Page 1

The Calculus Bible www.calcbible.com June 8, 2010

Introduction This document is not meant to be a textbook, but rather a review guide or a study aide for high school students who have completed the Advanced Placement AB Calculus curriculum. All other BC Calculus topics are separately outlined in The Calculus Bible, The New Testament by the same author.

Part I

Functions 1 Domain, Range, Asymptotes, and Fundamental Properties 1.0.1 Polynomials •

1.0.2

D =

R

Rational

f (x) D = g(x)

R

 {x:

g(x)

= 0}

Vertical asymptotes •

x = a, where a is an exclusion from the domain

Horizontal symptotes •

y = 0 if

y = ratio of leading coecients if

g(x)is

a higher order than

f (x)

f (x)

and

1

g(x)

are of same order


1.0.3

Trigonometric

y = sin(x) •

D =

R

R =

−1 ≤ y ≤ 1

y = sec(x)  • D = x : x 6= •

R = | y |

R =

odd integer

nπ 2

odd integer

where n is an

≥1

y = tan(x)  • D = x : x 6= •

nπ 2

where n is an

R

Identities • cos2 (x) + sin2 (x) = 1 • sin(2x) = 2sin(x)cos(x) • cos(2x) = cos2 (x)sin2 (x) •

1.0.4

2

tan

(x) + 1 = sec2 (x)

Inverse Trigonometric

y = sin−1 (x) •

D =

−1 ≤ x ≤ 1

R =

−π 2

≤y≤

π 2

y = sec−1 (x) •

D =

|x| ≥ 1

R =

0≤y≤

π or 2

π≤y≤

3π 2

y = tan−1 (x) •

D =

R

R =

−π 2

<y<

π 2

2


1.0.5

Exponential

y = bx , b > 0 •

D =

R

R =

y>0

Remember:

1.0.6

y = bx ⇔ x = logb y

Logarithmic

y = logb x •

D =

x>0

R =

R

Properties • logb (m + n) = logb m + logb n • logb ( m ) = logb mlogb n n • logb mr = rlogbm • logb 1 = 0 • logb bx = x • blogb x = x

2 Absolute Value • |f (x)|

ips all points above the x-axis

• f (|x|)

makes the function even

3 Inverses •

Found by switching the places of

Remember:

y = logb x

and

y = bx

x

and

y

and solving for

y

are inverses, which means

inverses

3

y = ln(x)

and

y = ex

are


4 Odd and Even Functions 4.1 A function is odd if f (−x) = −f (x) ∀x ∈ D •

Symmetrical with the origin

Polynomials with all odd powers (no constants, because

4 = 4x0 )

4.2 A function is even if f (−x) = f (x) ∀x ∈ D •

Symmetrical with the y-axis

Polynomials with all even powers

4.3 Remember, many functions are neither odd nor even

4


5 Graphs You Must Know y = bx , b > 1

y = bx , b < 1

5


y = logbx (includes y = ln(x))

2 2 2 2 2 x â&#x2C6;&#x161; + y = r (note: r is the radius, so 3x + 3y = 15 is a circle of radius 5

6


x2 a2

+

y2 b2

= 1, a > b

x2 a2

+

y2 b2

= 1, a < b

7


x2 a2

 yb2 = 1 2

y = x3

8


6 Zeroes of a function â&#x20AC;˘

Set the function

â&#x20AC;˘

For cubics, nd any rational root, r, and synthetically divide by



=0

and solve

(xr)

If you cannot nd a rational root, use Newton's method of approximation: n) xn  ff0(x (xn )

xn+1 =

7 Symmetry â&#x20AC;˘

y-axis:

(x, y) â&#x2020;&#x2019; (x, y)

â&#x20AC;˘

x-axis:

(x, y) â&#x2020;&#x2019; (x,y)

â&#x20AC;˘

origin:

(x, y) â&#x2020;&#x2019; (x, y)

(non-function)

â&#x20AC;˘ y = x: (x, y) â&#x2020;&#x2019; (y, x)

8 Limits 8.1 Theorems 1. Limit of a sum or dierence = sum or dierence of the limits 2. Limit of a product = product of the limits 3. Limit of a quotient = quotient of the limits (unless the limit of the denominator = 0) 4. Limit of the n

th

th root = n root of the limit

8.2 Limits you must know 1. 2. 3. 4. 5.

6.

