CALCULUS 2 FIVES SHEET F=Fun at home I=Incunabula … · · 2016-02-27CALCULUS 2 FIVES SHEET...

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

Friday, February 26 NO SCHOOL! Staff Development Day!! Monday, 2/29 F #111 p.545 (5,8) , p.551 (127 - show both the slope field and approx. using Euler's method) AND THE FOLLOWING:

1. Find the area bounded by the spiral r = ln on the interval π ≤ ≤ 2π. A) 2.405 B) 2.931 •C) 3.743 D) 4.810 E) 7.487

2. Find the slope of the curve defined parametrically by x = et, y = t – t3

9 at its smallest x-intercept.

•A) –40.171 B) –2 C) 0.050 D) 0.999 E) 1

3. x – 6

x2 – 3x dx = A) ln | x2 (x – 3) | + C B) – ln |x2 (x – 3)| + C

•C) ln

x2

x – 3 + C D) ln

x – 3

x2 + C E) none of these

p.545: 5. y1 = 2, y2 = 2.0202, y3 = 2.0618; exact = 2.0942 8. y1 = 2.4, y2 = 2.8, y3 = 3.2, y4 = 3.6, y5 = 4; exact = 4 p.551: 127. y5 = –3.4192; exact = –4.4817 = – e3/2

I xlim 3x 2 4

2 7x x 2 is A) 3 B) 1 •C) –3 D) E) 0

V A cell phone plan costs $20 each month, plus 5 cents per text message sent, plus 10 cents for each minute used over 30 hours. In January Michelle sent 100 text messages and talked for 30.5 hours. How much did she have to pay?

E Learning to solve differential equations numerically using Euler's method. (See Euler’s method notes) S Consider the differential equation y ' = (sin x)(sin y). (a) Calculate approximate values of y using Euler's

method with three steps and dx = 0.1 starting at (0,2). Use EULERT 2.027 Tuesday, 3/1 – BRING THE NOTES FOR LOGISTIC GROWTH TO CLASS, Late Start Day! F #112 Worksheet #33 I COMMON CORE Math – at the back of the FIVES sheet (1)

V The line y = 3x + k is tangent to the curve y = x3 when k is equal to ? E Solving logistic differential equations and using them to solve problems. (See logistic growth notes)

S 1. Solve the following differential equation: dydx = 3y(2 – y) and y(0) =

12 .

2. Riverdale High has 1000 students. On day 0, 20 students start a rumor, which spreads logistically (is jointly proportional to the students who know the rumor and to those who do not know it, that is: dPdt = kP(1000 – P) ). A day later, 50 students know it. What happens over the next 10 days? When is the

rumor spreading the fastest?

Note: if dydt = ky(A – y), then y =

A1 + Ce–Akt


Wednesday, 3/2 F #113 Worksheet #34 I COMMON CORE Math – at the back of the FIVES sheet (2) V Last summer 30% of the birds living on Town Lake were geese, 25% were swans, 10% were herons, and

35% were ducks. What percent of the birds that were not swans were geese? E 1. Integrating using partial fractions and parts. 2. Solving simple differential equations using variables

separable. 3. Solving problems involving exponential growth and decay. S The population of a city increases continuously at a rate proportional, at any time, to the population at that

time. The population doubles in 50 years. After 75 years the ratio of the population P to the initial

population Po is (A) 94 (B)

52 (C)

41 •(D)

2 21 (E) none of these

Thursday, 3/3 F #114 Worksheet #35 I COMMON CORE Math – at the back of the FIVES sheet (3) V The area, in square inches, of the surface of a zone cut from a sphere of radius 4 in. by two parallel planes,

one through the center of the sphere and the other 1 in. away, is equal to ? E 1. Practice using your calculator with integrals.

S Let S = e x2

dx0

1

Of the following, which best approximates S? A)0.333 B)0.632 •C)0.743 D)1.457

E)1.676 Friday, 3/4 – TEST TODAY!! F #115 Worksheet #36 I COMMON CORE Math – at the back of the FIVES sheet (4) V In the eight-term sequence A, B, C, D, E, F, G, H, the value of C is 5 and the sum of any three consecutive

terms is 30. What is A + H ? E Everyone repeat after me: Calculus BC –– Just Do It! S On a certain day, the fly population is 1000 and growing at the (net) instantaneous rate of 50 flies per day.

