Limits & Continuity Questions and Answers

3 2 5 4 3 2 1 f m 0 Please show all your work NO WORK NO CREDIT Write an equation for the transformed logarithm shown below Your answer should include a vertical scaling and will be in the form f x a log x c 2 3 ans 1 2 3 4 1

1 Examine the graph shown here Then answer the questions that follow Amount Inches of Rainfall in St Vallis 16 14 12 10 8 6 4 2 0 Jan Feb Mar Apr May June July Aug Sept Oct Nov Dec a What type of graph is this b In which month did the 2nd heaviest amount of rainfall occur c Which month received nine inches of rain d How much more rainfall fell in November than in June 5 marks r What is the midrange of this data

2 x 8 and g x 3 4 Find the area of the region bounded between the curves of the function f x 8 2008

Exercise 10 0 22 C level Find a basis for the kernel of 80 using column reduction The dimension of this subspace will tell you how many clusters the diagram forms

Use the Integral Test to determine whether the or divergent series is convergent sum n 1 infty n e 4 n Evaluate the following integral int 1 infty x e 4 x d x Since the integral finite the series is

You estimate that you can save 7 100 by selling your home yourself rather than using a real estate agent What would be the future value of that amount if invested for five years at 9 percent Refer to Exhibit 1 A Round time value factor to 3 decimal places and your final answer to 2 decimal places Future value

2 For the function f x x 36 find f x To find f x first find f x f x X x 36 2

Evaluate the following sum 16n 40 n 3n 2n

a Find an formula for an in terms of n valid whenever n 1 an SN a an IN 4 IN 10 b Evaluate S Eligible answers include real numbers and the strings inf inf and DNE where for n 1 B N SN a

Determine whether the sequences are increasing decreasing or not monotonic If increasing enter as your answer If decreasing enter Das your answer If not monotonic enter N as your answer 1 an 2 0 3 an 4 On Unid 7014 90 45 An 7

6 Given the graph of f x Sketch f x on the san

Write the following expression in

If we know that g 0 1 and g 0 7 evaluate the numerical value of f 0 where Note Only enter a number as your answer e g x e g x f x

9 7 4 Ex 2 Find all solutions to the equation 2 2 Cos 0 10 7 4 25 Find all solutions given the equation cos 0 1 0 11 7 4 34 Find all solutions given the equation 4 sin 0 3 03 stb

3 7 2 16 Use an addition or subtraction formula to write the expression as a trigonometric function of one number and then find its exact value cos 10 cos 80 sin 10 sin 80 4 7 3 9 Find tan 2x given 1 tan x cos x 0 5 7 3 Use the appropriate half angle formula to find the exact value of the expression after first converting from radians to degrees Show ALL work TL sin sin 0

The series a c 2 2 Converges absolutely Diverges b Converges conditional

Find the sum of the series a c 2 10 IMB b d 3 315 11 10

Determine whether the series converges k 1 k 2 a Converges b Diverges

2 Determine whether the series a Converges b 5 32 converges or diverges Diverges

PRADESTEHE FASTVARKOER ERMOTETE C ESEORAN a JEAN Qani TENTATATER P HERCHE DES NAT POTREGU MOR VARE PERBAnontra URASSHMEE415330 SHERBIMET EE Converges BERGRED HAPATE KRAW OF PETRARCIENSO BARNECK HERHANA OG HIVIRAN MARGIESST S b Use the root test to determine if the series converges or diverges Diverges E 1 7k 9 K

f x 2x 3 g x x 4 and h x 2x 2 5x 12 Write a formula for each of the llowing functions and then simplify

t r 3 2 4 6 5 4 X 3 a 1 N 3 4 R q The function graphed above is decreasing on the interval 6 The inflection point is at x

li ap 0 12 jou op dat we sa ZEE ZL ons zo s n mb s for at je stana pod adwea tap n papunoj aq plnous Shamsur B LDWIN

5 a Evaluate the sum of the scries 1 n T e 27 1 n 2 b Determine whether or not the the series n n 1 In In n 2 converges If it converges then find the sum of the series Hint Properties of logarithms might be useful

65 66 Difference Quotient Let f x cos x and g x sin x Use Addition or Subtraction Formulas to show the following f x h f x h 65 COS X 1 cos h h sin x si sin h h

