Limits & Continuity Questions and Answers

5 Evaluate each of the following limits or state that the limit diverges does not exist Give a cle justification for each answer 2 points each x 6x a lim x 6 2x 72 b lim 2 9 4 g 0 2g c lim x 80 2ln x x

8 The annual change in the population of New York State in people per year is given in the table below Year Rate 2014 26 943 2016 20 930 2018 49 751 2020 645 165 2022 180 341 Create the best estimate for the total change in population of NYS between 2014 and 2022 Write a sentence interpreting what your estimate means in this context

lim ln sin x check what is happening first to sin x when x then plug into In x X T

5 The Willis Tower in Chicago is 1453 feet tall and is located 1 mile inland from the shore of Lake Michigan as indicated in the figure An observer in a boat on the lake directly in from of the Willis Tower looks at the top of the tower and measures an angle of elevation of 5 How far offshore is the boat Give the answer in miles rounded to 2 decimal places 10 pts 1454 ft Lake Michigan

2 Determine the solution to the inequality x 4x 12x 0 6 pts

Determine the solution to 2cos 0 cos 0 0 on the interval 0 0 2 6 pt

that would h Given the positon function f x 16x 80x simplify the difference quotient 3 h f 3 represent the average velocity of the object from time x 3 to time x h by eliminating the variable h from the denominator 6 pts

etermine whether the sequence converges or diverges 3n 11 5 3n elect one a It diverges O b It converges

If the sum of a blue die and a pink die is 8 what is the probability that one of them is at least 5 Select one O a 5 36 O b 1 9 O c 4 5 O d 2 3

Write the equation of the ellipse graphed below 16 4 3 4 Select one O a x 4 y 3 16 49 16 O b x 4 3 49 16 1 O c x 4 y 3 16 49 O d x 4 2 y 3 49 16 1 1

For the function f x graphed below which of the following is the lim f x x 3 10 5 Select one O a O b O C O d Oe 10 3 lim f x 10 x 3 lim f x 1 x 3 lim f x 2 5 lim f x 3 lim f x does not exist x 3

Examine the table for f z and the graph for g z f a 0 1 2 1 0 2 2 1 2 3 1 8 4 2 5 1 8 6 1 2 10 2 B 6 4 2 2 4 6 8 4 6 8 10 12 14 2020 StrongMind Created using GeoGebra g x b A Which statement is true for f x and g z over the interval 0 x 4 Of z increases at a faster rate Og z increases at a faster rate Of z decreases and g a increases

In an effort to control vegetation overgrowth 74 rabbits are released in an isolated area free of predators After 2 years it is estimated that the rabbit population has increased to 666 Assuming exponential population growth what will the population be after another 6 months Round to the nearest rabbit Answer How to enter your answer opens in new window Keyp Keyboard Short

Jose hopes to earn 600 in interest in 2 3 years time from 12 000 that he has available to invest To decide if it s feasible to do this by investing in an account that compounds monthly he needs to determine the annual interest rate such an account would have to offer for him to meet his goal What would the annual rate of Interest have to be Round to two decimal places Interest formulas

5 sketch the curve and find a polar equation b y 3 a y x c x y 8

Martha invests 7600 in a new savings account which earns 3 8 annual interest compounded monthly What will be the value of her investment after 5 years Round to the nearest cent Interest form

The function C 1 Co 1 r models the rise in the cost of a product that has a cost of Co today subject to an average yearly inflation rate of r for 1 years If the average annual rate of inflation over the next 14 years is assumed to be 2 5 what will the inflation adjusted cost of a 111 300 boat be in 14 years Round to two decimal places

The function C Cell r models the rise in the cost of a product that has a cost of Co today subject to an average yearly inflation rate of r for years If the average annual rate of inflation over the next 14 years is assumed to be 2 5 what will the inflation adjusted cost of a 111 300 boat be in 14 years Round to two decimal places

Maria has recently inherited 8900 which she wants to deposit into an IRA account She has determined that her two best bets are an account that compounds semi annually at an annual rate of 4 Account 1 and an account that compounds daily at an annual rate of 3 4 Account 2 Step 1 of 2 Which account would pay Maria more interest

b lim ln sin x check what is happening first to sinx when x then plug into In x X T

lim In sin x check what is happening first to sinx when x then plug into lnx PIR

e lim 6 16 4 6 166 6 multiply top and bottom by conjugate

d lim 1 2 1 201 4 x 4 4 x Yes its fractions and yes you need to get a common denominato

Find the radius of convergence R of the series R n 9x 1 n 1 Find the interval I of convergence of the series 01 0 01 0 or 1 01 1 I 01 1 21 1 OI 9 9

Seton Tell whether each sentence below is true or false I A half angle identity relates an expression like tan 57 individually 2 The sum and difference identities can be used to find the exact value of trig functions whose ratios we haven t 54 memorized like cos 75 to an expression that involves trig ratios of angle a

