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CALCULUS REFERENCE 3/18/03 10:49 AM Page 1

SPARKCHARTSTM

CALCULUS REFERENCE

SPARK

CHARTS

TM

THEORY

SPARKCHARTS

TM

DERIVATIVES AND DIFFERENTIATION
f (x+h)−f (x)
h
h→0

Definition: f � (x) = lim

is continuous and differentiable on the interval and F � (x) = f (x)
...
Sum and Difference: dx f (x) ± g(x) = f � (x) ± g � (x)
�
�
d
2
...
Product: dx f (x)g(x) = f � (x)g(x) + f (x)g � (x)

�

Mnemonic: If f is “hi” and g is “ho,” then the product rule is “ho d hi plus hi d ho
...
Quotient:

Mnemonic: “Ho d hi minus hi d ho over ho ho
...
The Chain Rule

• First formulation: (f ◦ g)� (x) = f � (g(x)) g � (x)
dy
dy
• Second formulation: dx = du du
dx

dy
dx

and then cos y − x(sin y)y � − 2yy � = 3
...

x sin y+2y

=0

2
...
Powers:

d
(xn )
dx

4
...
95 CAN

f (x0 )+4f (x1 )+2f (x2 )+· · ·+2f (xn−2 )+4f (xn−1 )+f (xn )

�

�

�

d
(log a
dx

• Cosine:
• Cotangent:
• Cosecant:

x) = sec 2 x

x) = sec x tan x

• Definite integrals: reversing the limits:

d
(cos
dx

d
(cot
dx

d
(csc
dx

• Definite integrals: concatenation:

= a ln a

x) =

x) = − sin x

x) = − csc 2 x

x) = − csc x cot x

• Arccosine:

d
(cos −1
dx

x) =

1
1+x2

• Arccotangent:

1
x) = − √1−x2

d
(tan −1
dx

d
(cot −1
dx

• Arcsecant:

d
(sec −1
dx

x) =

√1
x x2 −1

• Arccosecant:

d
(csc −1
dx

f (x) dx = −

p

f (x) dx +
a

�

�

a

f (x) dx
b

b

f (x) dx =
p

b
a

�

f (x) dx ≤

�

�

b

f (x) dx
a

b

g(x) dx
...
Substitution Rule—a
...
a
...

•
�

3
...


1
x) = − x√x2 −1

INTEGRALS AND INTEGRATION

b
a

If f (x) ≤ g(x) on the interval [a, b], then

1
x) = − 1+x2

√ 1
1−x2

�

�

• Definite integrals: comparison:

1
x ln a

d
(sin −1
dx

�
� Indefinite Integrals:
�
�
•
f (x)g � (x) dx = f (x)g(x) − f � (x)g(x) dx or u dv = uv − v du

• Definite Integrals:

�

b

b

f (x)g � (x) dx = f (x)g(x)]a −

a

�

b

f � (x)g(x) dx
a

4
...
Simpson’s Rule: Sn =

+ · · · + 2a2 x + a1

• Arbitrary base:

1
x

x) =

�

∆x
2

�
∆x

�

• Arbitrary base:

x

x) = cos x

d
(tan
dx

d
(sec
dx

=

xk + xk+1
2

• Constant multiples: cf (x) dx = c f (x) dx
2

=e

f

k=0

= nx n−1 (true for all real n �= 0)

d
(an xn
dx

d
(ex )
dx

k=1

�

�
�
1
...
Constants:

d
(c)
dx

• Arcsine:

$3
...
Exponential
• Base e:

2
...
Trapezoidal Rule: Tn =

dy
cos y + x d(cos y) − 2y dx = 3 dx ,
dx
dx

8
...
Left-hand rectangle approximation:
n−1
�
Ln = ∆x
f (xk )
k=0

dx
dx

f (t) dt

a

3
...


Ex: x cos y − y 2 = 3x
...
Differentiate both sides of the equation with respect to x
...
Then, rewrite

�

Part 2: If f (x) is continuous on the interval [a, b] and F (x) is any antiderivative of f (x),
� b
f (x) dx = F (b) − F (a)
...
Implicit differentiation: Used for curves when it is difficult to express y as a function

• Tangent:
• Secant:

50395

f (x) dx = F (x) + C if F � (x) = f (x)
...
Trigonometric
• Sine:

Copyright © 2003 by SparkNotes LLC
...

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of SparkNotes LLC
...
Logarithmic
• Base e:

ISBN 1-58663-896-3

antiderivatives:

b

√

±a2 ± x2
...


• Formal definition: Let n be an integer and ∆x =

For each k = 0, 2,
...
The expression ∆x
k
k
k=0
� b
n−1
�
∗
f (x) dx is defined as lim ∆x
f (xk )
...
The definite integral
b−a

...

�
• Indefinite integral: The indefinite integral f (x) dx represents a family of

π
2
3π
2

π
2

1 + tan 2 θ = sec 2 θ

APPLICATIONS
GEOMETRY
Area:

�

a

b�

Volume of revolved solid (shell method):

f (x) − g(x) dx is the area bounded by y = f (x), y = g(x), x = a and x = b
�

if f (x) ≥ g(x) on [a, b]
...

� b
�
�2 �
�2
f (x) − g(x) dx is the volume of
Volume of revolved solid (washer method): π
a

the solid swept out between y = f (x) and y = g(x) as they revolve around the x-axis on

�

b

2πxf (x) dx is the volume of the solid
a

obtained by revolving the region under the curve y = f (x) between x = a and x = b
around the y-axis
...


