brose - november schedule

CALCULUS 2 FIVES SHEET

F=Fun at home I=Incunabula V=Variety E=Endeavors S=Specimens

Wednesday, 10/30

F #54 p.217 (10,11,17,20,26,79) Find intervals where curve is increasing, decreasing, concave up, concave

down, critical points, points of inflection, and graph the curve.

I

0

3

(x + 1)1/2 dx = A) 21

2 B) 7 C)

16

3 •D)

14

3 E) –

1

4

V

E Use the second derivative to determine concavity of a curve and to find inflection points.

S 1. y = 4 + 3x – x3 2. y = x4 + 4x3 + 5 3. y = x4 – 4x3 + 10 4. y = x5/3 – 5x2/3

5. See p. 217 (7): y = sin |x| on [–2π,2π]

Thursday, 10/31 - BOO!!!!

F #55 Worksheet #15

I If ƒ(x) = x + 1

x , then the set of values for which ƒ increases is

•A) (–∞, –1) (1, +∞) B) [–1,1] C) (–∞, +∞) D) (0, +∞) E) (–∞, 0) (0, +∞)

V

E Solving word problems that involve maximizing or minimizing a function.

S 1. Find two positive numbers whose sum is 20 and whose product is as large as possible. 2. A square piece

of cardboard has 12 inches in a side. An open box is formed by cutting out equal square pieces at the

corners and bending upward the projecting portions that remain. Find the maximum volume that can be

obtained.

3. Find all points on the curve x2 – y2 = 4 closest to (6,0).

4. Find all points on the curve y = x2 closest to (18,0).

Friday, 11/1

F #56 Worksheet #16

I The length of the curve y = 2x between (0,1) and (2,4) is

A) 3.141 •B) 3.664 C) 4.823 D) 5.000 E) 7.199

V

E 1. Determine the absolute maximum and minimum of a function.

S Find all absolute and relative maxima and minima, terrace pts, and inflection pts:

1. y = x3 – 3x + 2 on [0,2] 2. y = x3 – 12x + 1 on [–3,5] 3. y = x3 – 2x2 + x on [–1,2]

4. y = x3 + 3x2 – 24x + 20 on [0,5].



Monday, 11/4

F #59 Worksheet #17

I 1. If ƒ(x) = x2 ln(x3), then ƒ ' (x) =

A) 3x + ln(x3) B) 1

x C)

1

x2 D) 1

x + 2x ln(x3) •E) 3x + 2x ln(x3)

2. Find the derivative of cos3 2x. A) 3cos2 2x •B) –6 cos2 2x sin 2x C) 3 sin2 2x

D) –3 cos2 2x sin 2x E) 3 cos 2x sin 2x

V If the velocity of a particle moving along the x - axis is v(t) = 3t2 + 5, and if at t = 0 its position is 4, then

at any time t its position x(t) is:

A) 6t + 4 B) 3t2 + 5 C) 3t3 + 5t – 4 D) t3 + 5t + 4 E) t3 + 5t – 4

E Review for the test. Get ready for the test! Know the concepts, not just the mechanics.

S Some exciting "notes" problems.

Tuesday, 11/5

F #59.5 Do the following free response question without a calculator. 2009 BC Form B (5)

I During the worst 4-hour period of a hurricane the wind velocity in mph is given by

v(t) = 5t – t2 + 100, 0 ≤ t ≤ 4. The average wind velocity during this period is, in mph,

A) 10 B) 100 C) 102 •D) 104 2

3 E) 108

2

3

V

–2

2

4 dx

x2 =

E Get ready for the test! Know the concepts, not just the mechanics.

S 2004 BC Free Response #4



Wednesday, 11/6

F #60 Worksheet #18

I For what non–negative value of b is the line given by y = – 1

3 x + b normal to the curve

y = x3 ? A) 0 B) 1 •C) 4

3 D)

10

3 E)

10 3

3

V p.272 (80)

E Review for the test. Sketching curves given information about a function (an AP problem from the past).

Get ready for the test! Know the concepts, not just the mechanics.

