CALCULUS 2 FIVES SHEET F=Fun at home I=Incunabula V=Variety E=Endeavors S=Specimens

Monday, 1/4 – FIRST DAY OF SECOND SEMESTER!!!!! F #72 Read p.613-619 Do p.619 (2–6,8–10) For problems 2–6, find the first four terms of the sequence and,

if the sequence converges, what is its limit? For problems 8–10, find the first 6 terms of the sequence. I COMMON CORE Math (1) – at the back of the FIVES sheet V (no calculator)

E 1. Introduction to infinite sequences and their limits. 2. To investigate recursive sequences. S Write out the first 4 terms of the following sequences and, if they converge, find the limit:

1. an = 1n 2. an =

n + 12n + 1 3. an =

n – 1n 4. an = 3 5. an = (–1)n+1

1n 6. (–1)n+1

n – 1

n Write the first

6 terms of the sequences: 7. where x1 = 1 and xn+1 = xn + (2n + 1) 8. where x1 = 1 and xn+1 = xn + 1

2 n

1. 1, 1/2, 1/3, 1/4, . . . -> 0 2. 2/3, 3/5, 4/7, 5/9, ... -> 1/2 3. 0, 1/2, 2/3, 3/4, ... -> 1 4. 3,3,3,3,... -> 3 5. 1, -1/2, 1/3, -1/4, ... -> 0 6. 0, -1/2, 2/3, -3/4, ... -> diverges

Tuesday, 1/5 F #73 Read p.623-628. Pay close attention to Table 8.1 on p. 625 (and below). Do p.628 (1–10) Write the

first 4 terms for each sequence. Determine which converge and which diverge. Find the limit of each that converges.

I COMMON CORE Math (2) V (no calculator)

E 1. Find the convergence or divergence of an infinite sequence. 2. To find the limit of a convergent

sequence. If an L as n ∞ , then f(an) f(L).

S Write the first 4 terms of these sequences and, if they converge, find the limit: 1. an =

– 1

n

2. an = 4 – 7n6

n6 + 3 3. an =

n3 + 5nn4 – 6

4. an = n2 – 5n + 1 5. an =

5n2n

1. 0 2. –7 3. 0 4. diverges 5. 0 : use l'hopital's rule 5

2n ln 2

nlim lnn

n 0 n∞

lim nn n∞lim n1/n 1

nlim x1/n 1 if x 0 n

lim xn 0 if 1 x 1

nlim 1

x

n

n

ex nlim xn

n! 0

Examples:

1. an = ln(n2)

n 2. an = n

n2 3. an = n

3n 4. an =

–

12 n 5. an =

n – 2

n n 6. an = 100n

n!

Wednesday, 1/6 GUEST SPEAKER DAY!!! Go to the Golden Horseshoe for class today if you are periods 0-4

CALCULUS 2 FIVES SHEET F=Fun at home I=Incunabula V=Variety E=Endeavors S=Specimens

Thursday, 1/7 F #74 p.628 (11,15,21,22,24,25,27,28,33,34,35,38,42)

I If ƒ(n + 1) = 2ƒ(n) + 1

2 and ƒ(1) = 2, then ƒ(37) = A) 18 B) 19 C) 20 D) 21 E) 22

V (no calculator)

E 1. Find the convergence or divergence of an infinite sequence. 2. To find the limit of a convergent

sequence. 3. Pay attention to Table 8.1. S Determine if the sequence converges or diverges and find the limit if convergent:

1. an = n + 1

n 2. an = ln n

n 3. an = 1 + (–1)n 4. an = nπ cos(nπ) if n = 1,2,3,...

an = –π,2π,–3π,4π,–5π,... so diverges (“I” problem is “C”) Friday, 1/8 F #75 Read p.630–637. Do p.638 (7,9,10,11,14,24,27,30,35,39)

I If y = – x3

3 – x2, then the value of d2y

dx2 when x = 12 is A) –1 B) –

54 C)

49576 D) –5 E) –3

V (no calculator)

E 1. An introduction to infinite series. 2. Computing the sum of an infinite geometric series. 3. Use the Divergence Test (nth Term Test) to determine if a series diverges. Remember: A series

converges if its sequence of partial sums converges. S Find the ratio, then compute the sum of the series:

1. 1 + 12 +

14 + . . . +

12n–1

+ . . . 2. 1 – 13 +

19 – . . . + (–1)n–1

13n–1 + . . .

