Limits & Continuity Questions and Answers

Write an equation for the graph in the form y a x h k where a is either 1 or 1 and h and k are integers 1109

Verify that the Intermediate Value Theorem applies to the indicated interval and find the value of c guaranteed by the theorem 5 4 f x x x X 1 f c 6

0 58 1 Points Complete the table Round your answers to five decimal places Assume x is in terms of radian 3 sin x X lim X 0 lim X 0 X f x 0 1 2 99500 0 01 2 99995 0 001 2 99999 0 0 001 2 99999 X X 0 01 2 99995 0 1 Use the result to estimate the limit Use a graphing utility to graph the function to confirm your result Round your answer to five decimal places 3 sin x X 3

Discuss the continuity of the function f x x x 3 2 4 1 3 y 2 3 20x 3 Of is discontinuous at x 0 Of is discontinuous at every integer f is discontinuous at every rational number Of is continuous for all real x

Discuss the continuity of the function x 4 x 2 f x f is discontinuous at x 5 4 3 2 1 y 5 4 3 2 1 1 21 3 4 1 2 3 4 5

Consider the following f x 2 6x 3 Find the x value at which f is not continuous Is the discontinuity removable Enter NONE in any unused answer blan X XS4 X 4 Select

Describe the interval s on which the function is continuous Enter your answer using interval notation f x X x x 5

lim X 5 Create a table of values for the function and use the result to estimate the limit Use a graphing utility to graph the function to confirm your result Round your answers to four decimal places If an answer does not exist enter DNE x x 1 5 6 x 5 X f x 4 9 lim x x 1 5 6 X 5 X 5 Need Help Read It 4 99 4 999 5 001 5 01 MY NOTES 5 1 ASK YOUR TEACHI

Consider the following limit lim 4x 9 X 9 Find the limit L L a Find the largest 6 0 such that If x LI 0 01 whenever 0 x cl 6 Round your answer to five decimal places 8 b Find the largest 0 such that If x LI 0 005 whenever 0 x cl 8 Round your answer to five decimal places 8

Find the limit of the function if it exists If an answer does not exist enter DNE x 27 27 lim x 3 x 3 Write a simpler function that agrees with the given function at all but one point Use a graphing utility to confirm your result g x

Sketch a graph of a function f that satisfies the given values f 2 is undefined lim f x 5 24 2 f 3 7 lim f x 2 2 3 O 10 8 C 10 2 N 2 10 Maple Generated Plot y to 101 8 2 IN 2

Consider the following information lim f x 8 HIC Use the information to evaluate the limits a lim f x X C b c d f x lim HIC 12 lim f x PIC lim f x 2 3 PIC

Sketch the graph of f sin x x 0 f x 9 9 cos x 0 x cos x X y 201 2 15 10 y 20 15 10 S A 37 2 2 Identify the values of c for which lim f x exists x c O The limit exists at all points on the graph except where c x O The limit exists at all points on the graph O The limit exists at all points on the graph except where c 18 The limit exists at all points on the graph except where c 0 and c 20 15 10 5 y y 20 15 10 X 3r 2 x

Consider the following F x x 1 FX 1 if x 1 2 y 2 2 X Use the graph to find the limit below if it exists If an answer does not exist enter DNE lim f x

Consider the following x 3 3 lim 1 3 x 3 y 2 10 8 6 4 2 2 4 2 6 8 10 Use the graph to find the limit if it exists If an answer door

Find the limit If an answer does not exist enter DNE X 9 lim x 9 x 81 STEP 1 Factor the denominator X 9 lim X 9 x 9 x STEP 2 Simplify lim x 9 x 1 1 STEP 3 Use your result from Step 2 to find the limit X 9 11

