Limits & Continuity Questions and Answers

Cancelling T from the numerator and denominator gives us lim In y lim x 00 X 54

If y 1 2 x then In y Part 2 of 5 We can re write 6x Part 3 of 5 6x In 1 Now lim In y lim X 00 X 00 In In 1 6x 6x In 1 6x We can apply L Hopital s Rule and get lim X78 6x 1 6 L H 11 lim 818 m 1 6x

Part 1 of 5 an 2 1 If y 1 2 A Part 2 of 5 r n 1 2 then In y In 1 We can re write 6x 6x In 1 2 6x n 1 2

Tutorial Exercise Determine whether the sequence converges or diverges If it converges find the limit 6n Part 1 of 5 an If y 1 2 916x X then In y In 1 2 In 1 2 1

For the following right triangle find the side length x 12 X 35 0

Calculate the sum of the series sn n 1 5n 1 00 a whose partial sums are given n 1

Fill in the blanks in the statement below The represents the expected proportion of intervals that will contain the parameter if a large number of diffe samples of size n is obtained It is denoted The level of confidence represents the expected proportion of intervals that will contain the parameter if a large number of different samples of size n is obtained It is denoted 1 x 100 2 2x 100

Finding Cross Products In Exercises 7 8 2 and 10 find a u x v b v x u and C V X V 7 u 2i 4j v 3i 2j 5k

After how long will only 1 mg remain Step 1 To find the time at which only 1 mg remains we must solve 1 y t 50 2 t 30 and so we get the following t 30 log2 1 50 1 50 Step 2 Hence we conclude the following t 30 In In 2 yr Round your final answer to one decimal place

57 Braking Load A 48 000 pound truck is parked on a 10 slope see figure Assume the only force to overcome is that due to gravity Find Answer a the force required to keep the truck from rolling down the hill and b the force perpendicular to the hill a 8335 1 lb Weight 48 000 lb

Oil leaked from a tank at a rate of r t liters per hour The rate decreased as time passed and values of the rate at two hour time intervals are shown in the table t h r t L h 8 8 7 5 6 9 6 2 5 6 Find lower and upper estimates for the total amount of oil in liters that leaked from the tank over the interval 0 10 Use five equal subintervals lower estimate upper estimate 0 L L 2 10 5 1

Rewrite the equation in standard form then tell the positive b value of the following 9y 4x 18y 32x 91 0 4 9 01 156 None of these 3 12 5 6 2

3 Evaluate each limit if it exist a x 8x lim X 3 2x 1

the angle between the vectors a in radians and b in degrees 11 u 1 1 v 2 2

W S 0 01 n 1 Step 4 Recall the following theorem 8 n 1 For the sum 0 01 n 1 we have a 0 01 The geometric seriesar 1 a ar ar a ar ar is convergent if Irl 1 and its sum is n 1 7 1 0 01 n 1 0 81 81 n 1 1 10 01 0 01 100 99 0 01 100 99 the geometric series ar 1 n 1 Step 5 We have found the value of the geometric sequence with common ratio 0 01 Therefore we can now wice the repeating fraction 0 81 as a ratio of integers as follows 0 01 1 81 100 100 99 17 Inci 1 and r 0 01 1 Therefore we have the following

Choose ALL of the following that are equivalent to a Pythagorean identity Select 6 correct answer s O cot x csc x 1 Otan x sec x 1 csc a cot x 1 tan x sec x 1 Osin x 1 cos x Ocot x csc x 1 Osec x tan x 1 sin x 1 cos x cos r sin x 1 Ocot x csc x 1 O sin x cos x 1 Osec x 1 tan x K

Given the identity cos x sin x 1 select all of the following that are equivalent to it

Determine which of the following identity replacements is the best accurate choice in solving the equation 5 cos x sin x 1 5 1 sin x sin x 1 05 1 2 sin r sin x 1 05 1 sin x sin x 1 None of these O5 sin 2 sin x 1 D

1 2 Points DETAILS PREVIOUS ANSWERS SCALCET9 11 10 021 Find the Taylor series for f x centered at the given value of a Assume that f has a power series expansion Do not show that R x 0 x5 3x x a 4 f x 1220 1425 x 2 658 x 2 R Find the associated radius of convergence R 2 323 3 x 3 3 20 x a 4 x 55 0 X

lim 318 n 1 an lim 816 lim 818 x1 7 n 1 450 1 X 4 n n lim n 1 a n 1 lim 818 an 4 n n 1 4 Step 2 We have found the following limit from the Ratio Test x 7 n n 1 Submit Skip you cannot come back 71 By the Ratio Test the limit converges if Q n x FIX 1 and diverges if 1 Therefore the limit converges for Ixl if 21 The limit diverges for x Again recall that the radius of convergence is the positive value R such that the power series in this case centered at 0 converges if x R and diverges if x R Therefore the radius of convergence for the given power series is R

