I. Polar Equations What you are finding: r · I. Polar Equations What you are finding: Polar equations are in the form r=f(!). This generates a set of points that are at a ... Demystifying

© 2011 www.mastermathmentor.com - 24 - Demystifying the BC Calculus MC Exam

I. Polar Equations

What you are finding: Polar equations are in the form

r = f !( ). This generates a set of points that are at a radius r from the pole (origin) based on a function of

! , the central angle at the pole. How to find it: We normally need to find slopes of these curves, which means that we need to represent the polar curve parametrically. The formulas to do so are:

x = rcos! and y = rsin! .

To find

dy

dx, you use the formula:

dy

dx=dy d!

dx d!. Horizontal tangents occur when

dy

d!= 0 and vertical

tangents occur when

dx

d!= 0. If

dy

d! and

dx

d! are both zero simultaneously, no conclusion can be made.

Typical AP problems involve finding the area bounded by a polar curve between two angles

! and ".

A =1

2f !( )[ ]

2

"

#

$ d! . This assumes the function f is continuous and non-negative.

Arc length in polar form is given by the formula:

s = f !( )[ ]2

+ " f !( )[ ]2

#

$

% d! = r2

+dr

d!&

'

(

)

2

#

$

% d! .

43. What is the equation of the line tangent to the polar curve

r = 4! at

! =3"

2?

A.

y =3!x

2" 6! B.

y =2x

3!" 6! C.

y = !6"

D.

y =!2x

3"! 6" E.

y =!3"x

2! 6"

_______________________________________________________________________________________

44. (Calc) The graph shows the polar curve

r = 2 + cos 4!( ) for

0 !" ! 2# . What is the total area enclosed by the graph?

A. 6.283 B. 12.566 C. 14.137 D. 27.681 E. 28.274


45. Which of the following is equal to the area of the region inside the polar curve

r = 3sin! and outside the polar curve

r = 2sin! ?

A.

5 sin2!

0

" 2

# d! B.

5

2sin

2!0

2"

# d! C.

10 sin2!

0

" 2

# d!

D.

5 sin2!

0

"

# d! E.

5

2sin

2!0

"

# d!

_______________________________________________________________________________________

46. A particle moving along the polar curve

r =1! sin" has position

x t( ),y t( )( ) at time with

! = 0 when t = 0. The particle moves along the curve such that

dr

dt=dr

d!. Describe the motion of the

particle at

t =!

6.

I. Getting closer to the x-axis II. Getting closer to the y-axis III. Getting closer to the origin A. I only B. II only C. I and II only D. II and III only E. I, II and III

_______________________________________________________________________________________

47. (Calc) The polar graph in the figure to the right is

r = ! + 2sin! for

0 !" !5#

3.

Find the greatest distance of the curve from the origin. A. 2.094 B. 2.457 C. 3.504 D. 3.826 E. 5.236


48. The graph of the polar curve

r = 4cos3! is shown at the right. Which of the following does not represent the area of the shaded region?

I.

48 cos2

3

0

! 6

" # d# II.

24 cos2

3

!" 6

" 6

# $ d$ III.

8 cos2

3

0

2!

" # d#

A. I only B. II only C. III only D. I and II only E. I, II and III

_______________________________________________________________________________________

49. (Calc) A path around a park is in the shape of the polar graph of

r = ! as shown in the figure to the right. If a walker starts at the origin and walks around the path and then back to origin along the x-axis, how far does he walk?

A. 6.110 B. 6.197 C. 9.252 D. 13.477 E. 16.619

_______________________________________________________________________________________

50. (Calc) The graph of the polar curve

r = 2 ! 4sin" is a limaçon with two loops as shown in the figure to the right. Find the area between the two loops.

A. 25.688 B. 35.187 C. 35.525 D. 37.361 E. 37.699


J. Taylor Polynomial Approximations

What you are finding: Taylor polynomial approximations allow students to write a polynomial that approximates a function. They are important because transcendental functions like trig functions or log functions can be expressed with nothing but addition and multiplication. How to find it: The nth degree Taylor polynomial approximation for a function

f x( ) at x = c is given by:

Pn x( ) = f c( ) + ! f c( ) x " c( ) +! ! f c( ) x " c( )

2

2!+

! ! ! f c( ) x " c( )3

3!+ ...+

fn( )

c( ) x " c( )n

n!

