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
√ + 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
(xr)
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 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 = +∞,
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
b→±∞ 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
• ±∞us
•
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 Denition
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 × g , f ◦ 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)
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)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
[sin−1 (u)]
dx
=
du
√1
1u2 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 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
∃c
in
10.4 Dierentiability vs. Continuity
•
The fact that
discontinuous
f (x)
at c
For example,
•
For
•
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) =
−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
Inection points
•
If concavity changes, (f
•
If
f 0 (x0 ) = 0
•
If
f 0 (x0 ) = ±∞
•
If
(c, f (c)) is
•
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
•
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 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
•
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
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
•
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 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.
•
•
Disks with one curve (axis of revolution=axis of orientation=variable of integration):
b
x − axis : V = π
ˆ
d
y − axis : V = π
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 = π
ˆ
y − axis : V = π
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 =π
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.