27 5 Points DETAILS Test the series for convergence or divergence 2 2 4 6 8 S 6 7 SCALCET9 Evaluate the following limit lim bn 318 Identify b Assume the series starts at n 1 0 10 8 Since lim b 0 and bo 1 810 4 b for all n the series converges

9 0 1 Points DETAILS 8 7 PREVIOUS ANSWERS Determine whether the geometric series is convergent or divergent If it is convergent find its sum If the quantity diverges enter DIVERGES 49 343 8 64 SCALCET9 11 2 024 MY NO

Calculate the sum of the series Sn n 1 3n 1 00 n 1 an whose partial sums are given

a What is the difference between a sequence and a series b A sequence is an ordered list of numbers whereas a series is an unordered list of numbers A sequence is an ordered list of numbers whereas a series is the sum of a list of numbers A sequence is the sum of a list of numbers whereas a series is an ordered list of numbers A sequence is the sum of a list of numbers whereas a series is an unordered list of numbers A sequence is an unordered list of numbers whereas a series is the sum of a list of numbers What is a convergent series What is a divergent series A series is convergent if the sequence of Select is a Select sequence A series Select if it is not convergen

Calculate the first eight terms of the sequence of partial sums correct to four decimal places n 1 2 3 4 5 6 7 8 8 1 5n n 1 nent or divergent

Find the radius interval of convergency of the power series 1 22 1 05 0 1

Determine whether the sequence converges or diverges If it converges find the limit If the sequence diverges enter DIVERGES an e4n n 4 lim a

a What is a sequence A sequence is an ordered list of numbers A sequence is the product of an ordered list of numbers A sequence is an unordered list of numbers A sequence is the sum of an ordered list of numbers A sequence is the sum of an unordered list of numbers b What does it mean to say that lim 818 8 an The terms a approach infinity as n become large The terms an approach 8 as n becomes small The terms an approach infinity as 8 approaches n The terms a approach infinity as 8 approaches n n O The terms a approach 8 as n becomes large c What does it mean to say that liman n O The terms a The terms a The terms a n o become small as n becomes small become small as n becomes large become large as n becomes large become large as n becomes small The terms an The terms a approach zero as n becomes large

List the first five terms of the sequence 3 0 2 n n 22 23 24 25 a6 11 11 11 MI

Step 5 Now we must finde radius of convergence for the Maclaurin series First we set up the ratio test as follows 4 n 2 x 1 an 1 lim 816 an lim 510 Submit Skip you cannot come back n o 4 n 1 x That is the positive number R such that the series converges if x al R and diverges if Ix al R We can use the ratio test to find R

f x f 0 f 0 x 4 00 n 0 f 0 2 begins as 8 f 0 3 3 Submit Skin you cannot come back 8 x 12 4 0 4 12 x 16 V 16 x 20 20 4 x Step 4 Now that we have the beginning of the Maclaurin series continuing this pattern we find the general formula as follows f x 4 8x 12x 16x 20x

2 Points f x Find a power series representation for the function Center your power series representation at x 0 00 n 0 DETAILS f x SCALCET9 11 9 003 MI 1 9 x Determine the interval of convergence Enter your answer using interval notation

Answer Above the xy plane and above Quadrants II or IV or below the xy plane and below Quadrants I or III

Finding the Equation of a Sphere In Exercises 37 38 39 40 41 and 42 find the standard equation of the sphere with the given characteristics 37 Center 7 1 2 Radius 1 Answer x 7 y 1 z 2 1 38 Center 1 5 8 Radius 5 39 Endpoints of a diameter 2 1 3 1 3 1 Answer X 1 2 3 y 2 z 1 21 4

Plotting Points In Exercises 5 6 7 and 8 plot the points in the same three dimensional coordinate system 5 a 2 1 3 b 1 2 1

3 Use the data below for the temperature in degrees Celsius for a wood fire oven App Time in Minutes 058 10 13 15 19 21 25 290 285 285 Temp C 25 120 205 250 290 280 a find the average rate of change of the temperature between 0 and 25 minutes Show proper units and notation b Estimate the instantaneous rate of change of the temperature at 10 minutes Use 2 methods