Converge or Diverge: 3. n=1

∞

1

2n 4. n=1

∞

n + 1

n 5. n=1

∞

5(–1)n

4n 6. n=0

∞

tan π4

n 7. n=1

∞

n

2n + 5

8. n=1

∞

1n 9.

n=1

∞

1

n2 10. n=1

∞ n2 11.

n=1

∞ (–1)n+1 (“I” problem is “E”)

1. r = 1/2 conv 2. r = –1/3 conv 3. r = 1/2 conv 4. div 5. r = –1/4 6. 7. div 8. div 9. conv

CALCULUS 2 FIVES SHEET F=Fun at home I=Incunabula V=Variety E=Endeavors S=Specimens

Monday, 1/11 F #76 Read p.640–643 Do p.643 (1,3,4,5,6,7,9,11,13,17,18,25) I COMMON CORE Math (3) V What is the largest prime divisor of every 3-digit number with 3 identical non-zero digits? Show work. E 1. To use the Integral Test for convergence of an infinite series. 2. To learn about the p–series and its

convergence and divergence. (For p-series: ∑1ap converges if p>1, diverges if p≤1)

S Converge or diverge: 1. ∑ 1n2 2. ∑

1n (harmonic series) 3. ∑

k

ek2 4. ∑ 1

3n

5. ∑ 1

n + 6

6. ∑ n

n2 + 1 7. ∑

n

en2 1. p-ser, con 2. p-ser, div 3. int, con 4. p-ser, div 5. int, div 6. int, 7. con

NOTES:

nlim lnn

n 0 n∞

lim nn n∞lim n1/n 1

nlim x1/n 1 if x 0 n

lim xn 0 if 1 x 1

nlim 1

x

n

n

ex nlim xn

n! 0

Tuesday, 1/12 - LATE START TODAY! F #77 p.700 (1–13 odd, 23,25,26,30) For 23,25,26,30 – determine if the series converges or diverges and

state why. I COMMON CORE Math (4)

V What are all 3 ordered triples of integers ( , , )a b c , with 0 ,a b c for which 1 1 1

1?a b c

E 1. Review for the test.

S Do the following sequences converge or diverge? 1.

1

n2 2.

(–1)n

n 3.

3n

en 4.

n3

en 5.

ln n

n

Wednesday, 1/13 F #78 Worksheet #22

I The length of the arc given by x = 4cos3 t and y = 4 sin3 t 0 ≤ t ≤ π2 is

A) π2 B)

3π2 C) 3π D) 3 E) 6

V A distribution consists of the integers from 1 through 100, inclusive, such that the frequency of each integer n is 12n . What is the median of this distribution?

E Review for the test. S 1. The slope of a curve for all x equals 2x + 3 and (0,1) lies on the curve. Find the equation of the curve.

2. If a1 = 1, and an+1 = an

n + 3 , write the next 3 terms. 3. If an = 3

1 + n2 , then does the sequence converge

or diverge? 4. n=1

∞

3

1 + n2 : Converge or diverge? (“I” problem is “E”)

CALCULUS 2 FIVES SHEET F=Fun at home I=Incunabula V=Variety E=Endeavors S=Specimens

Thursday, 1/14 F #79 Worksheet #23

I The graph of y = 5x4 – x5 has a point of inflection at A) (0,0) only B) (3,162) only C) (4,256) only D) (0,0), and (3, 162) E) (0,0) and (4,256) V (no calculator)

E Final review for tomorrow's test. S Lots of practice problems. (“I” problem is “B”) Friday, 1/15 – TEST TODAY!!!! F #79.5 Read p.644–648(especially the part on the direct comparison test) DO p.649(2,3)

I 1

e x ln x dx = A)

e2 + 14 B) 2e2 – 1 C)

e2 – e – 12 D) 2e2 + 1 E) undefined

V (no calculator)

E 1. Easy test on sequences and series. 2. To use the Comparison Test for convergence of an infinite series. S None (“I” problem is “A”) Monday, 1/18 No School: Martin Luther King, Jr Day

THE AP TEST “SEASON” IS COMING UP! THE AP MOVIE WILL HELP YOU GET INTO THE RIGHT MODE

FOR GETTING A 5!!!!!

Thursday, 1/21 F #80 p.649(3,4,5,7,23,24) I COMMON CORE Math (5) V (no calculator)

E To use the Comparison Test for convergence of an infinite series.

S 1. ∑ 1

2n + 1 2.

