Limits & Continuity Questions and Answers

A bag with 10 marbles has 2 blue marbles and 8 yellow marbles A marble is chosen from the bag at random What is the probability that it is red Write your answer as a fraction in simplest form 8 08 X

Determine absolute or Convergence of conditional convergence 8W n 1 A 4 1 n 1 1

Let f x x f 3 4x 7 if 1 The graph of f is given below 5 4 3 2 1 5 4 3 2 1 1 2 3 4 if V 1 x 3 x 3 2 3 4 5 a Determine the following function value Enter DNE if the function is undefined at x ow

2 Analyze and then sketch the function Ax x BX E a Determine the asymptotes A 2 b Determine the end behaviour and the intercepts c Find the critical points and the points of inflection d Determine the interval of increasing decreasing and the concav e Sketch the graph K 2 K A

1 point Use the limit comparison test to determine whether a Choose a seriesbn with terms of the form bn n 7 For n 7 an n bn n x lim lim lim n 0 bn n n 7 an n 7 6n 5n 7 4 2n4 b Evaluate the limit in the previous part Enter as infinity and as infinity If the limit does not exist enter DNE an Note You can earn 40 partial credit for 2 3 correct answers converges or diverges and apply the limit comparison test Write your answer as a fully simplified fractio c By the limit comparison test does the series converge diverge or is the test inconclusive Choose

3 4 is on the graph of f x Move the point to another point that would also be on f x 8 6 4 2 8 6 4 2 2 4 6 8 Y 2 60 00 8

16 Which of the following inequalities can be used to study the convergence 1 of n 2 n Explain 1 n n 1 n 1 n n 1 2

Use the table to find lim X 0 X f x sin 10x X sin 10x X 0 03 0 02 0 01 0 01 0 02 0 03 9 8507 9 9335 9 98339 9833 9 9335 9 8507

1 x 2 The graph of f is given below Let f x 5 4 3 2 1 5 4 3 2 1 1 2 3 4 5 1 2 3 Determine the following function value Enter DNE if the function is undefined at x 2 f 2

4 points Use the ratio test to determine whether an 1 lim 8 8 an a Find the ratio of successive terms Write your answer as a fully simplified fraction For n 17 lim TL 0 n 17 lim 81 8 n6n n 2 b Evaluate the limit in the previous part Enter o as infinity and o as infinity If the limit does not exist enter DNE an 1 an converges or diverges c By the ratio test does the series converge diverge or is the test inc Note You can earn 40 partial credit for 2 3 correct answers usive Choose

The graph of f is given below 5 4 3 2 1 Hint 5 4 3 2 1 Submit Question 1 2 3 4 1 2 3 q Determine the intervals on which f is continuous Enter the solution using interval notation f is continuous on 4

m n Sio 5 4 3 2 1 5 4 30 h is continuous on Hint 7 1 2 3 2 3 4 5 La Determine the intervals on which h is continuous Enter the solution using interval notation 4 5

Question 23 The graph of h is given below 5 4 3 2 1 5 4 3 2 1 1 2 3 4 h is continuous on 1 2 3 4 q Determine the intervals on which h is continuous Enter the solution using interval notation

Let f x 2 5 4 3 2 1 4 The graph of f is given below T if if 3 20 H 2 3 x 0 x 0 2 3 4 S q Determine the following function value Enter DNE if the function is undefined at x 0 f 0

The graph of g is given below O 5 4 3 2 g is continuous on Hint 4 3 2 1 2 3 1 5 Determine the intervals on which g is continuous Enter the solution using interval notation 2 3 4

2 1 201 X1200 Which of the following limits is are equal to zero lim II lim 100 x e x III lim ze lim x 3x 5 e x 00 A II only B I and II only C II and III D I II and III C

1 point Evaluate the function f x 10 x 7 x f 2 1 6 f 2 1 3 f a at the indicated values f 1 46 f a 1

Calculate the derivative d dt t S 56 sec 14x 33 dx sec 14t 33 On what interval is the derivative defined

2 Sketch the graph of a function f x with domain 5 5 such that all three of the following conditions hold a f x is continuous everywhere except at x 1 b lim 1 f x 4 c f x 0 on 5 1 and f x 0 on 1 5

5 Calculator auth A 2 269 f x Let f be the function defined above where c is a constant For what value of c is f continuous for all x B 3 460 4 COS X c 5 ln x c for x 1 C 4 066 D There is no such value of c

1 point Write out the first five terms of the sequence with 1 7 Enter the following information for an a1 6 a2 25 4 a3 64 27 a4 81 256 a5 32 3125 lim 818 1 7 Enter DNE if limit Does Not Exist e 7 rect Does the sequence converge yes 1 3 determine whether the sequence converges and if so find its limit Enter yes or no

Find a power series for the function centered at c 3 2x 1 f x f x 80 n 0 1 1 2 37 c 2 x 2 X Determine the interval of convergence Enter your answer using interval notatic 1 7

A drone operator is on a road running north and south and using a drone to observe a herd of geese 1700 yards east and 2600 yards north of the operator s position If the drone flies so that the distance from the herd is three times to the distance to the road describe the path of the drone then determine how far will it be from the drone operator when it crosses an east west line through the operator s position The path of the drone has the shape of a parabola hyperbola X point straight line ellipse When the drone crosses the east west line through the operator s position it will be 1300 X yards from the operator enter your answer rounded to the nearest

y 52 22 the cursor until the graph is correct 7 6 5 Sketch the graph of 4 Clear All Draw 32 2 7 6 5 4 3 2 1 2 23 3 4 4557 6 7 1 Graph by selecting a type of graph then select the center then movin 2 3 4 5 6 YY G