If c = 0, then the Taylor polynomial is a Maclaurin polynomial and is given by:

Pn x( ) = f 0( ) + ! f 0( )x +! ! f 0( )

2!x2

+! ! ! f 0( )

3!x3

+ ...+f

n( )0( )

n!x

n

Typical problems involve giving the Taylor polynomial approximation and asking students to find the values of specific derivatives at some value of c. Sometimes, you will be given a formula for the nth derivative of a function in a Taylor series and you will be asked to write the Taylor polynomial. Taylor polynomial approximation problems go hand-in-hand with the second derivative test. If there is no first-degree x-term in the Taylor polynomial, then the value of c about which the function is centered is a critical value. Thus the coefficient of the

x2 is the second derivative divided by 2! Using the second

derivative test, we can tell whether there is a relative maximum, minimum, or neither at x = c.

51. Let f be a function having derivatives for all orders of real numbers. The first three derivatives of

f at x = 0 are given in the table below. Use the third-degree Taylor polynomial at x = 0 to approximate

f1

2

!

" #

$

% & .

x f x( ) ! f x( ) ! ! f x( ) ! ! ! f x( )

0 5 2 "8 24

A.

1

2 B.

11

2 C. 7 D.

17

3 E.

23

2

_______________________________________________________________________________________

52. Let

P x( ) =1+ 2 x !1( ) + 3 x !1( )2

! 4 x !1( )3

+ 6 x !1( )4 be the fourth-degree Taylor polynomial for the

function f about x = 1. What is the value of

! ! ! f 1( )?

A. -24 B. -12 C. -4 D.

!2

3 E. 1


53. The Maclaurin series for a certain function f converges to

f x( ) for all x in the interval of convergence. The nth derivative of f at x = -1 is given by

fn( )!1( ) =

!1( )n+1

n + 1( )!

1! 2n( )2

for n " 2

If the graph of f has a horizontal tangent at (-1, -4), describe the behavior of the graph of f at x = -1.

A. relative maximum B. relative minimum C. cusp point D. inflection point E. none of these

_______________________________________________________________________________________

54. Consider the differential equation

dy

dx= xy ! y . Let

f x( ) be a particular solution to this

differential equation with initial condition

f 2( ) = !1. Find the 2nd degree Taylor polynomial for y about x = 2.

A.

!1! x ! x2 B.

!7 + 7x ! 2x2 C.

!3+ 3x ! x2

D.

!1+ x !x2

2 E.

!1! x !x2

2

_______________________________________________________________________________________ 55. In the 50th degree Taylor polynomial for

f x( ) = 2ln x centered at x = 1, write the coefficient for

x !1( )50 .

A.

1

25! B.

!1

25! C.

2 49!( ) D.

1

25 E.

!1

25


K. Error Bounds

What you are finding: When the Taylor polynomial alternates in sign (+ - + - …) and converges to some limit, the error is easily found: it is simply the absolute value of the next term in the series as the terms get smaller and smaller. But when the Taylor polynomial doesn’t alternate in sign, we need to find the Lagrange error. No BC topic fills students (and teachers) with as much dread as the Lagrange Error. Typically there is only one free response question on the BC exam concerning Lagrange and students usually ignore it. When the nth degree Taylor polynomial is found for an approximation to a function

f x( ) at x = c, there is an error – the difference between

f c( ) and the result of the Taylor polynomial approximation. The Lagrange Error Bound gives the maximum error that using the Taylor polynomial approximation will yield. I liken it in my classes to asking instructions to some residential address. There are some detours right around the address. Rather than giving full directions (because you don’t know exactly where the detours are), you give directions to a road that is one mile from your house. You tell the people at that time to get out and ask directions. You know that at worse, they will only be one mile from your house (the error). How to find it: The nth degree Taylor polynomial approximation for a function

f x( ) at x = c is given by

Pn x( ) = f c( ) + ! f c( ) x " c( ) +! ! f c( ) x " c( )

2

2!+

! ! ! f c( ) x " c( )3

3!+ ...+

fn( )

c( ) x " c( )n

n! Remember that this is an approximation for

f x( ) . Stated a different way:

f x( ) = f c( ) + ! f c( ) x " c( ) +! ! f c( ) x " c( )

2

2!+

! ! ! f c( ) x " c( )3

3!+ ...+

fn( )

c( ) x " c( )n

n!+ Rn x( ) where

Rnx( ) is

the error that is created when using the Taylor polynomial. This

Rnx( ) is called the Lagrange form of the

remainder and is found by the formula:

Rn x( ) =f

n+1( ) z( )

n +1( )!x ! c

n+1. The z is some value between x and c.