3

n nn1

3. ∑

sin2 n

2n

1. conv: comp w/ 1/2n 2. conv: comp w/ 3/(2n) – if you chose 3/n it doesn’t work 3. conv: comp w/ 1/2n

REMEMBER: IF THE SMALLER ONE DIVERGES, THEN THE BIGGER ONE DIVERGES.

CALCULUS 2 FIVES SHEET F=Fun at home I=Incunabula V=Variety E=Endeavors S=Specimens

IF THE BIGGER ONE CONVERGES, THEN THE SMALLER ONE CONVERGES. Friday, 1/22 F #81 Do p.649 (6,8,10,11,13,27,28) I STAR Summative Math (6) V (no calculator)

E 1. To use the Limit-Comparison Test for convergence of an infinite series.

S 1. ∑ 2n + 1

n2 + 2n + 1 2. ∑

12n – 1

3. ∑ 1 + n ln n

n2 + 5 4. ∑

ln nn3/2

1. lim-comp w/ ∑2/n 2. dir comp w/ ∑1/2n 3. lim comp w/ 1/n 4. comp w/ 1/n5/4 Monday, 1/25 F #82 p.649 (1,9,12,15)

I The function f is continuous for all positive real numbers. If f(x) = ln x2 – x lnx

x – 2 when x 2, then f(2) is

(A) – 1 (B) – 2 (C) – e (D) – ln 2 (E) undefined V (no calculator)

E 1. More work with the Limit-Comparison Test. S When is a surprise not a surprise??? (“I” problem is “D”) Tuesday, 1/26 F #83 Read p.649-654. Do the following series converge or diverge? Show all work.

Ratio Test: 1. ∑3k

k! 2. ∑4k

k2 3. ∑k!k3

Root Test: 4. ∑

3k + 2

2k – 1 k 5. ∑k5k

Use any appropriate test:

6. ∑2k

k3 7. ∑7k

k! 8. ∑k2

5k 9. ∑k50e-k 10. ∑k2

3 k

I A point moves on the x–axis in such a way that its velocity at time t (t > 0) is given by

v = ln t t . At what value of t does v attain its maximum?

A) 1 B) e1/2 C) e D) e 3/2 E) There is no maximum value for v. V (no calculator)

E 3. Learn to use the ratio and root tests for convergence or divergence of a series.

S Converge or diverge? 1. ∑1k! 2. ∑

k2k 3. ∑

kk

k! 4. ∑(2k)!

4k

5. ∑

4k – 5

2k + 1 k 6. ∑1

(ln(k + 1))k (“I” problem is “C”)

CALCULUS 2 FIVES SHEET F=Fun at home I=Incunabula V=Variety E=Endeavors S=Specimens

Wednesday, 1/27 F #84 p.661-662 (1–7,9,45)

I At x = 0, which of the following is true of the function f defined by ƒ(x) = x2 + e–2x ? A) ƒ is increasing B) ƒ is decreasing C) ƒ is discontinuous. D) ƒ has a relative minimum E) ƒ has a relative maximum. V Of the following choices of , which is he largest that could be used successfully with an arbitrary in an

epsilon–delta proof of x2lim (1 3x)= –5? A) = 3 B) 6 = C) =

2 D) =

4 E) =

5

E 1. Determining the convergence or divergence of alternating series. 2. Approximating error of the sum of an alternating series.

S Converge or diverge? 1. ∑(–1)k+1 1k 2. ∑(–1)k+1

k + 3k(k + 1) 3. ∑

(–1)k+1

2k + 1 4. ∑(–1)k+1 k + 13k + 1

5. Estimate the sum if the first 5 terms of ∑(–1)k+1 1k are used to estimate S.

6. Also ∑(–1)k+1 1

2 k (“I” problem is “B”) 1. Con 2. Con 3. Con 4. Div 5. 47/60 6. 11/32

Thursday, 1/28 F #85 p.661 (13,15,16,17,21,46) and p.654 (1,3,8,9) I COMMON CORE Math (7) V (no calculator)

E 1. To determine if a series is absolutely convergent, conditionally convergent, or divergent. 2. With an alternating series, if the series diverges, you're done! If the series converges, then check the

series of absolute values –– if it converges then the original series converges absolutely –– if it diverges, then the original series converges conditionally.