lim a = +â&#x2C6;&#x17E;,

bâ&#x2020;&#x2019;0+ b

where

a>0

and nite

lim a = â&#x2C6;&#x17E;,

where

a>0

and nite

lim a = â&#x2C6;&#x17E;,

where

a<0

and nite

bâ&#x2020;&#x2019;0â&#x2C6;&#x2019; b bâ&#x2020;&#x2019;0+ b

lim a = +â&#x2C6;&#x17E;,

bâ&#x2020;&#x2019;0â&#x2C6;&#x2019; b

lim

a

bâ&#x2020;&#x2019;Âąâ&#x2C6;&#x17E; b

= 0,

where

where

a

a<0

and nite

is nite

1

lim(1 + x) x = lim (1 + x1 )x = e xâ&#x2020;&#x2019;0

xâ&#x2020;&#x2019;â&#x2C6;&#x17E;

9


8.3 Non-existent limits â&#x20AC;˘ Âąâ&#x2C6;&#x17E;us â&#x20AC;˘

limits which do not converge are non-existent limits

 â&#x20AC;˘

1 a non-existent limit (lim 2 doesn't exist) xâ&#x2020;&#x2019;0 x

If

Ex:

limsin( x1 ) xâ&#x2020;&#x2019;0

lim sin(x)

and

lim f (x) 6= limâ&#x2C6;&#x2019; f (x)

xâ&#x2020;&#x2019;a+



then the limit does not exist

xâ&#x2020;&#x2019;a

Ex:

lim |x| x xâ&#x2020;&#x2019;0

are non-existent (uctuates between 1 and -1)

xâ&#x2020;&#x2019;â&#x2C6;&#x17E;

does not exist

9 Continuity 9.1 Denition A function is continuous at

a

if

f (a)

exists and

lim f (x) = limâ&#x2C6;&#x2019; f (x) = f (a)

xâ&#x2020;&#x2019;a+

xâ&#x2020;&#x2019;a

9.2 Theorems 1. If

f (x)

(a)

and

g(x)

are continuous at

f + g , f g , f Ă&#x2014; g , f â&#x2014;Ś g

f (b) is continuous at g

f (x) [a, b]

2. If

is continuous on

f (x) is f (x) = c

3. If

c

continuous on

if

c

then

are continuous at

g(c)

g(c) 6= 0

[a, b]

, then

[a, b]

and

f (x)

has a maximum and a minimum value on

f (a) < c < f (b)

Part II

Dierential Calculus 10 The Derivative 10.1 Denitions (you must know them!) 1.

f 0 (x) = lim f (x+h)h  f (x)

2.

f 0 (a) = lim f (x)x  fa (a) = lim f (a+h)h  f (a)

hâ&#x2020;&#x2019;0

xâ&#x2020;&#x2019;a

(generates a slope function)

hâ&#x2020;&#x2019;0

10

then

â&#x2C6;&#x192;

at least one

x

in

[a, b] :


10.2 Theorems 1.

d = c dx [f (x)]

d [cf (x)] dx

2. Product rule:

d dx

h

f (x) g(x)

d [f (u)] dx

d [u(y)] dx

5. Implicit:

i

=

dy = u0 (y) dx

7. Rolle's Theorem: If then

â&#x2C6;&#x192;c

g(x)f 0 (x)  f (x)g 0 (x) (g(x))2

= f 0 (u) du dx

6. Logarithmic: If y = f (x) d f (x)g(x) dx g(x)ln(f (x))

f (a) = 0

= f (x)g 0(x) + g(x)f 0 (x)

d [f (x)g(x)] dx

3. Quotient rule: 4. Chain rule:

where c is a constant

in

g(x)

, then

ln(y) = g(x)ln(f (x))

f (x) is continuous (a, b) : f 0 (c) = 0

8. Mean Value Theorem: If (a, b) : f 0 (c) = f (b)bfa (a)

f (x)

on

d [ur ] dx

2.

d [sin(u)] dx

= cos(u) du dx

3.

