA b Determine the ERA of a relief pitcher who gives up 5 earned runs one third of an inning c Graph the level curve A e i 2 and describe the relationship between e and i in this case a The pitcher s ERA for the season was Round to two decimal places as needed b The relief pitcher s ERA was Round to two decimal places as needed c Graph the level curve A e i 2 Choose the correct graph below O A A 10 Q OB A 10 Describe the relationship between e and i in this case CICCESS Q O C A 10 O D A 10 Q