The graph of a function f t is shown below y 4 3 a What is A 3 b What is A 5 2 1 2 3 4 5 6 The following questions are above the function 4 x f t dt 8 9 10 7 c At which values of x if any does A z have a local maximum d At which values of x if any does A x have a local minimum e Find the absolute maximum value and the absolute minimum value of the function A z f t dt on the interval 0 10

Consider the following function f x 1 x 2 Determine whether f x approaches or as x approaches 2 from the left and from the right a lim f x X 2 b lim f x X 2

dy Find by implicit differentiation Then find the slope of the graph at the given point If an answer is undefined enter UNDEFINED dx dy dx xy 40 8 5 At 8 5 y

The height in h feet of a baseball t seconds after it was struck can be described by the function h 161 801 2 Which graph below best shows where the ball was higher than 66 feet high O OC 0110 01 4 00 0 1 4 0 2 3 910

Question 4 25 points A football player punts a football such that the graph at the right shows the height of the ball in feet t seconds after it had been kicked Which graph below best describes the time at which the ball was higher than 65 feet O O O O 01 00 01 4 0 4 0 104011 2 3 910 2 3 105 100 95 90 85 80 65 55 50 45 40 35 30 25 20 15 10

Given 4 3 2n n 1 a write this series in standard form b is the series convergent or divergent Explain c If convergent find the sum

et f x be a polynomial of degree 3 over F13 Given that f 1 1 and f 2 4 what is the probability that f 0 5 Give such a polynomial All coefficients of f x should be integers in 0 1 2 3 4 5 6 7 8 9 10 11 12

2 Identify the shape the directrix and the eccentricity of r 4 2 cos 0

Refer to the functions r p and q Find the function r x 6x Part 0 2 BROKENSAANUAR Part 1 of 2 Part 1 2 Part 2 of 2 The domain of p x x 9x q x 5 x 0 10 9 X x and write the domain in interval notation x in interval notation is 00 CO 30 2 0 0 0 0 0

3 A carnival Ferris wheel with a radius of 7 m makes one complete rotation every 16 s The bottom of the wheel is 1 5 m above the ground a Draw a graph to show how a person s height above the ground varies with time b Write an equation to model the path of a passenger on the Ferris wheel where height is a function of time c Determine Emily s height above ground after 2 3 minutes 85 1 51 too 8 R 2 L A an

In Problems 4 to 6 evaluate the given limit Justify your solution 4 lim tan 426 1 22 1 2x 8118

6 lim In 2 3 4x 3 2x 3 2 x x 6 3 In Problems 7 and 8 find intervals of continuity of each function and explain why each functi continuous on those intervals Also check for one sided continuity at the endpoints of each inte

2 Determine a and b is such a way that the following function is continuous at z 0 Justify your solution f x a sin sin 3x 2x b ax 2ax 16 if r 0 if r 0 if x 0

Evaluate the expression 2 x y x y f x 40994 and y 2 Jopa 2 701 be in

x x r l x 2 r 1 a Is f x continuous at x 0 How do you b Is f x continuous at x 1 How do you Let f x know know plato to

alyze the following alternating series for absolute and conditional convergence 1 n 1 n4n 1 n 1

Concavity CUP or frOWN a Find the intervals of concavity of the function f x x 6x

Use polar coordinates to find the limit of the function as x y approaches 0 0 7xy 2 f x y A 1 OB O C 1 y

Date Jan 1 Feb 1 Mar 1 Apr 1 May 1 June 1 July 1 Aug 1 Sept 1 Oct 1 Nov 1 Dec 1 a bin aa durtablew pl form 1 year 2015 task 1 state OR place medford H M 9 09 10 00 11 14 12 43 14 04 15 04 15 14 14 27 13 09 11 45 10 20 9 19 Minutes 549 600 674 763 844 904 914 867 789 705 620 559 1000 950 900 850 800 750 700 650 600 550 500 y X 1 2 3 4 5 6 7 8 9 10 11 12 Plot the points on the grid provided Let January 1 correspond to x 0 February 1 correspond to x 1 etc and let y minutes of daylight Connect the points with a smooth curve It should look like a sine wave b Use your graph to find the amplitude period horizontal shift and vertical shift of the function Note It s reasonable to assume the minutes of daylight in January 2016 is the same as in January 2015 The length of a cycle is 12 months so that s the period Then write a sine or cosine function to represent the data

ev Does the Intermediate Value Theorem guarantee that the following equation has a real solution between and 3 Answer 2 Points 582x2 138x3 6x4 798x 30 1 O YES O NO Keypad Keyboard Shortcu

6 Given the correlation coefficient r value determine the strength of the relationship Defend your answers 6 marks a r 1 b r 0 86 c r 0 12

How long will it take for 5800 to grow to 25 000 at an interest rate of 3 3 if the interest is compounded continuously Round the number of years to the nearest hundredth 4 43 yr 4427 33 yr 0 44 yr 44 27 yr

As on the previous page f x jele T 52 6 4 x d Find lim f x and lim f x I 0 8118

d If lim f x 0 then f x is continuous at x 0 1 0 TRUE FALSE

c If lim f x exists then x 1 is in the domain of f x 2 1 TRUE FALSE