d [cos(u)] dx

= sin(u) du dx

4.

d [tan(u)] dx

= sec2 (u) du dx

5.

d [cot(u)] dx

= csc2 (u) du dx

6.

d [sec(u)] dx

= sec(u)tan(u) du dx

7.

d [csc(u)] dx

= csc(u)cot(u) du dx

8.

d [sinâ&#x2C6;&#x2019;1 (u)] dx

=

du â&#x2C6;&#x161;1 1u2 dx

9.

d [tanâ&#x2C6;&#x2019;1 (u)] dx

=

1 du 1+u2 dx

10.

d [secâ&#x2C6;&#x2019;1 (u)] dx

=

du â&#x2C6;&#x161;1 u u2 1 dx

11.

d [au ] dx

= au ln(a) du dx

12.

d [eu ] dx

= eu du dx

13.

d [ln|u|] dx

= rurâ&#x2C6;&#x2019;1 du dx

=

dy dx

d = y dx g(x)ln(f (x)) =

and dierentiable on

(a, b)

and

f (b) =

is continuous on [a,b] and dierentiable on (a,b),

10.3 Dierentiation Formulas 1.

[a, b]

and

1 du u dx

11

â&#x2C6;&#x192;c

in


10.4 Dierentiability vs. Continuity â&#x20AC;˘

The fact that discontinuous



f (x) at c

For example,

â&#x20AC;˘

For

â&#x20AC;˘

Continuity

f (x)

is non-dierentiable at

c

is not sucient to conclude that

f (x)

is

y = |x3|

to be dierentiable at

does not

c, f (x)

must be continuous at

c

necessarily imply dierentiability

11 Applications of the derivative 11.1 Slope of curve (a, b) = f 0 (a)

The slope of a curve at point

. Remember that

(a, b)

is on both the curve and

the tangent line.

11.2 Slope of normal line The slope of the normal line at point

(a, b) =

â&#x2C6;&#x2019;1 f 0 (a)

11.3 Curve Sketching 11.3.1

Increasing/Decreasing

â&#x20AC;˘

If

f 0 (x) > 0

then

f (x)

is increasing

â&#x20AC;˘

If

f 0 (x) < 0

then

f (x)

is decreasing

11.3.2

Critical Points

â&#x20AC;˘

If

f 0 (c) = 0

â&#x20AC;˘

If

f 0 (c)

11.3.3

then

(c, f (c))

is a critical point

does not exist, then

(c, f (c))

is a critical point

Relative extrema

x = c is a critical c, then (c, f (c)) is a

point and

x = c is a critical c, then (c, f (c)) is a

point and

f 0 (x) > 0

to the left of

c

and

f 0 (x) < 0

to the right of

to the left of

c

and

f 0 (x) > 0

to the right of

â&#x20AC;˘

If

â&#x20AC;˘

If

â&#x20AC;˘

If

x=c

is a critical point and

f 00 (c) < 0

then

(c, f (c))

is a relative maximum

â&#x20AC;˘

If

x=c

is a critical point and

f 00 (c) > 0

then

(c, f (c))

is a relative minimum

relative maximum

f 0 (x) < 0

relative minimum

12


11.3.4

Inection points

â&#x20AC;˘

If concavity changes, (f

â&#x20AC;˘

If

f 0 (x0 ) = 0

â&#x20AC;˘

If

f 0 (x0 ) = Âąâ&#x2C6;&#x17E;

â&#x20AC;˘

If

(c, f (c)) is

â&#x20AC;˘

If

00

(x)

changes sign) at the point

(x0 , f (x0 )),

then

(x0 , f (x0 ))

is

an inection point then there is also a horizontal tangent at that point then there is also a vertical tangent at that point

a critical point and

f 0 (x)

does not change sign at

x=c

, then

(c, f (c)) is

an inection point

(x0 , f (x0 ))

f 00 (x0 ) = 0.

is an inection point, then

Note: the converse is not neces-

sarily true!