The environment can support 10,000 flies. (P ' = kP(A – P) where P is number of flies, A is carrying capacity of the environment.) (a) How many flies are there in 100 days? (b) When is the population of flies 5000? (c) When does the population grow the fastest? What is the population then?

(a) about 9700 (b) about 40 days (c) about 40 days - 5000 flies Monday, 3/7 F #117 Worksheet #38 I COMMON CORE Math – at the back of the FIVES sheet (5) V A pair of standard 6-sided fair dice is rolled once. The sum of the numbers rolled determines the diameter

of a circle. What is the probability that the numerical value of the area of the circle is less than the numerical value of the circle’s circumference?

E Everyone repeat after me: Calculus BC –– Just Do It! S If the half-life of a radioactive substance is 8 years, how long will it take, in years, for two thirds of the

substance to decay? A) 4.68 B) 7.69 C) 12 D) 12.21 •E) 12.68 t=0, N=No ; t=8, N=(1/2)No ; t=?, N=(1/3)No


Tuesday, 3/8 F #119 Worksheet #40 I COMMON CORE Math – at the back of the FIVES sheet (6) V The area in the first quadrant bounded by the curve with parametric equations x = 2a tan , y = 2a cos2 , and the lines x = 0 and x = 2a is equal to ? E Practice some applications of the definite integral.

S 1.

1

∞dxx2 2.

1

∞1x dx 3.

2

3dx

(3 – x)2 4.

0

2dx

(x – 1)2/3

Wednesday, 3/9 F #121 Worksheet #42 I COMMON CORE Math – at the back of the FIVES sheet (7) V Let f (x) = x5 + 3x – 2, and let f–1 denote the inverse of f. Then (f–1) ' (2) equals ? E FIVE – Are you up to the challenge? S Lots of problems for you to practice!!! Thursday, 3/10 – TEST TODAY!!!!! F #124 Worksheet #45 I COMMON CORE Math – at the back of the FIVES sheet (8) V At an elementary school, the students in the third grade, fourth grade, and fifth grade run an average of 12,

15, and 10 minutes per day, respectively. There are twice as many third graders as fourth graders, and twice as many fourth graders as fifth graders. What is the average number of minutes run per day by these students?

E 1. Review arc length and improper integrals. S 1. Find the length of the arc of y = x3/2 from x = 1 to x = 8.

2. Find the area enclosed by the cardioid r = 2(1 + cos ) [See p. 165 example 19]. 3.

1

∞dxx = 4.

1

∞dx

x =

Friday, 3/11 – ANOTHER TEST TODAY!!!!! F #125 Worksheet #46 I COMMON CORE Math – at the back of the FIVES sheet (9) V A small bottle of shampoo can hold 35 milliliters of shampoo, whereas a large bottle can hold 500

milliliters of shampoo. Jasmine wants to buy the minimum number of small bottles necessary to completely fill a large bottle. How many bottles must she buy?

E 5 –– Just do it! S Let f be the function defined for x > 0, with f(e) = 2 and f’, the first derivative of f, given by f '(x) x2 ln x. (a) Write an equation for the line tangent to the graph of f at the point (e,2). (b) Is the graph of f concave up or concave down on the interval 1 < x < 3? Give a reason for your answer. (c) Use antidifferentiation to find f(x).


Monday, 3/14

F #128 2011 Free Response (1) AND

1. Evaluate

x cos 2x dx. A)

12 x cos 2x –

14 sin 2x + C B)

12 x cos 2x –

14 cos 2x + C

C) 12 x sin 2x –

14 sin 2x + C D)

12 x cos 2x +

14 sin 2x+C •E)

12 x sin 2x +

14 cos 2x+C

2. The line tangent to the graph of y = x3 – 3x2 – 2x + 1 at x = – 1 will also intersect the curve at which of the following values of x?

A) x = 4 •B) x = 5 C) x = 6 D) x = 7 E) x = 8

3. Let f(x) = (cos x)3n

n1

. Evaluate f

2π

3 .