Suppose that fis a one to one function and f is its inverse Suppose also that h x 4 and g x x2 xsecx Then which of the following do we NOT know to be true X A fogof x x x secx B hogog x 4 OC gohof x 16 4 sec 4 D his f f ofofog x 4x 4x sec x E fof oh x 4

If f x x2 on the domain 2 2 then f OA has domain 4 4 OB equals C doesn t exist D has range 2 2 hos doma

Suppose P x and the F x hog x Then which of the following are possible definitions for hand g I h x x g II h x x g x X x 2 x 7 H III h 2 37 9 2 IV h x 7 9 x x x 7 x 7 8 OA II and III only B I and II only C I and IV only D I III and IV only E I II III and IV

Tutorial Exercise Find the Maclaurin series for f x using the definition a Maclaurin series Assume that f has a power series expansion Do not show that R x 0 Find the associated radius of convergence R f x 4 1 x

If f x b x 3 for all x write a formula for b n 0 O b O b O b Ob 7 Ob F 7 3 f 3 3 f 3 F 7 7 f 3

4 Evaluate the following 100 4 2 0 2 i 3 i a b 8 2i 3 21 5 2

Finding a Unit Vector In Exercises 35 36 37 and 38 find the unit vector in the direction of v an verify that it has length 1 35 v 3 12 Answer 17 4 17

F x 5 4x3 Describe any relative extrema Select the correct choice below and if necessary fill in the answer box es to within your choice OA The relative minimum point s is are and the relative maximum point s is are Simplify your answers Type ordered pairs using integers or fractions Use a comma to separate answers as needed OB The relative minimum point s is are and there are no relative maximum points Simplify your answer Type an ordered pair using integers or fractions Use a comma to separate answers as needed OC The relative maximum point s is are and there are no relative minimum points Simplify your answer Type an ordered pair using integers or fractions Use a comma to separate answers as needed O D There are no relative minimum points and there are no relative maximum points Identify any intervals over which the function is increasing or decreasing Select the correct choice below and fill in the answer box es within your choice and is not decreasing anywhere or fractions for any numbers in the expression Use a comma to separate answers as needed and decreasing over the interval s OA The function F x is increasing over the interval s Type your answer in interval notation Use integers OB The function F x is increasing over the interval s Type your answers in interval notation Use integers or fractions for any numbers in the expressions Use a comma to separate answers as needed and is not increasing anywhere OC The function F x is decreasing over the interval s Type your answer in interval notation Use integers or fractions for any numbers in the expression Use a comma to separate answers as needed Choose the correct graph below OA 10 Q OB Ay DDD Q O C OD Q Q G

Slope 4 passing through 2 17 What is the point slope form of the equation of the line Simplify your answer Use integers or fractions for any numbers in the equation What is the slope intercept form of the equation of the line Simplify your answer Use integers or fractions for any numbers in the equation Use the figure below to list the slopes m m m3 and m4 in order of decreasing size Use the graph to identify the a x intercept or state that there is no x intercept b y intercept or state that there is no y intercept slope my OA The y intercept is Type an integer OB There is no y intercept slope m slope m slope m Q a Select the correct choice below and fill in any answer boxes within your choice OA The x intercept is Type an integer OB There is no x intercept b Select the correct choice below and fill in any answer boxes within your choice HIDD Choose the correct answer bele O m3 M M M4 m3 M M M4 O m M3 M4 M O m4 M M M3 OOO 10

The graph of f x 5sin 2 x 1 is obtained from the graph of y sin x by Shifting left 2 units and down 1 unit Stretching vertically by a factor of 2 horizontally by a factor of 5 and down 1 Shifting right 1 unit and up 5 units Vertical stretch by a factor of 5 horizontal shrink by a factor of 2 and a phase shift of 1 unit right Shifting right 5 units and stretching vertically by a factor of 2

H possible 60 30 2 30 this gra 60 90 SI SIXP X 120 150 x

To find the distance between two cities a satellite calculates the distances and angle shown not to scale Find the distance between the cities Round answers to the nearest tenth To get full points show your work for this problem on your scratch paper 250 km 1 8 210 km

Find the measure of ZR to the nearest degree The picture is not neccessarily drawn to scale P 90 145 02 5 7 R 3 Q

If the period of a function is 5 9 radians what is it s frequency