Limits & Continuity Questions and Answers

10 5 TT 10 5 0 5 11 X 1 VE 1 LO 5 vs 10 10 3 If the black graph is f x x then which of the following represents the dashed blue

How did you decide If f x x 5 2 then which of the following statements is true The 5 is a tight number and moves the graph horizontally to the right while the 2 is a loose number and moves the graph vertically up The 5 is a tight number and moves the graph horizontally to the left while the 2 is a loose number and moves the graph vertically up The 5 is a tight number and moves the graph horizontally to the left while the 2 is a loose number and moves the graph vertically down The 5 is a tight number and moves the graph horizontally to the right while the 2 is a loose number and moves the graph vertically down

7 Let h a 25 2 and j x 7x 1 a What is the domain of h x b What is the domain of j x c Let k x h x j x What is the domain of k x 2 h 2 What is the domain of f x d Let f x

1 10 pts Use the graph to find the following limits and function value 10 10 5 5 LO 10 5 10 a b C d e f 2 7 lim f x x 7 lim f x x 2 lim f x x 2

Intervals of Increase and Decrease and Local Extrema Answer the following questions using the function f x The derivative of the given function is f x The critical point s of f are B In x x2 NOTE Enter values as a comma separated list if needed If you believe the answer is none type none that occurs at z f has a relative local maximum value of f has a relative local minimum value of NOTE Enter values as a comma separated list if needed If you believe the answer is none type none that occurs at z f is increasing on and decreasing on NOTE State the open intervals of increase and or decrease as a comma separated list if needed

2 Use the limit comparison test to determine if 2 3 n converges or diverges

1 Use the comparison test to determine if 8 n 1 n 1 n 2 converges or diverges

8 4 points Determine ALGEBRAICALLY if the function f x 3x 8 is even odd or neither

blem 5 For the rational function R x a State the domain of R x b Find all zeros 3x 6 x 3 C find c Find all intercepts d Find all poles and identify all vertical asymptotes e Generate a sign chart to determine where R x is positive and where it is nega f Identify all horizontal asymptotes Sketch the monk of DU

5 Solve the inequality State your answer in interval notation 5r 2 0

The function f is defined as follows f x 2x 3 O A if 3 x 0 if x20 a Find the domain of the function b Locate any intercepts c Graph the function d Based on the graph find the range OB Ay OB The range does not have any isolated values It can be described by Type your answer in interval notation OC C The range consists exclusively of one or more isolated values It can be described as co a comma to separate answers as needed d Based on the graph find the range Select the correct choice below and fill in the answer box es to complete your choice A The range has at least one isolated value It can be described as the union of the interval s Use a comma to separate answers as needed AY 40 and the set OU

Find a formula for the nth partial sum of the series and use it to find the series sum if the series converges 5 5 36 216 5 5 6 5 67 1 The formula for the nth partial sum sn of the series is

4 Which Series converge or diverge give reasons for your answer If it converges find the sum 3 a 1 1 b 1 n 2 8 n 1 n 1

3 Tell if the sequence converges or diverges If it converges find it s limit n 1 a a b an n 2n 1 1 3 n 1 1 n c a 70 4n d a 2 11 2 2

3 Use the nth Term Test for divergence to show that the series is divergent or state that the test is inconclusive a 1 b ln n 1 n 8 n 1 n 2 n 3

2 Write out the first 5 terms of the sequence with its first term along with the recursion formula for the remaining terms a 2 an 1 na n 1

Cooper Section 3 2 52 On the planet Golgafrincham there are two systems for measuring temperature Using the Celia scale water freezes at 0 C and boils at 50 C Using the Furryhat scale water freezes at 10 F and boils at 110 F Both systems divide the temperatures in between into equal increments Letz be the temperature in degrees Celia and y be the temperature in degrees Furryhat a Find a formula which converts temperatures measured in Celia into Furryhat y x 9 5 F b Find a formula which does the inverse x y c What temperature is the same in both Celia and Furryhat C

Give the equations of any vertical horizontal or oblique asymptotes for the graph of the rational function x2 5x 6 2x x 21 f x OA There is one vertical asymptote Its equation is Type an equation Use integers or fractions for any numbers in the equation B There are two vertical asymptotes The equation of the leftmost one is x 3 and the equation of the 7 rightmost one is x 2 Type equations Use integers or fractions for any numbers in the equations OC There are no vertical asymptotes Identify the horizontal or oblique asymptote if one exists Select the correct choice below and if necessary fill in the answer box to complete your choice OA The horizontal asymptote is Type an equation Use integers or fractions for any numbers in the equation OB The oblique asymptote is Type an equation in slope intercept form Use integers or fractions for any numbers in the equation