S 1. ∑(–1)n+1 1n2 2. ∑(–1)n+1

1n 3. ∑(–1)n+1

n5n + 1 4. ∑(–1)n+1

n + 4n3

1. conv abs 2. conv cond 3. diverges 4. conv abs Friday, 1/29 F #86 p.700 Practice Exercises (25,26,27,28,30,34,35,37,38,40)

I 0

1

x2 – 2x + 1 dx is A) – 1 B) – 12 C)

12 D) 1 E) none of the above

V (no calculator)

E Review for the test on series. S Lots of review questions!! (C is the answer to the “I”)

CALCULUS 2 FIVES SHEET F=Fun at home I=Incunabula V=Variety E=Endeavors S=Specimens

Monday, 2/1 F #87 Worksheet #24

I Which of the following is a point of discontinuity for ƒ(x) = x2 – 4

x2+2x–3 is A) –3 B) 2 C) 0 D) –1 E) –2

V

2xx + 5

dx = (show all work)

E Review for the test on series. S Lots of review questions!!! (“I” problem is “A”) Tuesday, 2/2 F #89 Worksheet #25

I

2

+∞

dxx2 is (A)

12 (B) ln 2 (C) 1 (D) 2 (E) nonexistent

V (no calculator)

E 1. Review for tomorrow's test. Study material from the last test as well as material since the last test to succeed on the test. (“I” is “A”) Wednesday, 2/3 – TEST TODAY!!! F #90 Read p.663-670 Do p.671 (1a)

I If dydx = tan x, then y =

A) 12 tan 2x + C B) sec2x + C C) ln |sec x| + C D) ln |cos x| + C E) sec x tan x + C

V (no calculator)

E 1. Take an easy test on the convergence and divergence of infinite series and sequences using all of the

tests that we have covered. 2. Begin the study of power series. S None (“I” is “C”)

CALCULUS 2 FIVES SHEET F=Fun at home I=Incunabula V=Variety E=Endeavors S=Specimens

ANSWERS Assignment #72 2) 1, 1/2, 1/6, 1/24 lim=0 3) 1,–1/3,1/5,–1/7 lim=0 4) 1,3,1,3 lim=no limit 5) 1/2,1/2,1/2,1/2 lim=1/2 6) 1/2, 3/4, 7/8, 15/16 lim=1 8) 1, 1/2, 1/6, 1/24, 1/120, 1/720 9) 2,1,–1/2,–1/4,1/8,1/16 10) –2, –1, –2/3, –1/2, –2/5, –1/3 Assignment #73 1) 2.1,2.01,2.001,2.0001 conv to 2 2) 0,3/2,2/3,5/4, converges to 1 3) –1/3,–3/5,–5/7,–7/9 conv to –1 4) –3/2,5/(1–3 2 ),7/(1–3 3 ),9/–5 diverges 5) –.444, –.9875, –1.36, –1.665 conv to –5 6) 1/3,1/4,1/5,1/6 converges to 0 7) 0,1,2,3 diverges 8) 0,–7/54,–13/17,–21/2 diverges 9) 0,2,0,2 diverges 10) 0, 1/2, –2/3, 3/4 diverges Assignment #74 11. conv 1/2 15. conv 2 21. conv 0 22. diverges 24. conv 1 25. conv 1 27. conv e7 28. conv e–1 33. diverges 34. conv 0 35. conv 4 38. conv 0 42. conv 1 Assignment #75

7. 45 9.

73 10. 4 11.

232 14.

103 24. diverges: geom series r = 2 > 1

27. diverges: lim n->∞(–1)n ≠ 0 30. diverges since lim ≠ 0 35. diverges

39. converges: geom S=ππ–e

7. 1–1/4+1/16–1/64+1/256–... 9. 7/4+7/16+7/64+7/256+ ... 10. –5/4+5/16–5/64+5/256– ... 11. 6+7/6+49/36+143/216+... 14. 2+4/5+8/25+16/125+... Assignment #76 1. conv: geo 3. diverges: lim≠0 4. diverges: integral 5. diverges: p-ser 6. conv: p-ser 7. conv: geo 9. diverges: integral 11. conv: geo 13. diverges: integral 17. diverges: lim=∞ 18. diverges: lim≠0 25. conv: integral Assignment #77 1. 1 3. –1 5. diverges 7. 0 9. 1 11. e–5 13. 3 23. conv (geo) 25. diverges (p-ser) 26. diverges (mult of harmonic or p-ser) 30. conv (integral) Assignment #79.5 2. div; comp w/ ∑1/n 3. conv; comp w/ ∑2/2n Assignment #80 3. conv; comp w/ ∑2/2n 4. conv; comp w/ ∑1/n2 5. div; lim≠0 7. conv; comp w/ ∑(1/3)n 23. conv; comp w/ ∑1/3(n–1) 24. div; lim ≠ 0 Assignment #81 6. conv; lim comp w/ ∑1/n3/2 8. conv; lim comp w/ ∑1/n3/2 10. div; lim comp w/ ∑1/n 11. conv; lim comp w/ ∑1/n2 13. div; lim comp w/ 1/n 27. conv; lim comp w/ 1/n2 28. conv; lim comp w/ ∑1/n2