S 2004 BC Form B Free Response #4

Thursday, 11/7

F #60.5 Worksheet #18.5

I (x–2)3

x2 dx = A)

(x–2)4

4x2 + C B)

x2

2 – 6x + 6 ln|x| –

8

x + C

C) x2

2 – 3x + 6 ln|x| +

4

x + C D) –

(x–2)4

4x + C •E) none of these

V

E Get ready for the test! Know the concepts, not just the mechanics.

S Something exciting!

Friday, 11/8 – End of Grading Period

F #61 Worksheet #19

I If V = 4

3 πr3, what is

dV

dr when r = 3? A) 4π B) 12π C) 24π ••D) 36π E) 42π

V

h0

lim sin(1 h) sin1

his approximately A) 0 B) 0.54 C) 0.63 D) 0.89 E) none of these

E Lots of review will help on the test.

S 2002 AB Free Response (1)

Monday, 11/11 - No School Today

Tuesday, 11/12 – LATE START TODAY!!!

F #62 Worksheet #20

I x

xdx

1

2

is A) –3 •B) 1 C) 2 D) 3 E) nonexistent

V

x

lim 1 x2x A) 0 B) 1 C) 2 D) e E) nonexistent

E Final review for tomorrow's test.

S 1. Find the minimum slope of y = x5 + x3 – 2x. –2 2. Find the minimum value of

f(x) = x2 + 2

x on

1

2 , 2 . 3 3. Find the relative maximum value of y =

ln x

x .

1

e



Wednesday, 11/13 – TEST TODAY!!!

F #63 1. Gas is escaping from a spherical balloon at the rate of 2 ft3 per min. How fast is the radius shrinking

when the radius is 12 ft? 1/(288π) 1/(288π) ft/min

2. A ladder 20 ft long leans against a house. Find the rate at which the top of the ladder is moving

downward if its foot is 12 ft from the house and moving away at the rate of 2 ft per second. 3/2 ft/sec3/2

I If y=ln(x2+y2), then the value of dy

dx at the point (1, 0) is A) 0 B)

1

2 C) 1 •D) 2 E) undefined

V

E 1. Take an easy test. 2. Start working related rates problems.

S None

Thursday, 11/14

F #63.5 You may use a calculator with this problem. This is 2009 BC (2).

I If x2y + xy2 = 2, then dy

dx at (1,1) = ? A) 0 B) 1 •C) – 1 D) 2 E) – 2

V

E Reviewing rate problems

S Further review of related rates problems.



Friday, 11/15

F #64 Read p.172-176. Do p.176-177 (11,21,22) AND THE FOLLOWING:

1. A 26 ft long ladder leans against a vertical wall. The foot of the ladder is pulled away from the wall at

the rate of 4 ft per second. How fast is the top sliding down the wall when the foot is 10 ft from the wall?

2. How fast is the radius of a spherical soap bubble changing when air is blown into it at the rate of 10 cm3

per second at the instant the radius is 1 cm? 5/(2π) cm per sec.

I If ƒ(x) = x – 1

x + 1 for all x ≠ –1, then ƒ '(1) = A) –1 B) –

1

2 C) 0 •D)

1

2 E) 1

V

E To review related rates problems.

S 1. Gas is escaping from a spherical balloon at the rate of 2 ft3 per min. How fast is the radius shrinking

when the radius is 12 ft? 1/(288π) 2. A ladder 20 ft long leans against a house. Find the rate at which the

top of the ladder is moving downward if its foot is 12 ft from the house and moving away at the rate of 2 ft

per second. 3/2

3. 2005BC form B Free Response (2)

Monday, 11/18

F #65 p.176-177 (18–answer as m/min ,23,27) AND THE FOLLOWING: Water runs into a conical tank

at the rate of 2 ft3 per minute. The tank stands point down and has a height of 10 ft and a base radius of 5

ft. How fast is the water level rising when the water is 6 ft deep?

I The area of the region between the graph of y = 8x – 8x3 and the x-axis from x = 0 to

x = 1 is A) 0 •B) 2 C) 4 D) 6 E) 8

V

E Further review of related rates problems.