A) – 17 •B) –

19 C)

17 D)

89 E) The series diverges.

4. Which if the following is equal to 1

3 (2x2 – 5)3 x dx ?

(A) 14

1

3u3 du • (B)

14–3

13u3 du (C)

–3

13u3 du (D) 4

1

3u3 du (E) 4

–3

13u3 du

5. The series x + x3 + x5

2! + x7

3! + ... + x2n + 1

n! + ... is the Maclaurin series for

(A) x ln (1 + x2) (B) x ln (1 – x2) (C) ex2 • (D) xex2 (E) x2ex

6. Let f(t) = sin t – 2 cos t2, where 0 ≤ t ≤ 4. For what value of t is f(t) increasing most rapidly? (A) 1.76 (B) 2.81 (C) 3.32 (D) 3.56 •(E) 3.77

ans: 1. E 2. B 3. B 4. B 5. D 6. E

FR 2: a. (–.767,.588), (2,4), (4,16) b. .767

2(2x – x2)dx +

2

4(x2– 2x)dx c. π

–.767

2[(5 – x2)2 – (5 – 2x)2]dx

I COMMON CORE Math – at the back of the FIVES sheet (10) V The area bounded by the lemniscate with polar equation r2 = 2 cos 2 is equal to ? E 1. Review arc length and improper integrals. S 1. Find the length of the arc of y = x3/2 from x = 1 to x = 8. 2. Find the area enclosed by the cardioid

r = 2(1 + cos ) [See p. 165 example 19]. [6π] 3.

1

∞dxx = 1 4.

1

∞dx

x = +∞


Tuesday, 3/15 F #129 2011 Free Response (2) AND

1. A particle is moving along the x-axis according to the equation x(t) = 4t2 – sin 3t where x is given in feet

and t is given in seconds. Find the acceleration at t =

2

.

•A) – 1ft/sec2 B) 5 ft/sec2 C) 11 ft/sec2 D) 17 ft/sec2 E) 2π ft/sec2

2. Let f(x) = 0

2x

et2+5 dt. Find f ' (x). A) ex2 +5 B) e4x2 +5 C) 2ex2 +5 •D) 2e4x2 +5 E) 4ex2 +5

3. Use implicit differentiation to find dydx for the equation exy + 4y = 7.

(A) – 14 exy (B)

y x + 4e–xy (C)

7 – yexy

4 + xexy (D) – y4 exy • (E) –

yexy

xexy + 4

4. Find the average value of the function y = x cos x on the closed interval [5,7]. (A) 4.4 • (B) 5.4 (C) 6.4 (D) 7.4 (E) 10.8 5.

A 15-foot ladder is leaning against a building as shown, so that the top of the ladder is at (0, y) and the

bottom is at (x, 0). The ladder is falling because the ground is slippery; assume that

dydt = – 12 feet per second at the instant when x = 9 feet. Find

dxdt at this instant. (A) 6 feet per second

(B) 9 feet per second (C) 12 feet per second • (D) 16 feet per second (E) 20 feet per second

I (x–2)3

x2 dx= A) (x–2)4

4x2 + C B) x2

2 – 6x + 6 ln|x| – 8x + C C)

x2

2 – 3x + 6 ln|x| + 4x + C

D) – (x–2)4

4x + C •E) none of these

V The integral that represents the area inside the circle r = 3 sin

and outside the cardioid r = 1 + sin

is given by ?

E GO 4 5!!!! S GO 4 5!!!!


Wednesday, 3/16 F #130 2011 Free Response (3 and 4) AND

1. The area bounded by the curve x = 3y – y2 and the line x = – y is represented by

A) (2y y2 )dy0

4

B) (4y y2 )dy0

4

C) (3y y2 )dy0

3

ydy0

4

D) (y2 4y)dy0

3

E) (2y y2 )dy0

3

2. The region bounded by y = ex, y = 1, and x = 2 is rotated about th x-axis. The volume of the solid generated is given by the integral:

