Limits & Continuity Questions and Answers

1 point The three series An Bn and Cn have terms 1 1 n10 n5 0 1 2 3 n n 1 An Use the Limit Comparison Test to compare the following series to any of the above series For each of the series below you must enter two letters The first is the letter A B or C of the series above that it can be legally compared to with the Limit Comparison Test The second is C if the given series converges or D if it diverges So for instance if you believe the series converges and can be compared with series C above you would enter CC or if you believe it diverges and can be compared with series A you would enter AD n 1 5n n 1309n15 Bn 10 9n5 5 5n 8n 5n10 9n35 8n n 8n 9n13 5n 2 7 2 Cn 1 n Note To get full credit all answers must be correct Having all

For the following exercises start with the graph of f x 2 Then write a function g x that results from the given transformation Shift f x 1 units upward g x help formulas Shift f x 8 units downward g x Shift f x 5 units left g x Shift f x 4 units right g x Reflect f x about the x axis g x Reflect f x about the y axis g x B help formulas help formulas help formulas help formulas help formulas

Evaluate the exponential function h x 4 for h 3 Round to 4 decimal places if necessary 3 help formulas h 3

2 Evaluate the indicated limits If it does not exist is the limit or neither lim x 2x x 2x b lim x 6x x I 118 X48 a X x 2 2x c lim d lim x118 4 3x x6 9

Follow the steps for graphing a rational function to graph the function G x OD The domain is the set of all real numbers Write G x in lowest terms Select the correct choice below and if necessary fill in the answer box within your choice OA G x Simplify your answer B G x is in lowest terms X ZA 4 Find the intercepts of the graph Select the correct choice below and if necessary fill in the answer box es within your choice OA The graph has x intercept s and y intercept Simplify your answers Type integers or fractions Use a comma to separate answers as needed Type each answer only once OB The graph has y intercept and no x intercept Simplify your answer Type an integer or a fraction OC The graph has x intercept s and no y intercept Simplify your answer Type an integer or a fraction Use a comma to separate answers as needed Type each answer only once OD The graph has neither x intercepts nor y intercepts

Find the formula for an exponential equation that passes through the points 3 8 and 3 1 The exponential equation should be of the form y ab help equations Round a and b values to at least 5 decimals where appropriate

Let f x 3x 9x 12 4x 8x L 12 Domain Find the domain vertical asymptote s horizontal asymptote and select the correct graph help intervals Vertical Asymptote s Horizontal Asymptote If more than one enter values as a comma separated list help numbers help numbers

Let f x 9x 2 9x 9 Domain Find the domain vertical asymptote behavior of the function near the vertical asymptote and horizontal asymptote help intervals Vertical Asymptote a As x1 f x As a 1 f x Horizontal Asymptote y help numbers enter infinity or infinity help numbers enter infinity or infinity help numbers help numbers

Find an equation for a rational function f x that satisfies the following Vertical asymptote at x 8 and x 4 x intercepts at 3 0 and 6 0 y intercept at 0 36 f x help formulas

Find the formula for an exponential equation that passes through the points 0 5 and 1 1 The exponential equation should be of the form y ab help equations

Let f x x 2x 8 x 6x 40 Identify the removable discontinuity Removable Discontinuity x help numbers

First Floor Second Floor Third Floor 13 15 14 16 11 9 22 38 40 39 The two way table shown above gives data on the connection preferences available from a local cable company in reference to the floor in which the apartment is located What is the marginal distribution of second floor connections Round to the nearest whole percent Internet Only Cable Only Internet Cable 11 6 39 36 42 117

5 Waylon is back to school shopping with his mom at a large department store He s picked o the following items 3 pair of pants for 87 total 4 t shirts for 48 total 1 backpack for 35 1 hoodie for 26 1 pair of shoes for 37 When they get to the register they see this sign BIGGEST BACK TO SCHOOL SALE OF THE SEASON Spend 100 Spend 200 Spend 300 Spend 400 Save 10 Save 15 Save 20 Save 30 a How much will Waylon s purchases cost with the correct discount 1pt