11.3.5

Concavity

â&#x20AC;˘

If

f 00 (x) > 0

on

(a, b)

then the graph of

f (x)

is concave up on

â&#x20AC;˘

If

f 00 (x) < 0

on

(a, b)

then the graph of

f (x)

is concave down on

(a, b) (a, b)

12 Max-min word problems 1. Evaluate the function at the endpoints and at all critical points. 2. The largest value will be the absolute maximum and the smallest value will be the absolute minimum. 3. Remember, continuous functions have both an absolute maximum and an absolute minimum on a

closed interval.

13 Motion along a line Go from

x(t) â&#x2020;&#x2019; v(t) â&#x2020;&#x2019; a(t)

by dierentiating.

14 Related Rates 1. Find an equation which relates the quantity with the unknown rate of change to quantitie(s) whose rate(s) of change are known. 2. Dierentiate both sides of the equation implicitly with respect to time. 3. Solve for the derivative that represents the unknown rate of change. 4.

Then

evaluate this derivative at the conditions of the problem.

13


Part III

Integral Calculus 15 Antiderivatives 15.1 Denition F (x)

is an antiderivative of

f (x)

if and only if

F 0 (x) = f (x)

15.2 Applications 15.2.1 â&#x20AC;˘

Exponential growth and decay

If you know that

â&#x20AC;˘ k

dy dt

= ky

then

y = Cekt â&#x2021;&#x201D; y = y0 ekt

where

y0 = y(0)

is the growth rate and may be given as a percent

â&#x20AC;˘ Tdouble =

15.2.2

ln2 , k

Thalve =  ln2 k

Dierential Equations

1. Separate the variables so that the equation reads

f (y)dy = g(x)dx

.

2. Integrate both sides, adding the constant to the side with the independent variable. i.e.

´

f (y)dy =

3. Solve for

y

´

f (x)dx + C

if possible.

16 Techniques of Integration 16.1 Integrals you must know 1. 2. 3. 4. 5. 6. 7. 8.

´

du = u + C

´

ur du =

´ ´

a du = au + C

1 u

ur+1 r+1

+C

du = ln|u| + C

eu du = eu + C ´ r r+1 u du = ur+1 + C ´ sin(u) du = cos(u) + C ´ cos(u) du = sin(u) + C 14


9. 10. 11. 12. 13. 14. 15. 16. 17.

´

sec2 (u) du = tan(u) + C

´

sec(u)tan(u) du = sec(u) + C

´

tan(u) du = ln|cos(u)| + C

´

√1 1u2

´

csc2 (u) du = cot(u) + C

´

csc(u)cot(u) du = csc(u) + C

´

cot(u) du = ln|sin(u)| + C

´

1 1+u2

´

du = sin−1 (u) + C

du = tan−1 (u) + C

√1 u u2 1

du = sec−1 (u) + C

16.2 U-substitution The purpose of a u-substitution is to make a dicult integral look like one of the integrals above.

16.3 Inverse chain rule theorem ˆ

f (g(x))g 0(x) = F (g(x)) + C

This theorem can be used to solve u-substitution integrals like

´

√ 2x 19x2 dx

16.4 Integration by parts ˆ ˆ

ˆ

0



0

f (x)g(x)|ba 

f (x)g (x) dx = f (x)g(x) b

f (x)g (x)dx = a

ˆ

f 0 (x)g(x) dx b

f 0 (x)g(x)dx a

17 The Denite Integral 17.1 Denition ˆ

b

f (x) dx = a

lim

max∆xk →0

15

n X k=1

f (xk n )∆xk

.


17.2 Fundamental theorems • • • • •

´b

f (x) dx = F (b)F (a) ´ d f (x) = f (x) dx ´ d f (x)dx = f (x) + C dx ´x f (t)dt = F (x) where F (a) = 0 a ´x d f (t)dt = f (x) dx a a

17.3 Approximations to the denite integral (n will be given) 17.3.1 Riemann sums using midpoints ˆ

17.3.2

b a

f (x)dx ≈

ba [y1 + y2 + ... + yn ] n

Trapezoids ˆ

b

a

f (x)dx ≈

ba [f (a) + 2f (x1 ) + 2f (x2 ) + ... + f (b)] 2n

17.4 Denite integral as area •

´b

´d

f (x)dx =area between f (x) and the x-axis on [a, b] if f (x) ≥ 0 on [a, b]

´

b

• a f (x)dx =area between f (x) and the x-axis on [a, b] if f (x) ≤ 0 on [a, b] a

u(y)dy =area between u(y) and the y-axis on [c, d] if u(y) ≥ 0 on [c, d]

´

d

• c u(y)dy =area between u(y) and the y-axis on [c, d] if u(y) ≤ 0 on [c, d] c

18 Applications of the Denite Integral 18.1 Motion along a line •

Go from

a(t) → v(t) → x(t)

Remember, the given conditions will determine the values of the constants generated

by integrating.

by integration.