Find the limits f x x 5 g x x a lim f x b c X4 5 lim g x lim g f x

Complete the table Round your answers to four decimal places X 4 lim x 4 x 5x 4 3 9 X f x Need Help 3 99 Read it 3 999 4 4 001 4 01 Use the result to estimate the limit Use a graphing utility to graph the function to confirm your result Round your answer to four decimal place X 4 lim x 4x 5x 4 4 1

Find the period of the function y 0 5 sin 0 5 12 F 12

Find the exact value of the real number y y sin 1 O2TL 07 3

Find the rectangular coordinates of the point with the given polar coordinates 4 3 O 2 2 3 O 2 2 3 O 2 24 3 O 2 2 3

Which of the following polar point DOES NOT represent the yellow dot below 3 4 57 6 18 7 77 6 O 2 7m 4 57 4 2 7n 4 27 3 6 54 47 3 3 2 O 4 2 37 2 M 2 3 100 T 1 3 5 57 3 1 4 No 1 6 D 70 4 O OT 27 117 6

Convert from degrees to radians Use the value of found on a calculator and round answers to four decimal places as needed 450 0 572 O 57 5TL 574

Find the amplitude of the function y 2 cos 6x AMA O 6 OF 04

aluate the determinant 2 8 h The de Simpl

Use a graphing calculator to evaluate A A 3 4 3 8 1 3 4 A 8 1 0 7 6 5 3 0 7 6 5 11

Consider the following 2 x 1 f x Find the x values at which fis not continuous Which of the discontinuities are removable Enter your answers from smallest to largest Enter NONE in am unused answer blanks X Select H Select

S CURRICULUM Luations like these 4 99 30 C 22 2 3 48 10 2 92 eat of solution of the ammonium nitrate ans 26 kJ mol ed in 1 0 L water the water decreases in temperature from 21 0 C NH4NO3 6 3C 3c E 3c 4 30 441 8 2B 4 8 9 0 2 14 7 16 28 2 14 7 16 4 8 907 2B 2 4 14 87 916 07 2 167 8 1 3 1 87 24 33 4 18 11 3 heat of combustion of benzene ans 3 1 x 103 kJ mol s enough heat to raise the temperature of 5 0 kg of wall

34 37 C 8 48 6 3C 3c 3c 4 3c 8 2B 4 8 9 0 2 14 7 16 2B 2 14 7 16 4 8 907 2B 2 4 14 8 7 9 16 0 2 B 2 G 2 16 B 1 3 1 8 26 kJ mol NH4NO3 erature from 21 0 C 24 33 11 2 10 2 3 18 6 2

8 Which of the following are the side lengths of a right triangle d 6 8 10 a 6 7 8 b 5 9 12 c 4 6 8

them to do equations like these ch equation 6 6 2 4 9 9 B H i 21 36 3 4 11 2 26 2 adj B 18 2 0 30 7 AO 1 72 174 42 264 48 6 3C lution of the ammonium nitrate ans 26 kJ mol NH4NO3 OL water the water decreases in temperature from 21 0 C 3c4 34 49 3c 24 33 10 4 18 combustion of benzene ans heat to raise the teff 447 8 28 4 8 9 0 2 14 7 16 2B 2 14 7 16 4 8 907 2B 2 4 14 8 7 9 16 07 2 B 2 6 2 16 B 1 3 1 8

Determine the Cartesian equation of the plane 0 0 3 1 4 2 1 5 EER

You can retry this quest Solve the following equation for over the interval 0 2 giving exact answers in radian units If an equation has no solution enter DNE Multiple solutions should be entered as a comma separated list sin 4x 2 sin 2x 0 Step 1 Use an appropriate double angle formula to rewrite the equaiton as 2 sin 2x cos 2x 2 sin 2x 0 Step 2 Solve the equation for over the interval 0 2 3 x 0 TT 2