An equation is given 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 COS a Find all solutions of the equation 0 1 0 b Find the solutions in the interval 0 21 8

Use a Double or Half Angle Formula to solve the equation in the interval 0 2 Enter your answers as a comma separated list sin 20 sin 0 0 0

Problem 8 Consider the following function f x est 5 8 5e x Problem 8 a Problem 8 b a Find the two horizontal asymptotes of f b Find the vertical asymptote of f Enter your answer symbolically as in these examples Enter your answer symbolically in th separate your answers do not include y in y

Problem 11 Evaluate the following limits 3 lim X 0 x5x 5x 1 b lim x5 tan 7 x5 X18 lim 1 X 8 7x

Find the absolute extreme values of the function shown below 10 4 Absolute Minimum Value 5 10 x

Find the exact solutions of the equation in the interval 0 2 sin 2x cos x 0 X X X X smallest value largest value

A quantity x is distributed through a population with probability density function p x and cumulative distribution function P x Decide if the statement is true or false If p 12 2 then half the population has x 12 True False

Use power series operations to find the Taylor series at x 0 for the following function 20xex The Taylor series for ex is a commonly known series What is the Taylor series at x 0 for ex Type an exact answer

Use substitution to find the Taylor series at x 0 of the function In 1 5x7 What is the general expression for the nth term in the Taylor series at x 0 for In 1 5x7 O n 1

6 The graph of f x 4x 5 x 2 domain and range of f 1 is shown Give the 12 10 8 6 4 204 y 18 2 16 14 12 10 8 6 2 10 2 2 4 6 8 10 12 X

A zoo needs to build a rectangular enclosure that is made up of a 15 foot brick retaining wall on one side and a 10 foot chain link fence topping a two foot high brick wall on the other three sides The brick retaining wall costs 280 dollars per foot and the chain link brick fence costs about 73 per foot What dimensions will minimize the cost to build an enclosure that holds an area of 1900 square feet The brick retaining wall will be about feet long The chain link fence just one of the sides adjacent to the brick wall not opposite will be about feet long

The campus of a college has plans to construct a rectangular parking lot on land bordered on one side by a highway There are 680 ft of fencing available to fence the other three sides Let x represent the length of each of the two parallel sides of fencing a Express the length of the remaining side to be fenced in terms of x b What are the restrictions on x X c Determine a function A that represents the area of the parking lot in terms of x d Determine the values of x that will give an area between 20 000 and 40 000 ft e What dimensions will give a maximum area and what will this area be a The length of the remaining side to be fenced in terms of x is Do not factor HH

1 Consider the function B t that gives the temperature of Boston Massachusetts on a particular day where t is measured in hours after midnight Is B t a one to one function How do you know

818 2 7 0002 9

Consider the growth of a cell assumed to be spherical in shape Suppose that the radius of the cell r increases at a constant rate per uni time Call the constant k Express your answers in terms of the radius of the cell r and its growth rate k a At what rate does the volume V increase dV dt b At what rate does the surface area S increase dS dt c At what rate does the ratio of surface area to volume S V change d S dt V Does the ratio S V decrease or increase as the cell grows

Question A car is travelling due north at a speed of 46 miles per hour If the car started off 153 miles directly west of the city of Smithville how fast is the distance between the car and Smithville changing when the car has travelled 137 miles Do include units in your answer and round to the nearest hundredth Provide your answer below

3 Find the Taylor series for f x 1 a 3 x

If g x ln x 8 x 8 find lim g x X 8

Question g x x 5 ln x 5 What is lim g x X 5 Provide your answer below 7 8 9 4 X V y 6 4 VO

Question Determine lim 2x e x X4 8

What is lim p x 818 2x P x 1 4

Determine algebraically whether the given function is even odd or neither g x 8x 6 O Even ONeither O Odd

Write an equation of a circle with diameter AB A 5 5 B 15 15 Choose the correct answer below OA x 10 y 10 15 OC x 5 y 5 15 O E x 10 y 10 5 OG x 10 y 10 50 OB x 10 y 10 50 OD x 10 y 10 50 OF x 15 y 15 5 OH x 10 y 10 50