=

´ t=b

Displacement

Distance traveled is the sum of the distances traveled right and left.

t=a

position at time=

a,

v(t)dt at every value of time where

the distanced traveled on each time interval.

16

v(t) = 0,

Evaluate the

and at time=

b.

Add up


18.2 Mean value (average value of a function) on [a, b] ´b a

fav =

f (x)dx ba

18.3 Area between two curves When nding the area between two curves, either axis can be used for orientation. the variable of integration

â&#x20AC;˘

X-axis:

R=

â&#x20AC;˘

Y-axis:

R=

´b a

´d c

must

However,

be the same as the axis of orientation.

[f (x)g(x)] dx [u(y)v(y)] dy

18.4 Volume of a solid of revolution 18.4.1 Around an axis The axis of revolution is determined in the problem. However, either axis can be used for orientation.

must

If the axis of orientation is dierent from the axis of revolution, the volume

be found using shells.

In general, it is simplest to use disks/washers for volumes of revolution around the x-axis, and shells for volumes of revolution around the y-axis.

â&#x20AC;˘

â&#x20AC;˘

Disks with one curve (axis of revolution=axis of orientation=variable of integration):

b

x â&#x2C6;&#x2019; axis : V = Ď&#x20AC;

Ë&#x2020;

d

y â&#x2C6;&#x2019; axis : V = Ď&#x20AC;

Ë&#x2020;

[f (x)]2 dx

a

[u(y)]2 dy

c

Washers with two curves (axis of revolution=axis of orientation=variable of integration)

18.4.2

b

x â&#x2C6;&#x2019; axis : V = Ď&#x20AC;

Ë&#x2020;

y â&#x2C6;&#x2019; axis : V = Ď&#x20AC;

Ë&#x2020;

a d

c

  [f (x)]2  [g(x)]2 dx   [u(y)]2  [v(y)]2 dy

Around a line

If the line is parallel to the x-axis, then your variable of integration must be parallel to the y-axis, then your variable of integration must be

V =Ď&#x20AC;

Ë&#x2020;

a

b

 R2  r 2 dr

Both radii must be in terms of the appropriate variable. 17

y.

x.

If the line is

In either case, remember:


18.4.3

Volumes of known cross sections

If the cross-sections are perpendicular to the x-axis, then your variable of integration is If the cross-sections are perpendicular to the y-axis, then your variable of integration is In either case, remember:

V = V =

Ë&#x2020;

b

A(x)dx a

Ë&#x2020;

d

A(y)dy

c

18

x. y.


The Calculus Bible: The New Testament Deborah A. Foley June 8, 2010

Part I

Advanced Techniques of Integration 1 Trigonometric Integrals 1.1 Reduction formulas (n is an integer ≥ 2) •

´

sinn (u)du =

´

tann (u)du =

−1 cos(u)sinn−1 (u) n

+

n−1 n

´

sinn−2 (u)du

´ cosn−2 (u)du cosn (u)du = n1 sin(u)cosn−1 (u) + n−1 n ´ ´ n−2 n−2 secn−2 (u)du • secn (u)du = secn−1(u) + n−1 •

´

secn−1 (u) ´  n−1

tann−2 (u)du

1.2 Power formulas (m and n are integers ≥ 2) 1.2.1 Powers of sine and cosine [sinm (x)cosn (x)] dx •

If

n

If

m

is odd, let

If

m

and

is odd, let

n

u = sin(x)

´

u = cos(x)

are both even, use the identities below, and then apply the appropriate

reduction formula

sin2 (x) =

1 cos(2x) 2

cos2 (x) =

1 + cos(2x) 2

1


1.2.2 Powers of sec and tan [tanm (x)secn (x)] dx •

If

is even, let

u = tan(x)