A) 0

2e2x dx B) 2

1

e2

(2 – ln y)(y – 1) dy C) 0

2(e2x – 1) dx D) 2

0

e2

y(2 – ln y) dy E) 0

2(ex – 1)2

dx 3. A particle moves on a straight line so that its velocity at time t is given by v = 4s, where s is its

distance from the origin. If s = 3 when t = 0, then, when t = 12 , s equals

A) 1 + e2 B) 2e3 C) e2 D) 2 + e2 E) 3e2 4. Bacteria in a culture increase at a rate proportional to the number present. An initial population of 200

triples in 10 hours. If this pattern of increase continues unabated, then the approximate number of bacteria after one full day is A) 1160 B) 1440 C) 2408 D) 2793 E) 8380

5. Using the substitution x = 2t – 1, the definite integral 3

5 t 2t – 1 dt may be expressed in the form

k (x 1) xa

b

dx , where {k,a,b} = A) 1

4,2, 3

B) 1

4, 3,5

C) 1

4,5,9

D) 1

2,2, 3

E) 1

2,5,9

6. The curve defined by x3 + xy – y2 = 10 has a vertical tangent line when x =

A) 0 or 1

3 B) 1.037 C) 2.074 D) 2.096 E) 2.154

I

0

1dx

4 – x2 = A)

π3 B) 2 – 3 C)

π12 D) 2( 3 – 2) • E)

π6

V Let ƒ be the function given by ƒ(x) = x4

16 – x3 + 3x2

8 – 2x. The function ƒ has a relative minimum at x =

E 1. Did you know that the length of an average person's hand is exactly inches? S 1. Let y=ƒ(x)=sin(Arctan x). Then the range of ƒ is

A) {y|0<y≤1} B) {y|–1<y<1} C) {y|–1≤y≤1} D) {y|–π2 <y<

π2 } E) {y|–

π2 ≤y≤

π2 }

2. The function ƒ(x) = 2x3 + x - 5 has exactly one zero. It is between A) –2 and –1 B) –1 and 0 C) 0 and 1 D) 1 and 2 E) 2 and 3

3. The derivative of ƒ(x) = x4

3 – x5

5 attains its maximum value at x = A) –1 B) 0 C) 1 D) 4

3E)

53


Thursday, 3/17 – TEST TODAY!!!!! F #131 2011 Free Response (5)

AND BC Practice exam 3 from 9th edition Barron’s book (1,5,9,13,17,21,25,29,33,37,41,45)

I A function f(x) equals x2 – xx – 1 for all x except x = 1. In order that the function be continuous at x = 1, the

value of f(1) must be A) 0 •B) 1 C) 2 D) ∞ E) none of these

V If the curve of the equation ky2 + xy = 2 – k passes through the point (– 2, 1), then k equals ? 2 E 5 –– GOTTA HAVE IT !!!!! S 1. You MAY use your graphing calculator on the Part B multiple choice problems and you MUST NOT

use it on the Part A problems. Friday, 3/18 F #132 Do the following free response problem 2011 (6)

AND BC Practice exam 3 from 9th edition Barron’s book (2,6,14,18,22,26,30,34,38,42)

I 1. The first four terms of the Taylor series about x = 0 of 1 + x are A) 1 – x2 +

x2

(4)(2) – 3x3

(8)(6)

B) x + x2

2 + x3

8 + x4

48 •C) 1 + x2 –

x2

8 + x3

16 D) 1 + x4 –

x2

24 + x3

32 E) –1 + x2 –

x2

8 + x3

16

2. Air is escaping from a balloon at a rate of R(t) = 60

1 + t2 ft3/min, where t is measured in minutes. How

much (in ft3) escapes during the first minute? A) 15 •B) 15π C) 30 D) 30π E) 30 ln 2 V If ƒ(x) = 5x and 51.002 5.016 , which is closest to ƒ ‘ (1)? (no calculator) E LET'S DRIVE FOR 5!!!!!

A look at the error in a power series if the series is not an alternating series. It’s called the Lagrange error bound!! How exciting!!!

S Look on the next page for this practice problem! S BC Free Response 2004 (6) – Lagrange Error Bound


Let f be the function given by f (x) sin 5x 4

, and let P(x) be the third-degree Taylor polynomial for f about x 0.

(a) Find P(x).

(b) Find the coefficient of x22 in the Taylor series for f about x 0.

(c) Use the Lagrange error bound to show that f1

10

P

1

10

1

100.

