The Calculus Bible

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 coe cients 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 m logb 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 √ + 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 •

Set the function

•

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

=0

and solve

(x r)

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

xn+1 =

7 Symmetry •

y-axis:

(x, y) → ( x, y)

•

x-axis:

(x, y) → (x, y)

•

origin:

(x, y) → ( x, y)

(non-function)

• y = x: (x, y) → (y, x)

8 Limits 8.1 Theorems 1. Limit of a sum or di erence = sum or di erence 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 = +∞,

b→0+ b

where

a>0

and nite

lim a = ∞,

where

a>0

and nite

lim a = ∞,

where

a<0

and nite

b→0− b b→0+ b

lim a = +∞,

b→0− b

lim

a

bâ†’Âąâˆž b

= 0,

where

where

a

a<0

and nite

is nite

1

lim(1 + x) x = lim (1 + x1 )x = e x→0

x→∞

9

8.3 Non-existent limits • Âąâˆžus •

limits which do not converge are non-existent limits

•

1 a non-existent limit (lim 2 doesn't exist) x→0 x

If

Ex:

limsin( x1 ) x→0

lim sin(x)

and

lim f (x) 6= lim− f (x)

x→a+

then the limit does not exist

x→a

Ex:

lim |x| x x→0

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

x→∞

does not exist

9 Continuity 9.1 De nition A function is continuous at

a

if

f (a)

exists and

lim f (x) = lim− f (x) = f (a)

x→a+

x→a

9.2 Theorems 1. If

f (x)

(a)

and

g(x)

are continuous at

f + g , f g , f Ă— g , f â—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

Di erential Calculus 10 The Derivative 10.1 De nitions (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)

h→0

x→a

(generates a slope function)

h→0

10

then

∃

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

∃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)b fa (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 [sin−1 (u)] dx

=

du √1 1 u2 dx

9.

d [tan−1 (u)] dx

=

1 du 1+u2 dx

10.

d [sec−1 (u)] dx

=

du √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

= rur−1 du dx

=

dy dx

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

and di erentiable on

(a, b)

and

f (b) =

is continuous on [a,b] and di erentiable on (a,b),

10.3 Di erentiation Formulas 1.

[a, b]

and

1 du u dx

11

∃c

in

10.4 Di erentiability vs. Continuity •

The fact that discontinuous

f (x) at c

For example,

•

For

•

Continuity

f (x)

is non-di erentiable at

c

is not su cient to conclude that

f (x)

is

y = |x 3|

to be di erentiable at

does not

c, f (x)

must be continuous at

c

necessarily imply di erentiability

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) =

−1 f 0 (a)

11.3 Curve Sketching 11.3.1

Increasing/Decreasing

•

If

f 0 (x) > 0

then

f (x)

is increasing

•

If

f 0 (x) < 0

then

f (x)

is decreasing

11.3.2

Critical Points

•

If

f 0 (c) = 0

•

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

•

If

•

If

•

If

x=c

is a critical point and

f 00 (c) < 0

then

(c, f (c))

is a relative maximum

•

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

In ection points

•

If concavity changes, (f

•

If

f 0 (x0 ) = 0

•

If

f 0 (x0 ) = Âąâˆž

•

If

(c, f (c)) is

•

If

00

(x)

changes sign) at the point

(x0 , f (x0 )),

then

(x0 , f (x0 ))

is

an in ection 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 in ection point

(x0 , f (x0 ))

f 00 (x0 ) = 0.

is an in ection point, then

Note: the converse is not neces-

sarily true!

11.3.5

Concavity

•

If

f 00 (x) > 0

on

(a, b)

then the graph of

f (x)

is concave up on

•

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) → v(t) → a(t)

by di erentiating.

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. Di erentiate 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 De nition F (x)

is an antiderivative of

f (x)

if and only if

F 0 (x) = f (x)

15.2 Applications 15.2.1 •

Exponential growth and decay

If you know that

• k

dy dt

= ky

then

y = Cekt ⇔ y = y0 ekt

where

y0 = y(0)

is the growth rate and may be given as a percent

• Tdouble =

15.2.2

ln2 , k

Thalve = ln2 k

Di erential 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 1 u2

´

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 di cult 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 1 9x2 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 De nite Integral 17.1 De nition 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 de nite integral (n will be given) 17.3.1 Riemann sums using midpoints ˆ

17.3.2

b a

f (x)dx ≈

b a [y1 + y2 + ... + yn ] n

Trapezoids ˆ

b

a

f (x)dx ≈

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

17.4 De nite 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 De nite 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 b a

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

•

X-axis:

R=

•

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 di erent 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.

•

•

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

b

x − axis : V = Ď€

ˆ

d

y − axis : V = Ď€

ˆ

[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 − axis : V = Ď€

ˆ

y − axis : V = Ď€

ˆ

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€

ˆ

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 =

ˆ

b

A(x)dx a

ˆ

d

A(y)dy

c

18

x. y.

Turn static files into dynamic content formats.

Create a flipbook
Issuu converts static files into: digital portfolios, online yearbooks, online catalogs, digital photo albums and more. Sign up and create your flipbook.