S A circular conical reservoir has depth 20 ft and radius of the top 10 ft. Water is leaking out so that the

surface is falling at the rate of 1

2 ft/hour. Find the rate, in ft3/hour, at which the water is leaving the

reservoir when the water is 8 ft deep. 8π



Tuesday, 11/19

F #65.5 Do the following free response problem with a calculator. 2009 Form B BC (2)

I (x + 1 ) sin x dx = ? A) –(x – 1) cos x + sin x + C B) – (x2/2 + x) cos x + C

•C) –(x + 1) cos x + sin x + C D) (x + 1) cos x – sin x + C E) x cos x – sin x + C

V The length of the arc given by x = 4cos3 t and y = 4 sin3 t , 0 ≤ t ≤ π

2 is (no calculator)

A) π

2 B)

3π

2 C) 3π D) 3 E) 6

E Further review of related rates problems.

S Further review of related rates problems.



Wednesday, 11/20

F #102 Free Response Problem 1: The figure below represents an observer at point A watching balloon

B as it rises from point C. The balloon is rising at a constant rate of 3 meters per second and the observer is

100 meters from point C.

(a) Find the rate of change in x at the instant when y = 50.

(b) Find the rate of change in the area of right triangle BCA at the instant when

y = 50.

(c) Find the rate of change in at the instant when y = 50.

Free Response Problem 2: Two particles move in the xy-plane. For time t ≥ 0, the position of particle A is

given by x = t – 2 and y = (t – 2)2, and the position of particle B is given by x = 3t

2 – 4 and y =

3t

2 – 2.

(a) Find the velocity vector for each particle at time t = 3.

(b) Set up an integral expression that gives the distance traveled by particle A from t = 0

to t = 3.

(c) Determine the exact time at which the particles collide; that is, when the particles are

at the same point at the same time. Justify your answer.

(d) Sketch the paths of particles A and B from t = 0 until they collide. Indicate the

direction of each particle along its path.

I Let R be the region enclosed by the curve y = ln x, the x-axis, and the line x = 5. If four equal subdivisions

of the closed interval [1,5] are used, then the trapezoidal approximation of the area of R is

(A) 3.18 •(B) 3.98 (C) 4.02 (D) 4.05 (E) 4.79

V

E 1. A look at vectors and parametric equations.

S The position of a particle at any time t ≥ 0 is given by x(t) = t2 – 3 and y(t) = 2

3 t3. (a) Find the magnitude

of the velocity vector (speed) at t = 5. (b) Find the total distance traveled by the particle from t = 0 to t = 5.

(c) Find dy/dx as a function of x.

(a) 4t2 + 4t4 = 2600 (b)

0

5

4t2 + 4t4 dt = 2

3 ( )263/2 – 1 (c) x + 3



Thursday, 11/21

F #103

Free Response Problem

During the time period from t = 0 to t = 6 seconds, a particle moves along the path

given by x(t) = 3 cos (πt) and y(t) = 5 sin (πt).

(a) Find the position of the particle when t = 2.5.

(b) On the axes provided below, sketch the graph of the path of the particle from

t = 0 to t= 6. Indicate the direction of the particle along its path.

(c) How many times does the particle pass through the point found in part (a) ?

(d) Find the velocity vector for the particle at any time t.

(e) Write and evaluate an integral expression, in terms of sine and cosine, that gives the distance the particle

travels from time t = 1.25 to t = 1.75.

and also do the following:

p.755-756 (Do not graph – 6a,b,c,d,e,23,29,31,43,49,53) and

p.763 (Sketch graph with graphing calculator – 3)

I The Mean Value Theorem guarantees a special point on the graph of y = x between x = 1 and x = 2000.

The x-coordinate of this point is closest to which of the following integers?

(A) 22 (B) 30 (C) 500 (D) 516 •(E) 523

V

n

limn( )21/n – 1 =

E 1. Reviewing polar coordinates.

S BC 2007 Form B (2)

Polar Coordinates: x = rcos y = rsin r2 = x2 + y2 , tan = y

x

Convert to rectangular form: 1.

2,5π

4

2, 2 2.