(d) Let G be the function given by G(x) f (t)dt. Write the third-degree Taylor polynomial for G about x 0.0

x

Monday, 3/21 F #133 2004 Free Response (6)

AND BC Practice exam 3 from 9th edition Barron’s book (3,7,11,15,19,23,27,31,35,43)

I 1. If y = ln ( )x2 + 1 (x > 0) , then the derivative of y2 with respect to ln x is

A) 2x

x2 + 1 B) 2

x2 + 1 C) 2x

ln x( )x2 + 1 •D)

2x2

x2 + 1 E) none of these

V The base of a solid is the region bounded by x2 = 4y and the line y = 2, and each plane section perpendicular to the y-axis is a square. The volume of the solid is ?

E 5 –– LET'S ROLL !!!!! A look at the error in a power series if the series is not an alternating series. It’s called the Lagrange error bound!! How exciting!!!

S 1. You MAY use your graphing calculator on the Part B multiple choice problems and you MUST NOT use it on the Part A problems or the I problems. Spend this weekend doing problems and think 5 . . . 5 . . . 5. If you are still not sure, see Coach Alcosser.

1. A cup of coffee at temperature 180°F is placed on a table in a room at 68°F. The differential equation

(d.e.) for its temperature at time t is dydt = –0.11(y – 68); y(0) = 180. After 10 minutes the temperature

(in °F) of the coffee is (calculator) A) 96 B) 100 C) 105 D) 110 E) 115

2. Let ƒ be the function given by ƒ(x) = x4

16 – x3 + 3x2

8 – 2x. The function ƒ has a relative minimum at x=

(A) 0.00 (B) 8.00 (C) 11.80 (D) 12.00 (E) 15.74 3. If you use Euler's method and two steps with ∆x = 0.1 for the differential equation (d.e.) y ' = y, with

initial value y(0) = 1, then, when x = 0.2, y is approximately A)1.100 B)1.120 C) 1.331 D) 1.464 E) none 4. Suppose the amount of a drug in a patient's bloodstream t hours after intravenous administration is

30(t + 1)2 mg. The average amount in the bloodstream during the first 4 hours is

A) 6.0 mg B) 11.0 mg C) 16.6 mg D) 24.0 mg E) none of these

5. Find the area in the first quadrant under the curve of y = e–x. A) 1 B) e C) 1e D) 2 E) none of these


I PROBLEMS

Practice for the COMMON CORE Math Test 1

Ans: -4.2


2

Ans: 53.13


3

Ans:

Exemplar: First response box: (17 + 2w)(13 + 2w) = 396 or an equivalent equation Second response box:

or w =

or equivalent values


4

Ans:

Exemplar: (shown at right) Other correct examples are possible. Rubric: (1 point) Student creates a correct example and determines that the claim is false.


5

Ans: 1 to 1.4


6

Ans: D


7

Ans: 64


8

Ans:

Exemplar: I would buy the ticket from Airline P. Both airlines are likely to have an on-time arrival since they both have median values at 0. However, Airline Q has a much greater range in arrival times. Airline Q could arrive anywhere from 35 minutes early to 60 minutes late. For Airline P, the flights arrived within 10 minutes on either side of the scheduled arrival time about 2/3 of the time, and for Airline Q, that number was only about 1/2. For these reasons, I think Airline P is the better choice. Rubric: (2 points) Student chooses Airline P and clearly explains that both airlines have the same center but that Airline P has a smaller spread. (1 point) Student states that either airline could be chosen because they have the same median, but does not address the issue of spread; OR The student states that both airlines have the same median and chooses Airline P, but does not justify the choice based on spread; OR The student explains that Airline P would be the better choice based on the smaller spread, but does not identify that both airlines have the same median.


9

Ans: D


10

Ans:

Exemplar: 2t + 3t = 24 or or or equivalent equation

Rubric: (1 point) Student enters a correct equation that can be used to find the time.

CALCULUS 2 FIVES SHEET F=Fun at home I=Incunabula … · · 2016-02-27CALCULUS 2 FIVES SHEET...

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Transcript of CALCULUS 2 FIVES SHEET F=Fun at home I=Incunabula … · · 2016-02-27CALCULUS 2 FIVES SHEET...