If m is odd, let

u = sec(x)

If

n

m

is even and

n

´

is odd, reduce to a power of

sec(x)

and then use the

sec

reduction

formula

1.2.3 Simplication formulas • • •

´

´

[sin(mx)cos(nx)] dx =

1 2

[sin(mx)sin(nx)] dx =

´ 1

´

2

´ 1

[sin(mn)x + sin(m + n)x]dx] [cos(mn)x  cos(m + n)x]dx], m > n

´

[cos(mx)cos(nx)] dx =

´

sec(u)du = ln |sec(u) + tan(u)| + C

2

[cos(mn)x + cos(m + n)x]dx], m > n

1.2.4 Integrals to remember • •

´

tan(u)du = ln |cos(u)| + C

2 Trigonometric subsitutions 2.1 Used for integrals that cannot be solved with a u-substitution •

If an integral contains

a2  x2 ,

If an integral contains

x2 + a2 ,

If an integral contains

x2  a2 ,

let

x = asin(θ)

let

let

and

x = atan(θ)

x = asec(θ)

dx = acos(θ)dθ

and

and

dx = asec2 (θ)dθ

dx = asec(θ)tan(θ)dθ

Solve the integral using reduction formulas where necessary, and then make the appropriate substitutions to convert back to

x.

3 Partial Fractions Used for 1. If

´

p(x) dx q(x)

p(x)

is a higher order polynomial than

q(x),

then divide rst.

2. Factor the denominator. 3. For every linear factor

(ax + b)m , introduce m terms A B R + + ... + 2 (ax + b) (ax + b) (ax + b)m

4. Solve for all constants. 5. Integrate each partial fraction separately.

2


4 Advanced integration by parts • •

The integral

´

f 0 (x)g(x)dx,

which results from simple integration by parts, may also

have to be integrated by parts. If the integral resulting from the second integration by parts is identical to the origi´ ´ 0 f (x)g (x)dx = h(x)  f (x)g 0 (x)dx, then simplify the equation to nal integral (e.g. ´ ´ 2 f (x)g 0 (x)dx = h(x), and nally f (x)g 0(x)dx = 21 h(x)

5 Improper integrals 1. If

f (x)

is continuous on

exists. 2. If

f (x)

is continuous on

limit exists. 3. If

(a, ∞),

then

(∞, b),

´∞

f (x)dx = lim

a

then

´b

b→∞ a

´b

f (x)dx,

f (x)dx = lim −∞

´b

a→−∞ a

provided the limit

f (x)dx,

f (x) is discontinuous at x = c, a < c < b, but is otherwise continuous on (a, b), then:

(a)

´b a

f (x)dx = lim x→c

´c a

f (x)dx + lim+ x→c

´b c

f (x)dx

(b) If either limit is non-nite, then the integral is non-nite.

Part II

Further applications of the denite integral 6 Volumes of known cross-sections 1. Express the area of the cross-section as a function of 2.

provided the

V =

´b a

A(x)dx

7 Length of a plane curve L=

ˆ bq

1 + [f 0 (x)]2 dx

a

3

x.


8 Work 1. An object must move over

[a, b]

while subject to a variable force

F (x)

of the motion. 2.

W =

´b a

F (x)dx

Part III

Parametrically dened equations 9 Denition x = f (t) y = g(t)

10 Derivatives dy = dx

dy dt dx dt

d2 y = dx2

dy 0 dt dx dt

11 Length of the plane curve L=

ˆ

b

s

a

dx dt

2

+



dy dt

2

dt

Part IV

Polar Functions 12 Lines 1.

rcos(θ) = a (x = a)

vertical line

2.

rsin(θ) = a (y = b)

vertical line

3.

Arcos(θ) + Brsin(θ) = C (Ax + By = C)

4.

θ = a (y = mx) (a) Ex.