You will not find z. For any nth degree Taylor Polynomial, you will need to find the maximum value of the (n +1)st derivative to calculate the Lagrange Error. There are generally two types of problems in the AP exam that ask for the Lagrange error. • the maximum value of the (n +1)st derivative is easily determined (usually trig functions). • the maximum value of the (n +1)st derivative is given to you.

56. The function f has derivatives of all orders for all real numbers and

f5( ) x( ) = e

cosx . If the fourth—degree Taylor polynomial for f about x = 0 is used to approximate f on the interval

0,1[ ], what is the Lagrange error bound for the maximum error on the interval

0,1[ ]?

A.

1

120 B.

e

120 C.

e

24 D.

1

24 E.

1

720


57. The second-degree Taylor polynomial for f about x = 1 is given by

P2x( ) = 8 ! 2 x !1( ) +12 x !1( )

2. The third derivative of f satisfies the inequality

f3( ) x( ) ! 24 for all x in the interval 0,1[ ]. Find the

Lagrange error bound for the approximation to

f 1.5( ) .

A.

1

8 B.

1

4 C.

1

2 D.

3

4 E. 1

_______________________________________________________________________________________

58. (Calc) Let f be a function having derivatives for all orders of real numbers. The function and its first three derivatives of f at x = 0 are given in the table below. The fourth-derivative of f satisfies the inequality

f4( ) x( ) ! 60 for all x in the interval 0,1[ ]. Find the maximum value of

f 0.5( ) .

x f x( ) ! f x( ) ! ! f x( ) ! ! ! f x( )

0 3 10 "12 15

A. 0.234 B. 6.708 C. 6.969 D. 14.031 E. 17.625

_______________________________________________________________________________________

59. Find the Lagrange error in calculating

f !0.1( ) for the third degree Taylor polynomial for

f x( ) = xex

about x = 0.

A.

1

24 B.

!0.1( )3

6 C.

!0.1( )3

24 D.

!0.1( )4

6 E.

!0.1( )4

24


L. Series Convergence Intervals

What you are finding: In the multiple-choice section, asking whether a series is convergent makes little sense. Either it is or it isn’t, making five possible answers meaningless. Typically, students will be given a series and asked to find the interval of convergence. That means a range of values of x such that when the infinite series terms are added, the sum converges to some limit rather than diverging to infinity or negative infinity. Overwhelmingly, the test for convergence and thus finding the interval of convergence uses the ratio test but students must also know and recognize the geometric series, p-series and alternating series. How to find it: First, students must know that any series whose terms do not approach zero as n approaches infinity cannot converge. This, the nth term test, can only prove divergence. The most common rules for convergence are the following. Again, the test that is used the most is the ratio test. The only series form that students need to be able to actually sum is geometric.

Type Geometric p ! series Alternating Ratio

Series form arn

n= 0

"

# =a

1! r

1

np

n=1

"

# !1( )nan

n=1

"

# ann= 0

"

#

Convergence r < 1 p > 10 < an+1 < an

and limn$"

an = 0limn$"

an+1

an< 1

Students should also know that the harmonic series

1

nn=1

!

" diverges while the alternating harmonic series

!1( )n 1

nn=1

"

# and p-series

1

n2

n=1

!

" converge. Finally, when students find the interval of convergence, they need

to check out the endpoints to determine convergence or divergence at these values as well.

60. Which of the following series converge?

I.

2

n +13

!

" #

$

% &

2

n=1

'

( II.

e

!"

#

$

%

n

n=0

&

' III.

2n

2n

+1

!

" #

$

% &

n=1

'

(

A. I only B. II only C. III only D. II and III only E. None


61. Which of the following series converge?

I.

n2

2n

n=0

!

" II.

e

100nn=1

!

" III.

sinn!

2

n3 2

n=1

"

#

A. I only B. I and II only C. I and III only D. II and III only E. I, II and III

_______________________________________________________________________________________

62. For what positive integers k > 0 will both

!1( )kn

nn=1

"

# and k

9

$

%

&

' n= 0

"

#2n+1

converge?

A. all odd B. all even C. k < 9 D. odd integers < 9 E. no integers

_______________________________________________________________________________________

63. For what values of p does the infinite series

n

n2p!1

+ 2n=1

"

# converge?

A.

p > 0 B.

p >1 C.

p !1 D.

p >1

2 E.

p >3

2


64. The ratio test is applied to the series

en 2

n !1( )!n=1

"

# to show convergence. Which of the following

inequalities results?