2,

4

Convert to polar form: 3. r = sin

x2 (y 2)

2 4

Friday, 11/22

F #104 Worksheet #30

I The function f given by f(x) = x3 – 3x2 – 2x – 1 has a relative maximum when x is

(A) –2.291 •(B) –0.291 (C) 0.423 (D) 1.577 (E) 2.291

V limx0

ln(x 1) xcos x

x2 2 = (A) (B) 2 (C) 2/5 (D) 0 (E) – 1/5

E 1. Converting between polar and rectangular coordinates. 2. Finding area bounded by polar curves.

Area inside a polar curve =

1

2r2

1

2

d Area between polar curves:

1

2(R2 r2)

1

2

d

S 1. Find the area inside r = 1 – sin .

1

2(1 sin)

2d

0

2

2. Find the area inside the circle r = –2cos and outside the circle r = 1. 4 3

–

2 3

( ) d

2 21 3

2cos 12 3 2



Monday, 11/25


I Find the volume if the curve y = 1

x bounded at the left by x = 1, and below by y = 0, is revolved about the

x – axis. A) π

2 •B) π C) 2π D) 4π E) none of these

V p.405 (16)

E 1. Review of area involving polar curves. 2. Test soon. S Lots of "notes" problems.

Tuesday, 11/26

THIS ASSIGNMENT IS DUE THE MONDAY AFTER THANKSGIVING AND DOES NOT HAVE

AN ASSIGNMENT NUMBER!!!!

F An exciting Thanksgiving Adage!!

I None

V None

E Definitely a surprising day!!!

S None

SURPRISE TESTS

1. AP-type exam, usually multiple choice questions so guessing equalizer applies (minus 1/4 for an

incorrect answer)

2. There will be more questions than you can answer in time allotted. There may be problems that we

haven't covered yet. Work as quickly as you can doing the easiest problems first.

3. Sometimes with, sometimes without calculators

4. Will count like a normal test – if you miss it, you will take the Comprehensive Make-up Exam

5. No homework the night of a Surprise Test. That day on the FIVES sheet is moved to the next day or

eliminated. If the day is moved, then all subsequent days on the FIVES sheet are moved also.



Monday, 12/2

F #106

In questions 1 – 4, x =

3 cos π

3t , 2 sin

π

3t is the position vector, where

x = 3 cos π

3 t and y = 2 sin

π

3 t , from the origin to a moving point P(x,y) at time t.

1. A single equation in x and y for the path of the point is

(A) x2 + y2 = 13 (B) 9x2 + 4y2 = 36 (C) 2x2 + 3y2 = 13 (D) 4x2 + 9y2 = 1

(E) 4x2 + 9y2 = 36

2. When t = 3, the speed of the particle is (A) 2π

3 (B) 2 (C) 3 (D) π (E)

13

3 π

3. The magnitude of the acceleration when t = 3 is (A) 2 (B) π2

3 (C) 3 (D)

2π2

9 (E) π

4. At the point where t = 1

2 , the slope of the curve along which the particle moves is

(A) – 2 3

( )9

B – 3

2 (C)

2

3 (D) –

2 3

3 (E) none of these

5. The locus of the polar equation r = 2 sec is (what does the graph look like!!!!)

A) a circle B) a vertical line C) a horizontal line D) a parabola

E) an oblique line through the pole

6. The base of a solid is the region in the first quadrant bounded by the axes and the line

2x + 3y = 10, and each cross-section perpendicular to the x-axis is a semicircle. Find the volume of the

solid.

I 1. A particle moves along the curve given parametrically by x = tan t and y = sec t. At the instant when

t = π

6 , its speed equals

Note: |v| =

dx

dt2 +

dy

dt2

A) 2 B) 2 7 •C) 2 5

3 D)

2 13

3 E) none of these

2. The curve of y = 1 – x

x – 3 is concave up when

A) x > 3 B) 1 < x < 3 C) x > 1 D) x < 1 •E) x < 3

V For what prime p is 2003p + 16 the square of an integer?

E 1. 5-Gotta Have It! 2. Magnitude of speed and acceleration vectors. 3. Preparing for the AP test means

getting down to work. 4. Get ready for test.