Surface area:

a

�

b

2πf (x)
a

�

2

1 + (f � (x)) dx is the area of the surface swept out by

revolving the function y = f (x) about the x-axis between x = a and x = b
...


CONTINUED ON OTHER SIDE

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...
Increasing p leads to decrease in revenue
...
Percentage change in p leads to similar percentage
change in x(p)
...


• Demand is inelastic if E(p) < 1
...
Increasing p leads to increase in revenue
...


¯
¯
• Market equilibrium is x units at price p
...
k
...
expectation or mean) of X : E(X ) = µX = −∞ xf (x) dx
�2
�
�2
�∞ �
2
2
• Variance: Var(X ) = σX = −∞ x − E(x) f (x) dx = E(X ) − E(X )
�
• Standard deviation: Var(X ) = σX
�m
�∞
• Median m satisfies −∞ f (x) dx = m f (x) dx = 1
...
Normal distribution (or Bell curve) with mean µ and
(x−µ)2
1
variance σ:
f (x) = √ e− 2σ2
σ 2π
• P (µ − σ ≤ X ≤ µ − σ) = 68
...

�
�
r m
• Interest compounded m times a year: reff = 1 + m − 1
• Interest compounded continuously: reff = er − 1

• Cost function C(x): cost of producing x units
...

x
�
• Marginal average cost: C (x)
If the average cost is minimized, then average cost = marginal cost
...


REVENUE, PROFIT

• Demand (or price) function p(x): price charged per unit if x units sold
...

�
�
r −mt
• Interest compounded m times a year: PV = A 1 + m
• Interest compounded continuously: PV = Ae−rt

PRESENT VALUE OF ANNUITIES AND PERPETUITIES
Present value of amount P paid yearly (starting next year) for t years or in perpetuity:
1
...
Interest compounded continuously
• Annuity paid for t years: PV = rP (1 − e−rt ) = erP (1 − e−rt )
−1
eff

If profit is maximal, then marginal revenue = marginal cost
...

PRICE ELASTICITY OF DEMAND

• Perpetuity: PV =

• Demand curve: x = x(p) is the number of units demanded at price p
...
K
...
LEARNING

EXPONENTIAL DECAY MODEL

CURVE) MODEL

= rP

• Solution:
•

P (t) = P0 ert
If r > 0, this is
exponential
growth; if r < 0,
exponential decay
...


INTEREST
σ

COST

dP
dt

∂g
∂p

• P (t): the amount after t years
...

• r: the yearly interest rate (the yearly percentage is 100r%)
...


x

1

FINANCE

95%
68%

• P (µ − 2σ ≤ X ≤ µ + 2σ) = 95
...
χ-square distribution: with mean ν and variance 2ν:
x
ν
1
f (x) = ν � ν � x 2 −1 e− 2
22Γ 2
�
• Gamma function: Γ(x) = 0∞ tx−1 e−t dt

BIOLOGY

0

SUBSTITUTE AND COMPLEMENTATRY COMMODITIES

COMMON DISTRIBUTIONS

C (x)
x

y = L(x)
completely
equitable
distribution

X and Y are two commodities with unit price p and q, respectively
...

• The amount of Y demanded is given by g(p, q)
...
X and Y are substitute commodites (Ex: pet mice and pet rats) if ∂f > 0 and
∂q
2
...


Cov(X, Y )
σ
Correlation: ρ(X, Y ) = X Y = �
σX σY
Var(X)Var(Y )

for x in

y
1

The quantity L is between 0 and 1
...


f (y) dy

g(x, y) dy
...
Then
�∞

x

x

dP
dt

P

P0
r<0

This downloadable PDF copyright © 2004 by SparkNotes LLC
...
95 CAN

−∞

0

p = D(x)

$3
...

1
...

2
...
Curve is nondecreasing: L� (x) ≥ 0 for all x
4
...
k
...
Gini Index):
� 1
�
�
L=2
x − f (x) dx
...

1
...
−∞ f (x) dx = 1
...


consumer
surplus
producer
surplus

3

1
b−a

p

¯
x
x
(So p = D (¯) = S (¯)
...


+

ACCELERATION

3
...
time t graph:
• The (signed) area under the graph gives the change
� t
in velocity:
a(τ ) dτ
v(t) − v(0) =

TM

CONSUMER AND PRODUCER SURPLUS

–

v(τ ) dτ

0

7

t

• Formula relating elasticity and revenue: R� (p) = x(p) 1 − E(p)

SPARKCHARTS

+

Writer: Anna Medvedovsky
Design: Dan O
...
Williams
Series Editor: Sarah Friedberg

2
...
time t graph:
• The slope of the graph is the accleration: v � (t) = a(t)
...
Percentage change in p leads to larger percentage

POSITION

1
...
time t graph:
• The slope of the graph is the velocity: s� (t) = v(t)
...


�
= rP 1 −

P
K

• K : the carrying

P0 > A

P

�

k

capacity

P0 < A

• Solution:
t

P (t) =

0

P0 > k

k
< P0 < k
2

P0 <

k
2
t

K

1+

�

K −P 0
P0

�

e−rt

SPARKCHARTS™ Calculus Reference page 2 of 2



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