A.

limn!"

e

n<1 B.

limn!"

en 2

n #1( )!<1 C.

limn!"

e

n<1

D.

limn!"

e

n #1<1 E.

limn!"

e

n!<1

_______________________________________________________________________________________

65. The function f is defined by the power series

f x( ) =1! 2x +1( ) + 2x +1( )2

! 2x +1( )3

+ ... for all real numbers for which the series converges. What is the interval of convergence for f ?

A. (0, 1) B. (0, 1] C. [0, 1] D. (-1, 0) E. (-1, 0]

_______________________________________________________________________________________

66. Find the interval for values of x in which the series

5x( )n

n +1n= 0

!

" converges.

A.

!15,1

5

"

# $

%

& ' B.

!1

5,1

5

"

# $

%

& ' C.

!5,5( ) D.

!5,5[ ) E. All values of x



f x( ) =1+ x ! 7( ) + x ! 7( )2

+ ...+ x ! 7( )n

+ ... for all real numbers x for which the series converges. What is the range of

f x( ) within the interval of convergence?

A.

1

2,!

"

# $

%

& ' B.

1

2,!

"

# $

%

& ' C.

0,![ ) D.

0,!( ) E.

!","( )

_______________________________________________________________________________________

68. Let

f be the function given by

f t( ) = 4 ! 4t2

+ 4t4! 4t

6+ ...= !1( )

n4t

2n

n= 0

"

# and

F be the function

given by

F x( ) = f t( )

0

x

! dt . Find the interval of convergence of the power series for

F x( ) about

t = 0.

A.

!1,1( ) B.

!1,1( ] C.

!1,1[ ] D.

!4,4[ ] E.

!4,4( ]


M. Taylor Series

What you are finding: There is little distinction between Taylor series problems and Taylor polynomial problems other than the fact that the series is an infinite sum while the Taylor polynomial has a degree and stops at some value of n. Thus the Taylor series is an exact sum, which converges to

f x( ) while the Taylor polynomial will have an error associated with it. How to find it: Students should know the Taylor series for the following functions, the first three of which have an interval of convergence as

!","( ) . These crop up all the time.

sin x = x !x

3

3!+x

5

5!!x

7

7!+ ...+

!1( )n

x2n+1

2n + 1( )!+ ... e

x= 1+ x +

x2

2+x

3

3!+x

4

4!+ ...+

xn

n!+ ...

cos x = 1!x

2

2!+x

4

4!!x

6

6!+ ...+

!1( )n

x2n

2n( )!+ ...

1

1! x= 1+ x + x

2+ x

3+ ...+ x

n+ ... Conv : !1,1( )

Typical problems involve finding the radius or interval of convergence of Taylor series using the general term. Mostly, the ratio test is used.

69. What is the coefficient of

x3 in the Taylor series for

f x( ) = 4 2x +1 about x = 0?

A. 12 B. 6 C. 2 D.

2

3 E.

3

2

_______________________________________________________________________________________

70. The Maclaurin series for

f x( ) is given by

1

3!+x

4!+x2

5!+x3

6!+ ...+

xn

n + 3( )!+ .... Find

f12( )0( ).

A.

1

3! B.

1

4! C.

11!

14! D.

11!

15! E.

12!

15!


71. The Maclaurin series for a certain function f converges to

f x( ) for all x in the interval of convergence.

The nth derivative of f at x = 0 is given by

fn( )

0( ) =n + 1( )!

0.5( )nn

3 for n ! 0. Find the radius of

convergence for f, if it exists, about x = 0.

A.

1

2 B. 1 C. 2 D. All reals E. does not converge

_______________________________________________________________________________________

72. Find the 6th degree term of the Taylor series for

sin x2( ) ! cos x 3( ) about x = 0.

A.

2x6

3 B.

x6

3 C.

x6

6 D.

!2x

6

3 E.

!x6

_______________________________________________________________________________________

73. (Calc) If

f x( ) = 3 + 5x +x

2

2!!x

3

3!+x

4

4!!x

5

5!+ ...

!1( )nxn

n!+ ... if n " 2 , find

f x( ) dx

0

4.5

!