49 Apply integration to the expansion 1 1 x n 0 1 x 1 x x x to prove that for 1 x 1 2 1 In 1 x f 1 n n m 1 X x 2 3 24 4

Biologists use and create mathematical models in order to predict outcomes of various systems they are studying For example A population biologist studying a specific population will want to predict how fast the population is growing and what is the maximum population that can be reached An epidemiologist studying a disease virus needs to make predictions as to how fast the virus is spreading and what the total number of infected might be One model that is used in both those scenarios is the so called Logistics function In this project we will show how to use this function to help model the spread of a virus The logistics equation is given as the solution to the following differential equation T N t rN t No dt where r is the growth rate N t is the total number of people infected at time t No is the maximum number of people that will be infected The solution to this differential equation is called the Logistics Function and is given by the formula Noo N t N t 1 e rt No where No is the initial number of people infected and t is time measured in days past the initial starting time To prepare medical institutions to function in an optimal way during an epidemic it is important to know the number of infected people per day This is so we can find the maximum number of people that will be infected per day so the institutions can expect when the peak infections will be The time T that corresponds to the maximum number of people infected per day is called the epidemic peak We can find the infected number of people per day by differentiating the Logistics function N t from this we can then determine the epidemic peak time T N 2

4 The period of the function shown is approximately M A 8 B6 C4 D2 6 AM 20 2 0 M 8

MIMIMINIMINIMI MIMIMIMIMIMIN MIMIMIMINIAI PIT

A solid is produced with the region in the first quadrant bounded by the graph of TU y 1 sin x the line X 2 the x axis and the y axis is revolved about the x axis Its volume is found by evaluating which of the following integrals A 1 B 1 sin4 x dx S 0 T 2 1 sin x dx 1 sin x dx C T S 1 1 1 sin x dx D TU

Given that log 3 1 58 and log 5 2 32 evaluate each of the following a log 15 b log 1 8 c log 0 6 d log 5 e log 81 f log 45 B

with terminus in the second quadrant In what quadrants can the terminus of angle 2 lie Check all that apply 0 1 0 IO II O III In what quadrants can the terminus of angle Check all that apply OIO II Part 2 If cos p cos 26 Q III 2 IV 7 DIV determine the following two values as illustrated 2 lie

Part 1 Shown is the graph of the a rational function 1 y f x obtained from the graph of g x means of a horizontal and a vertical translation 10 9 8 7 6 10 9 8 7 6 5 4 3 2 x 1 4 3 2 1 1 2 3 4 5 6 7 8 9 10 1 For what values is f x not defined X 2 3 4 5 6 7 8 9 10 X Give equation s for all vertical asymptotes of the graph of f Give equation s for all horizontal asymptotes of the by

13 Find a value that satisfies the equation log x 2 log x 3 log 6

The parabola shown is the graph of f x Ax 4x C The x intercepts of the graph are at 2 and and 4 Find the exact value of the y intercept and the coordinates of the vertex of the graph expressed in terms of rational numbers and radicals y intercept the coordinates of a point vertex coordinates of a point Enter Enter the 6 5 4 3 1 1 2 3 4 5

3 State whether the functions below are one to one or not Explain your conclusion a f x x b f x 2

En el tri ngulo ABC a 3 b 4 c 6 Halla la medida del A en grados y redondeado a 1 lugar decimal O a 117 3 Ob 62 7 OC 26 4 Od 36 3

The graphs of y f x solid and y g x dashed are shown 6 5 4 3 2 1 Let h x h x f x g x 0 when x 1 2 h x is positive in the interval interval notation 4 5 6 Enter your answer in

Part 1 Suppose log p 1 The value of log p q 5 is 2 The value of logk 7 and log 9 V 4 2 Pq is 3 One way to express k in terms of p and q is kll 4 If log r 15 one way to express r in terms of p and q is r

R Find the measure of ZTGH M G 81 T

8 csc 0 A C 2 3 2 3 3 2 3 3 J2ST Iubivibal S 2 11 B 6 D 3 7 6

Use sigma notation to write the sum 6 6 6 5 1 n n 1