θ=

line through the origin,

π 4

4

skewed line,

m = tan(θ)

A, B, C 6= 0

in the direction


13 Circles • r=a

centered at the origin, radius

• r = 2acos(θ)

centered at

• r = 2acos(θ) • r = 2asin(θ)

centered at

centered at

• r = 2acos(θ)

(a, 0),

centered at

radius=

(−a, 0),

(0, a),

a

radius=

radius=

(0, −a),

a,

a,

radius=

tangent to the y-axis

a,

tangent to the y-axis

tangent to the x-axis

a,

tangent to the x-axis

14 Rectangular hyperbolas r 2 sin(2θ) = 2a (xy = a)

5


15 Limacons

• r = a + bsin(θ), 1 <

a b

• r = a + bsin(θ),

a b

=1

• r = a + bsin(θ),

a b

<1

• r = absin(θ)

above curves reected over x-axis

• r = a + bcos(θ) • r = abcos(θ)

<2

above curves rotated

above curves rotated

90o clockwise

90o

counterclockwise

6


16 Lemniscates Note that

a =length

of one loop

• r 2 = a2 cos(2θ)

• r 2 = a2 cos(2θ)

• r 2 = a2 sin(2θ)

7


• r 2 = a2 sin(2θ)

17 Spirals r = aθ

8


18 Roses Note that

a =length

of one loop

• r = asin(nθ)

• r = acos(nθ) •

If

n

is odd, then

If

n

is even, then

n =number

of petals

2n =number

of petals

9


19 Area â&#x20AC;˘ f (θ)

must be continuous and non-negative

â&#x20AC;˘ A=

1 2

â&#x20AC;˘

´β ι

[f (θ)]2 dθ =

1 2

´β ι

(r)2 dθ

Warning: in the equation of a lemniscate,

r

is already squared!

20 Length of the plane curve 1. Dene the curve parametrically

2.

(a) Since

r = f (θ)

and

x = rcos(θ), x = f (θ)cos(θ)

(b) Since

r = f (θ)

and

y = rsin(θ), y = f (θ)sin(θ)

L=

´ β q dy 2 ι

dθ

+

 dx 2 dθ

dθ

Part V

Sequences and series 21 Sequences Convergence 

1. Find the limit as

n â&#x2020;&#x2019; â&#x2C6;&#x17E;

of

an .

You may use limit theorems previously proved,

including L'Hopital's rule.

(a) If the limit exists, then the sequence converges to that limit. (b) If the limit is non-existent, including

Âąâ&#x2C6;&#x17E;,

the sequence diverges.

2. Show that the sequence is monotone and that it is bounded in the proper direction. (a) Monotone defense i. Ratio rule

an+1 an an+1 B. an

A.

â&#x2030;¤1 â&#x2030;Ľ1

non-increasing non-decreasing

ii. Dierence rule A. B.

an  an+1 â&#x2030;Ľ 0 an  an+1 â&#x2030;¤ 0

non-increasing non-decreasing

iii. Derivative rule A. B.

f 0 (x) < 0 f 0 (x) > 0

decreasing increasing

10


(b) Convergence i. If ii. If

{an }is {an }is

non-decreasing and is bounded above, it is convergent. non-increasing and is bounded below, it is convergent.

{an }

iii. In all other cases,

diverges.

22 Series Convergence 

A series

Pâ&#x2C6;&#x17E; uk

converges to a sum

S

i the sequence of partial sums

k=1 limit. The limit of the sequence of partial sums is called the sum Geometric series converge i

|r| < 1.

S

{Sn }â&#x2C6;&#x17E; n=1

converges to a

of the series.

If the series converges, then the sum

S=

a 1â&#x2C6;&#x2019;r

22.1 Tests for convergence 22.1.1 Divergence test If

lim uk 6= 0,

then the series

kâ&#x2020;&#x2019;â&#x2C6;&#x17E; prove convergence.

Pâ&#x2C6;&#x17E; uk

diverges. Be careful! Showing that

k=1

lim uk = 0

kâ&#x2020;&#x2019;â&#x2C6;&#x17E;

does not

22.1.2 Integral test â&#x2C6;&#x17E; P

uk

converges i

k=1 and decreasing on

嫉&#x2C6;&#x17E; 1

f (x)dx converges (is nite), provided that f (x) is continuous, positive,

[1, â&#x2C6;&#x17E;)

and

uk = f (k).