For the given function find all asymptotes and the coordinates of any holes in its graph f x X 7 x 49 complete your choice A There is one vertical asymptote Its equation is x 7 Use integers or fractions for any numbers in the equation OB There are two vertical asymptotes The equation of the leftmost one is rightmost one is Use integers or fractions for any numbers in the equations OC There are no vertical asymptotes OA The equation of the oblique asymptote is Find the horizontal or oblique asymptotes Select the correct choice below and if necessary fill in the answer box to complete your choice Use integers or fractions for any numbers in the equation OB The equation of the horizontal asymptote is and the equation of the Use integers or fractions for any numbers in the equation C There are no horizontal or oblique asymptotes

Select all that apply DA f x B 1 x c f x 2x 1 X 8 DD f x a graph that does not have a vertical asymptote

domain of f f x 2 X 3 asymptotes for the graph of the rational function State the Identify any vertical asymptotes Select the correct choice below and if necessary fill in the answer box to complete your choice OA There is one vertical asymptote Its equation is Type an equation OB There are two vertical asymptotes The equation of the leftmost one is one is Type equations C There are no vertical asymptotes and the equation of the rightmost

J and g given by the following graph 6 2 19 6 8 10 12 Let F z f g z a Find F 1 and F 8 b Use linear approximation to estimate F 0 9 c There are three values of z in 0 10 at which F is not differentiable Find them explain your reasoning

a fog b gof a fog x x 2 Simplify your answer b gof x x 2 Simplify your answer c fof The domain of fog is 4 00 Type your answer in interval notation Use integers or fractions for any numbers in the expression S your answer d gog c f of x Simplify your answer The domain of g of is 00 Type your answer in interval notation Use integers or fractions for any numbers in the expression S your answer

Find and simplify the difference quotient f x 7x O A O B OC 1 OD O 1 7x x h 1 x x h f x h f x h h 0 for the given functio

Let f x cos x sinx a Find the absolute minimum and maximum values of on 0 27 This is asking for the minimum and maximum y values b Find the absolute minimum and maximum values of f on 0 00 Explain Hint How does the graph of f on relate to the graph on 0 27 If you re not sure try graphing the function on a calculator or computer How could you have figured out the behavior without a calculator c Reflect Back Akiko Batari and Carole are working on a All three have found the critical points of f on 0 27 and they have different ideas on what to do next Akiko says I ll use the First Derivative Test Batari says I ll use the Second Derivative Test Carole says I ll plug the critical points and endpoints 0 and 2 into f to see which have the highest and lowest values of f those are the absolute maximum and absolute minimum Do all three have complete plans or might they need to do more work beyond what they said liko best and why

Use the graphs of the rational functions in A to D on the right to answer the question Give all possible answers as there may be more than one correct choice Which choices have the x axis as a horizontal asymptote Select all the graphs that apply Graph C Graph B Graph D Graph A B C D Q 18 18 1 H Q Z

For f x x 3 and g x 4 x find a fog x b the domain of fog a fog x x 7 Simplify your answer b What is the domain of f o g The domain is Simplify your answer Type your answer in interval notation Use integers or fractions for any numbers in the e

1 1110 6 The graphs of the function f x given in blue thinner and g x given in red thicker are plotted above Suppose that u x f x g x and v x f x g x Find each of the following u 1 V 1

Find the horizontal asymptote s HA and the vertical asymptote s VA of the functia f x 3219 a HA lines y 3 and y 3 VA line x 0 b HA c HA line y 3 VA line x 0 d HA line 2 0 VA line y 3 e HA line y 0 VA line x 3 none VA lines 3 and x X 3

Given that x a x 3 and x b x 3 and x 1 2 c x 3 and x 1 d x 3 and x 1 is a zero of f x 2x 5x 4x 3 find the other zeros of f x biton bus ancilsmot e r 3 and x 1 Rewrite the polynomial f x in completely factored form

Which of the following functions is not one to one a f x x 2x on 1 00 b f x x x 1 on 0 0 c f x 2x 5 on 0 00 d f x x 4 on 0 00 e f x 2x 4 x on 00 00