S Some "notes" problems.



Tuesday, 12/3 – LATE START TODAY!

F #106.5 Worksheet #31.5

I

0

1dx

4 – x2 = A)

π

3 B) 2 – 3 C)

π

12 D) 2( 3 – 2) • E)

π

6

V

E Review for test tomorrow!!!

S None

Wednesday, 12/4 – TEST TODAY!!!


I 1.

dx

x2 – 6x + 8 =

1

2 ln

x – 4

x – 2 + C or ln

x – 4

x – 2 + C

2. If y is a differentiable function of x, then the slope of the curve of xy2 – 2y + 4y3 = 6 at the point

where y = 1 is •A) – 1

18 B) –

1

26 C)

5

18 D) –

11

18 E) 2

V

E Test today!!!

S None

ANSWERS

Assignment #54

10. max (–1,7), inc x<–1, dec x>–1, conc dn all x

11. rel min (1,1), max (–1,5), inf pt (0,3), inc x<–1,x>1, dec –1<x<1, conc up x>0, dn x<0

17. min: (–1,1) and (1,–1), max: (0,0), dec: x < –1, (0,1), inc: (–1,0), x > 1,

inf pt: (–1/ 3 ,–5/9) and (1/ 3 , –5/9), conc up: x < –1/ 3 and x > 1/ 3 , conc dn: (–1/ 3 ,1/ 3 )

20. min: (–3/2,–27/16), terrace: (0,0), inc: x > –3/2, except 0, dec: x < –3/2, inf pt: (0,0) and (–1,–1),

conc up: x<–1 and x>0, conc dn: (–1,0)

26. inc: all Reals except 0, conc up: x<0 conc dn: x>0, inf (terrace) pt: (0,0) 79. b = –3 when y"(1) = 0

Assignment #55, 56, 59, 60, and 62 Answers are on the worksheets

Assignment #63

1. 1/(288π) ft/min 2. 3/2 ft/sec

Assignment #63.5

(a) 980 people (b) t = 1.362 or 1.363 (c) 387.5 hours (d) 0.775 or 0.776 hour



Assignment #64

1. 5/3 ft per second 2. 5/(2π) cm/sec

p.176: 11. a) 14 cm2/sec, increasing b) 0 cm/sec, constant c) – 14/13 cm/sec, decreasing

21. dr/dt = 1 ft/min so dS/dt = 40π ft2/min 22. a) 2.5 ft/sec b) – 3/20 rad/sec

Assignment #65

p.176: 18. a) 8/(225π) m/min or .0113 m/min or 1.13 cm/min 18. b) – 4/(15π) m/min or – .0849 m/min

or – 8.49 cm/min 23. 11 ft/sec 27. 1 rad/sec and FR: 2/(9π) ft/min or 0.071

Assignment #102

1a. 3 5

5 1b. 150 1c. 3/125 2a. VA = (1,2) VB = (3/2, 3/2) 2b.

0

3

1 + 4(t – 2)2 dt 2c. t = 4

2d. Note: path of A: y = x2, of B: y = x + 2

Assignment #103

FR: (a) (0,5) (c) 3 (d) 3sin(t),5cos(t) (e)

1.25

1.75

(–3π sin(πt))2 + (5π cos(πt))2 dt = 5.392

p.755: 6a. (1,1) 6b. (1,0) 6c. (0,0) 6d. (–1,–1) 6e. (3 3 /2 , –3/2) 23. x = 2

29. x + y = 1 31. x2 + y2 = 1 43. x2 + (y – 4)2 = 16 49. r cos = 7 53. r2 = 4

p.763 graph with calculator – cardioid, symmetric about the y-axis

Assignment #104, 105 Answers are on the worksheets

Assignment #106

1. E 2. A 3. B 4. D 5. B 6. Area of semicircle = 1

2 π r2 =

1

2 π

y

2 2 =

π

8 y2

So, V = π

8

0

5

10

3 –

2x

32 dx =

π

72

0

5

100 – 40x + 4x2 dx = 125π

54

Assignment #106.5, 107 Answers are on the worksheet

brose - november schedule

Documents