Question Determine lim 2x X18

Determine whether the sequence is divergent or convergent If it is convergent evaluate its limit If it diverges to infinity state your answer as INF without the quotation marks If it diverges to negative infinity state your answer as MINF If it diverges without being infinity or negative infinity state your answer as DIV lim 7n 11 700 11 12

Determine whether the sequence is divergent or convergent If it is convergent evaluate its limit If it diverges to infinity state your answer as inf If it diverges to negative infinity state your answer as inf If it diverges without being infinity or negative infinity state your answer as div lim 1 sin 17 n 11 700

Write out the first five terms of the sequence with 1 n 1 converges and if so find its limit Enter the following information for an a1 a2 a3 a4 ab In n lim n on 1 Enter DNE if limit Does Not Exist Does the sequence converge ln n n 1 Enter yes or no n 1 determine whether the sequence

The parametric equations and parameter intervals for the motion of a particle in the xy plane are given below Identify the particle s path by finding a Cartesian equation for it Graph the Cartesian equation Indicate the portion of the graph traced by the particle and the direction of motion x 5 cosht y 5 sinh t costs Find a Cartesian equation for the particle s path y x 25 Type an exact answer using radicals as needed Graph the Cartesian equation below Indicate the direction of motion as t increases Use a dotted line to represent portions if any of the graph that are not traced by the particle Choose the correct graph below O A 10 10 10 OB 10 10 10 10 Q C 10 10 10 10 Q O D 10 10 10

Solve the multiple angle equation Enter your answers as a csc 2 2 X Need Help Submit Answer 9 1 Points DETAILS Approximate the x intercepts of the graph Use a graphing utility to check your solutions Enter your answers as a co y 4 sin x 4 cos x X Read It Need Help Submit Answer 1 5 Read It LARPRECALCRMRP7 5 3 070 Watch It 1 2 3

Find all solutions of the equation in the interval 0 360 Enter your answers from smallest to largest tan x 3 O

Piecewise Defined Functions Graph the functions in Exercises 25 28 Jx 25 f x 26 g x 27 F x 28 G x 2 x f1 x 12 X 0 x 1 1 x 2 4 x 33 2x 2x 53 93 1 x x 0 x 1 1 x 2 x 1 x 0 0 x

radius loses 3 inches every 30 seconds Do the following a Express the circumference in inches of the circle as a function of its radius in inches way that C r b Express the radius in inches of the circle as a function of time in minutes c Express the circumference in inches of the circle as a function of time in minutes C t d Choose the composition that represents your answer to part c O C t Cor t O r t ro C t O C r ro C r O C t ro C t Or C ro C t

estimate the total amount of infection needed to develop symptoms of measles NA N f t 1000 ZA 0 12 21 t days

The equation below gives parametric equations and parameter intervals for the motion of a particle in the xy plane Identify the particle s path by finding a Cartesian equation for it Graph the Cartesian equation Indicate the portion the graph traced by the particle and the direction of motion x 6t 1 y 24t 2 t y 4x 2 Graph the Cartesian equation below Indicate the direction of motion as t increases Choose the correct graph below O A OB O C 10 10 Indicate the portion of the graph traced by the particle Select the correct choice below and if necessary fill in the answer box to complete your choice OA The entire graph B The portion of the graph where t is less than D C None of the graph D The portion of the graph where t is greater than

6 2 x 7 X 2 OA OB O C O D 13 1 2 710 125 00 U 13 7 2 12 00 13 1 7 210 125 00 U 13 27 12 00 U

The horizontal asymptote is y 5 OA OB OC OD 5 f x x 5 f x x 5 X 9 f x 5x 1 x 9 5x X 9

Give the equations of any asymptotes of the type specified for the graph of the rationa 2x 11 x 2x 15 f x vertical OA x 3 x 5 OB x 3 x 5 OC y 3 y 5 OD y 2