22.1.3 P-series rule â&#x2C6;&#x17E; P

k=1

1 converges i kp

p>1

22.1.4 Comparison test If

uk â&#x2030;¤ vk Ifuk

for

â&#x2030;Ľ vk

â&#x2C6;&#x20AC;k â&#x2030;Ľ N ,

for

â&#x2C6;&#x20AC;k â&#x2030;Ľ

â&#x2C6;&#x17E; P

vk

is known to converge, then

k=1

22.1.5 Ratio Test If

â&#x2C6;&#x17E; P

uk

â&#x2C6;&#x17E; P

uk converges as well. k=1 â&#x2C6;&#x17E; â&#x2C6;&#x17E; P P N , and if vk is known to diverge, then uk diverges as well. k=1 k=1 and if

is a positively valued series and

k=1

Ď = lim uuk+1 , k

â&#x20AC;˘ Ď > 1,

the series diverges.

â&#x20AC;˘ Ď < 1,

the series converges.

â&#x20AC;˘ Ď = 1,

no conclusion can be reached.

kâ&#x2020;&#x2019;â&#x2C6;&#x17E;

11

then if:


22.1.6 Root test If

∞ P

uk

is a positively valued series and

ρ = lim

k=1

• ρ > 1,

the series diverges.

• ρ < 1,

the series converges.

• ρ = 1,

no conclusion can be reached.

k→∞

√ k

uk ,

then if:

22.1.7 Limit comparison test If

∞ P

uk

and

∞ P

vk

are positively valued series and if

ρ = lim uvkk ,

k→∞ k=1 k=1 then the two series either both converge or both diverge.

then if

ρ

is nite and6=

0,

22.1.8 Ratio test for absolute converegence If

∞ P

uk

is a series and

k=1

uk+1

ρ = lim uk , k→∞

then if:

• ρ > 1,

the series diverges.

• ρ < 1,

the series converges.

• ρ = 1,

no conclusion can be reached.

22.1.9 Alternating series test If

∞ P

uk

1.

uk+1 ≤ uk ∀k > N

is an alternating series and

k=1

2.

lim uk = 0

k→∞

then the series is said to be at least conditionally convergent.

22.2 Theorems 1. If a series converges absolutely, then it converges. 2. If, according to the ratio test for absolute convergence (22.1.8), a series diverges, then there is no chance it will conditionally converge.

22.2.1 Error Theorem ∞ P

uk is an alternating series and k=1 = |S Sn | ≤ un+1 If

n terms are used

12

to approximate the sum, then the error


22.2.2 Function approximations using innite series •

The function

f (x) is approximated in the neighborhood of zero by the series

The interval of convergence are the values of

x

∞ (x) P f (0)

k=0 for which the series converges.

k!

xk .

The function f (x) is approximated in the neighborhood of (x = a) by the series ∞ (x) P f (a) (xa)k . The interval of convergence are the values of x for which the series k! k=0 converges.

22.3 Function approximations by innite series to remember 1

=

1 x

∞ X xk

−1<x<1

k=0

x

e =

X xk k=0

sin(x) =

−∞<x<∞

k!

∞ X (−1)k x2k+1 k=0

cos(x) =

∞ X (−1)k x2k k=0

ln(1 + x) =

−∞<x<∞

(2k + 1)!

−∞<x<∞

(2k)!

∞ X (−1)k xk+1 k=0

(k + 1)

−1<x≤1

22.4 Derivatives and integrals of power series •

If

∞ P

ck (xa)k

is a power series, then

If

∞ P

ck (xa)k

is a power series, then

d dx

∞ P

ck (xa)k =

∞ P

kck (xa) k=0 k=0 k=1 only changes if the rst term of the original series is a constant ∞ ´ P

ck (xa)k =

k=0

k=0

∞ P

k=0

k−1

ck (xa)k+1 k+1

(the lower limit

+C

22.5 Derivation of a power series from another power series • •

If

∞ P

ck (xa)k

is a power series, then any monomial

k=0 can be substituted in for

x

m(x)

or binomial of form

axp + b

in the summation to serive another power series.

The interval of convergence can be determined by substituting interval of convergence.

13

m(x) for x in the original

The Calculus Bible  

The Calculus Bible is not meant to be a textbook, but rather a review guide or a study aide for high school students who have completed the...

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