Find the following limits if they exis a lim 11

Write a polynomial f x in complete factored form that satisfies the conditions Let the leading coefficient be 1 5 5 Degree 4 zeros 6 with multiplicity 3 and 8 with multiplicity 1 A f x x 8 x 6 C f x x 8 x 6 B f x x 6 x 8 D f x x 6 x 8

Consider the ellipse with equation The semimajor axis h s length 7 The semiminor axis has length 2 The vertices are located at 0 7 0 7 separated by commas The focal length is Question Heler The foci are located at 0 3 5 0 3 5 by commas Video 1 G o enter the coordinates of each vertex enter the coordinates of each focus separate

y 1 Graph by selecting a type of graph then select the center then moving 6 2 the cursor until the graph is correct 71 6 Sketch the graph of x 5 4 3 2 1 7 6 5 4 3 2 1 2 23 3 4 5 6 7 I 1 2 3 4 5 6 7 G

Show all the work to determine whether 4 3 converges or diverse

10 9 8 7 6 5 4 3 2 The focus F is at 3 2 3 1 9 8 7 654 3 321 2 2 m 3 4 S 6 7 8 10 OF The directrix has equation y 4 The line of symmetry has equation symmetry The y intercept s are 0 1 O 23 The equation of the parabola is y 12 H 22 3 1 4 7 8 9 IC and the vertex V is at Enter the equation of the directrix Enter the equation of the line of Enter the equation of the parabola The x intercept s are 3 2 3 0 3 2 3 0 Enter the exact coordinates of the x intercept s if there is more than one separate them with commas if none exist enter DNE 0 Enter the exact coordinates of the

For the ellipse x 132 y 7 1 the eccentricity is e

The graph of x 4 2 ellipse parabola hyperbola y 3 5 which has undergone the translations right 4 units then up 3 units left 4 units then down 3 units right 4 units then down 3 units 1 is a

For questions 9 10 the wait time of various rides at the amusement park is normally Histributed with a mean of 15 minutes and a standard deviation of 4 minutes 9 What is the probability that someone will wait for at least 20 minutes A 89 4 B 82 1 C 17 9 10 What is the probability that someone will wait between 8 and 18 minutes A 59 4 B 68 1 C 73 3 D 10 6 D 79 5

10 Use implicit differentiation to find dy dx for the ellipse xy y 1 b Check that the point 1 0 is on the ellipse c Is the tangent line to the ellipse horizontal at the point 1 0 CIRCLE ONE YES NO

A parabola is shown with the focus open circle and vertex closed circle marked 10 9 8 7 6 5 4 3 2 The focus F is at The directrix has equation 10 9 8 7 6 5 4 321 3 23 3 S 67 8 9 10 The line of symmetry has equation symmetry The equation of the parabola is 89 IC a and the vertex V is at Enter the equation of the directrix Enter the equation of the line of Enter the equation of the parabola The x intercept s are there is more than one separate them with commas if none exist enter DNE The 1 intercent s ar Enter the exact coordinates of the x intercept s if

The following parametric curve models the path driven by a race car driving around a 500m radius racetrack clockwise x 500 cos 0 y 500 sin 0 False True

Tutorial Exercise Find a power series for the function centered at c and determine the interval of convergence 3 f x 3x 2 C 6 Step 1 Rewrite the original function f x Step 2 f x 3 3x 2 18 18 20 3x18 20 3 3 after adding and subtracting 18 in the denominator as follows 3x 2 18 3 x 6 Divide the numerator and denominator of the function f x 20 20 x by 20 to put it in the form of a geometric series 6

If limno an False True 0 then an converges

Points State what the first and easiest step procedure should be in finding the integral x 7x 5 S O algebraic simplification Ou substitution dx O trigonometric substitution O parts O partial fractions

4 Evaluate the following integrals either using geometry or the fundamental theorem x e dx b 17 11 1 72 d dx c x x 5 dx

The point 1 3 in Cartesian coordinates may be modeled as 2 in polar coordinates False True 4

3 Sketch the graph of a function f x such that lim 3 2 f 3 4 and f x 0 when x 3

If the power series Cn x 1 diverges for a O False O True 0 then it diverges for x 6

Rewrite using sines and cosines sec y tan y sin x 2 cos x sec y tan y incorrect one You may lonvo lified syntax error Check your variables you might be using a

For the following series determine which test is most appropriate and easiest 3 O Geometric Series Test O Direct Comparison Test O Divergence Test n series test

Suppose sin 2x Then sin x 5 13 1 26 separated by commas If there is more than one possible value enter the value

Suppose sin Find the following sin 0 6 1 cos 0 6 3 5 sino 20 29 Moreover suppose is in Quadrant IV and is in Quadrant III

Suppose x 3 sec a Rewrite in terms of sines and cosines x 9

Let It be the finite region enclosed by the curve y6z 5 42 and the z axis Let V be the volume obtained by rotating R about the z axis Express the volume as an integral of the form V f x dz a b 10 f x Evaluate this integral to find the volume in question Give your answer in terms of Volume

Suppose sin 2x 5 13 Then sin x separated by commas If there is more than one possible value enter the value