46 lim x ln 1 8118 48 lim x3 2 sin 1 x 818

48 lim x3 2 sin 1 x

61 lim x 1 0 2 14 63 lim x

29 lim x 0 tan x sin x 31 lim x 0 tanh x x 33 lim X x3 x 0 3

Write the parametric equations as a function of a in Cartesian form S 1 with a 0 t x 3e y 6e 4

Find the derivative of 5 O 5 O In 5 5 O ln 5 5 O 5 In 5

Year between 2001 and 2004 Sales millions of dollars 1998 1999 201 219 2000 2001 2002 2003 2004 2005 2006 233 245 251 251 252 243 233 e table above gives the annual sales in millions of dollars of a product from 1998 to 2006 What was the average rate of change of annual sales between 2001 and 2002

Based on the graphs of f x and g z below answer the following questions You should not approximate any of your answers K 2 2 4 0 5 2 6 1 6 9 4 9 2 9 a What is the average rate of change of f x over the interval 2 2 x 6 1 b What is the average rate of change of g x over the interval 2 2 x 6 1 8 0 9

Find the intervals where the function y x 1 1 whose graph is given below is increasing and where it is decreasing and find the value of the absolute minimum Increasing Decreasing Click on graph to enlarge Absolute minimum help formulas help formulas help numbers

Score 11 17 answered Question 6 If f x is a linear function f 1 3 and f 3 5 find an equation for f x f x Question Help Video Message instructor

Identify the quadrant in which the following point is located 2 3 1 2 Choose the correct quadrant below that the given point is located quadrant I O quadrant III O quadrant IV O quadrant II

For the data given below answe ants 6 7 11 13 X Y 2 3 3 5 4 5 9 46 6 y 2 0357 x 1 3214 Type integers or decimals rounded to four decimal places as needed e What is the correlation coefficient r r 0 996 Type an integer or decimal rounded to three decimal places as needed f Use a graphing utility to draw the scatter plot and graph the line of best fit on it Choose the correct graph below OA Q 8 15 Xmin 0 Xmax 20 Xscl 2 Ymin 0 SOB 5 Xmin 0 Xmax 20 Xscl 2 Ymin 0 OC Xmin 0 Xmax 20 Xscl 2 Ymin 0 Vmov 0 Vede O D Q 15 3 Xmin 0 Xmax 20 Xsc Ymax 20 Ysd 2

The limit represents the derivative of some function f at some number a State such an f and a 1 h 1 h lim h 0 f x x x a 1 Of x x x a 0 Of r r a 2 Of x x a 0 Of x 26 a 1

Determine whether f 0 exists x sin f x Of 0 does not exist Of 0 does exist QUERY if x 0 if x 0

The limit represents the derivative of some function f and some number a State such an f and a cos h 1 h lim h 0 Of x cos x a 0 Of x cos 2 a 4 Of x cos r a 3 Of x cos x a Of x cos z a 6

For f x a fog 6 X 5 and g x b gof 5 X b gof x 1 find the following composite functions and state the domain of ea c fof d gog 6x a fog x 5 1 x Simplify your answer Use integers or fractions for any numbers in the expression The domain off og is 0 U 0 1 U 1 00 Type your answer in interval notation Use integers or fractions for any numbers in the expres your answer 5X 25 6 Simplify your answer Use integers or fractions for any numbers in the expression The domain of g ofis 5 U 5 00 Type your answer in interval notation Use integers or fractions for any numbers in the expre your answer c f of x Simplify your answer Use integers or fractions for any numbers in the expression

K The function f x x 2 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 a f x x 2 Simplify your answer Use integers or fractions for any numbers in the expression b The domain of f is 0 00 Type your answer in interval notation The range of f is 00 Type your answer in interval notation The domain of f is 0 00 1 Type your answer in interval notation The range of f is 0 00 Type your answer in interval notation c Graph f f1 and y x on the same coordinate axes The graph of y x is shown using da Choose the correct graph O A O C Q Q B D

K The function f x 6x 2 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 x 2 6 a f x Simplify your answer Use integers or fractions for any numbers in the expression b The domain off is 00 Type your answer in interval notation The range of f is 0 00 Type your answer in interval notation The domain of f is 0 00 Type your answer in interval notation The range is 0 00 Type your answer in interval notation c Graph f f and y x on the same coordinate axes of f1 Question Viewer

6x and g x 4x 6 evaluate the following expressions 4 b gof 2 c fof 1 c fof 1 d gog 0 3 4 420 Simplify your answer f 2 582 Simplify your answer f 1 36 Simplify your answer g 0 Simplify your answer