A. 50.489 B. 53.004 C. 67.235 D. 70.739 E. 72.218


N. Functions Defined by Power Series

What you are finding: Power series are similar to polynomials, but since they are series, they have an infinite number of terms. A power series is in the form:

a0

+ a1x + a

2x2

+ a3x3

+ ...+ anxn

+ ... , centered at zero. A power series in the form

a0

+ a1x ! c( ) + a

2x ! c( )

2

+ a3x ! c( )

3

+ ...+ anx ! c( )

n

+ ... is centered at c. How to find it: When you are given a formula for the nth term of a series, write out the first 4 or 5 terms to see if it is in the form of a power series. Taylor series for a function

f x( ) centered at c are special forms of

power series where the coefficient of each term has the special relation:

an =f

n( ) c( )

n!. All Taylor series are

power series but not all power series are Taylor series.

74. Let

f x( ) be the power series for

sin x , centered at x = 0. Which of the following is a power series?

I.

f x2( ) . II.

f x( ) III.

f ex( )

A. I only B. II only C. III only D. I and II only E. I, II and III

_______________________________________________________________________________________


f x( ) =!1( )

n+1x +1( )

n+2

n + 2( )!n=0

"

# for all real numbers x.

Describe the behavior of the curve. I. Relative Minimum at x = -1. II. Relative Maximum at x = -1 III. Inflection Point at x = -1.

A. I only B. II only C. III only D. I and III only E. None of these


O. Manipulation of Series or Taylor Polynomials

What you are finding: When you are given or asked to find a power series (or Taylor series) for some function

f x( ) , you are finding a polynomial with infinite terms using the variable x. That allows you to find the value of the function at any value or variable by replacing that x with that number or variable. When you are asked to find the derivative or integral of

f x( ) , you can take the derivative or integral of the power series allowing you to possibly integrate an expression that might not otherwise be integratable. How to find it: For instance, if you were asked to write a Taylor series for

f x( ) = ex3

centered at 0, rather than go through the tedious process of taking derivatives, you use the fact that

f x( ) = ex

=1+ x +x2

2+x3

3!+x4

4!+x5

5!+ ... so

f x3( ) = e

x3

=1+ x3

+x3( )2

2+

x3( )3

3!+

x3( )4

4!+

x3( )5

5!+ ....

You could find

ex3

dx! by integrating each term:

ex3

dx! = x +x4

4+

x7

7 " 2!+

x10

10 " 3!+

x13

13 " 4!+ ....

76. The hyperbolic sine function is defined as

sinh x =1

2ex! e

!x( ). Give the general term for the Maclaurin

series for

sinh x .

A.

x2n+1

2n +1( )! B.

!1( )n

x2n+1

2n +1( )! C.

x2n

2n( )! D.

!1( )n+1x2n

2n( )! E.

!1( )n

x2n

2n( )!

_______________________________________________________________________________________

77. A power series is used to approximate

e!x 3dx

0

1

" with a maximum error of 0.01. What is the minimum

number of terms needed to obtain this approximation? A. 2 B. 3 C. 4 D. 5 E. 6


78. If

f x( ) =sin2x

2

2x, which of the following is the Taylor series for f about x = 0?

A.

1!x2

2 " 3!+

x4

2 " 5!!2x

6

2 " 7!+ ... B.

x !x3

3!+x5

5!!x7

7!+ ...

C.

x !4x

6

3!+16x

10

5!!64x

14

7!+ ... D.

x !4x

5

3!+16x

9

5!!64x

13

7!+ ...

E.

2x2!8x

6

3!+32x

10

5!!128x

14

7!+ ...

_______________________________________________________________________________________

79. (Calc) Use the first three terms of a power series to approximate the area of Region R shown in the figure to the right. The function shown is

y =cos x

x.

A. 0.252 B. 0.254 C. 0.256 D. 0.342 E. 0.347

_______________________________________________________________________________________

80. The Maclaurin series for

1

1+ x is

!x( )n

n= 0

"

# . Which of the following is a power series expansion

for

x

1+ x2?

A.

1! x2

+ x4! x

6+ x

8+ ... B.

x2! x

4+ x

6! x

8+ ...

C.

x ! x3

+ x5! x

7+ x

9+ ... D.

1! x + x3! x

5+ x

7+ ...

E.

x ! x2

+ x3! x

4+ ...

I. Polar Equations What you are finding: r · I. Polar Equations What you are finding: Polar equations are in the form r=f(!). This generates a set of points that are at a ... Demystifying

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