CALCULUS 2 FIVES SHEET F=Fun at home I=Incunabula V=Variety E=Endeavors S=Specimens

Assignment #82 1. div; comp w/ ∑1/n1/2 9. div; lim or direct comp w/ ∑1/n since ∑1/n < ∑1/ ln n < ∑1/(ln(ln n)) 12. conv; lim comp w/ ∑1/n2 15. div; lim comp w/ ∑1/n Assignment #83 1. C 2. D 3. D 4. D 5. C 6. D 7. C 8. C 9. C 10. C Assignment #84 1. conv: alt ser, terms dec, lim=0 2. conv: alt ser, terms dec, lim=0 3. div: nth term, lim≠0 4. div: nth term: lim=∞ ≠ 0 5. conv: alt ser, terms dec, lim=0 6. conv: alt ser, terms dec, lim=0 Note f ' (x) < 0 for x ≥ e 7. div: nth term, lim≠0

9. conv: alt ser, terms dec, lim=0 45. |error| < |a5| =

(–1)6

1

5 = 15 = 0.2

Assignment #85 p.661: 13. Cond Conv 15. Abs Conv 16. Diverge 17. Cond Conv 21. Cond Conv

46. |error| <

(–1)6

1

105 = 0.00001 p.654: 1. conv(ratio test) 3. div(ratio) 8. conv(geometric)

9. div(div test - lim = e–1/3) Assignment #86 25. diverges(p-series) 26. diverges(p-series) 27. converges conditionally(alt/p) 28. converges absolutely(p/direct comp) 30. converges absolutely(integral) 34. converges conditionally 35. converges absolutely 37. converges absolutely 38. converges absolutely 40. converges absolutely Assignment #90 interval of conv: –1 < x < 1, radius of conv = 1

CALCULUS 2 FIVES SHEET F=Fun at home I=Incunabula V=Variety E=Endeavors S=Specimens

I PROBLEMS Practice for the COMMON CORE Math Test

1

2

3

4

The value of an antique has increased exponentially, as shown in this graph.

Based on the graph, estimate to the nearest $50 the average rate of change in value of the antique for the following time intervals:

$from 0 to 20 years

$from 20 to 40 years

CALCULUS 2 FIVES SHEET F=Fun at home I=Incunabula V=Variety E=Endeavors S=Specimens

5

6

David compares the sizes and costs of photo books offered at an online store. The table below shows the cost for each size photo book.

Book Size Base Price Cost for Each Additional Page

7-in. by 9-in. $20 $1.00 8-in. by 11-in. $25 $1.00 12-in. by 12-in. $45 $1.50

The base price reflects the cost for the first 20 pages of the photo book.

1. Write an equation to represent the relationship between the cost, y, in dollars, and the number of pages, x, for each book size. Be sure to place each equation next to the appropriate book size. Assume that x is at least 20 pages.

7-in. by 9-in.

8-in. by 11-in.

12-in. by 12-in.

2. What is the cost of a 12-in. by 12-in. book with 28 pages?

3. How many pages are in an 8-in. by 11-in. book that costs $49?

CALCULUS 2 FIVES SHEET F=Fun at home I=Incunabula V=Variety E=Endeavors S=Specimens

7

ANSWERS TO THE COMMON CORE PROBLEMS

1

2

3

4

5

6

7

Hannah makes 6 cups of cake batter. She pours and levels all the batter into a rectangular cake pan with a length of 11 inches,a width of 7 inches, and a depth of 2 inches.

One cubic inch is approximately equal to 0.069 cup.

What is the depth of the batter in the pan when it is completely 1poured in? Round your answer to the nearest of an inch. 8

1 point for the correct estimated average rate from years 0 to 20: $100

1 point for the correct estimated average rate from years 20 to 40: $150

1. 7-in. by 9-in. y = x

8-in. by 11-in. y = x + 5

12-in. by 12-in. y = 1.50x + 15

2. $57

3. 44 pages

11 point: For correct answer 1 or 1.125 inches. 8