Find the domain of the following function f x y 6xy 9x 4y Select the correct choice below and if necessary fill in the answer box to complete your choice OA x y 0 OB x y x y OC x y 0 OD x y x and y Use a comma to separate answers as needed

2 sin 4x x 0 1 cos 4x 5 points Find the limit lim

for z f x y shown below Note you can click on the graph for a larger view Determine whether the following values related to f x y at the point P are positive negative or zero begin array f x y f x x y f y x y f x x x y f y y x y f x y x y end array

2 The binomial theorem is useful in this question i Prove that for n N 2 1 m n m 1 an 1 ii Prove that an is increasing iii Prove that an e iv Prove that an converges Let e limnan

Evaluate the limit If the limit does not exist enter DNE Answers should be in exact form i e do not use a calculator to get a decimal approximation Rug Reu 8t 9 lim t 8t 5

A function is said to have a horizontal asymptote if either the limit at infinity exists or the limit at negative infinity exists Show that each of the following functions has a horizontal asymptote by calculating the given limit 7x lim x 4 2x 15x 14 lim 100 x 13x 12 x 8x9 lim 148 5 1472 lim 7 8 lim 8118 x 12x 9 13x x 12x 9 13x 10

Evaluate the limit lim x18 2x 2x 9x 10 6x 8x

Given that a is an unknown real number find each of the following limits If needed use oo for 00 for or DNE for does not exist a lim 818 a lim a x 8 8 xa x a a lim x a x a xxx a a lim x a 100 a x 11 00 00

1 34 Find the limit Use l Hospital s Rule where appropri ate If there is a more elementary method consider using it If l Hospital s Rule doesn t apply explain why x 1 1 lim x1 x x 2 1 xa 2 lim x 1 x 1

3 5 Pts Find the critical points of the function z y 2 x y and then pick JUST ONE of the critical points and use the discriminant D to determine if it is a local maximum a local minimum or a saddle point

a Sum the first two numbers in this series That is find a numeric value for 2 1 W k 1 k b Next add the first three numbers in the series

be is rolled es An is repr by d string of the sort OEE meaning an odd number on the first roll an even number on the second roll and an even number on the third roll The 8 outcomes are listed in the table below Note that each outcome has the same probability For each of the three events in the table check the outcome s that are contained in the event Then in the last column enter the probability of the event Event A No odd numbers on the first two rolls Event B An even number on the first roll Event C An odd number on each of the first two rolls OEO 0 0 0 EOO OOE U U O U 0 Outcomes EEO 0 U 0 EEE O 0 OEE U U U EOE 000 0 O O 0 Probability X

Without using a calculator find all the solutions of 1 2 where t cos t

8 9 marks Evaluate the following limits a sin 2 1 X b c lim x 0 lim x 3 x 0 lim log2z z 1 6x 2x x 1 X

Find the local quadratic approximation of x x 1 and use that approximation to find the approximation of 1 1

2 Classify each series as convergent conditionally convergent or divergent a 1 1 1 3k x 1

1 Determine whether the series converges a GO x 1 1 1 2k 1

Graph of f The figure above shows the graph of a function fwith domain0sxs4 Which of the following statements are true 1 lim f x exists 2 2 II lim f x exists 2 2 23 III lim f x exists B D E I only Il only I and II only I and III only I II and III D

ttry 0 of 1 pts See Details for mo Get a similar quest uestion hit using L Hospital s rule

Suppose f 5 3 f 5 10 g 5 2 g 5 1 and that f x g x are continuous for all x f x 3 lim x5 g x 2 O 0 O Cannot Be Determined

Question 10 The graph of y f x is shown Assume f x is continuous Find 8 7 6 xf x 16 lim x4 f x x 60 5 4 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 3 2 3 4 5 6 7 0

Consider lim e8x x X4 a lim 8x x X18 Question Tali

Question nslate the exponential statement into an equivalent arithmic statement 83536 points Points 0 of 1 The equivalent logarith 2000 ce 36 7167 Use integers or decim the equation

Question 6 4 3 12 he equation in equivalent logarithmic form 811 points O Points What is t 3

If 421 000 people each receive an average refund of 3 400 based on an interest rate of 3 percent what would be the lost annual income from savings on those refunds Do not round intermediate calculations Lost annual income