SPARK CHARTS CHARTS
TM
CALCULUS REFERENCE
SPARK SPARK CHARTS CHA RTS
TM
THEORY DERIV DERIVAT ATIVES IVES AND DIFFERENTIATION
S T R A H C K R A P S M T
f (x+ h)−f (x) Definition: f � (x) = lim h h→0
d dx
� ( ) ± ( ) = � ( ) ± �( ) � ( ) = � ( ) 2. Scalar Scalar Multiple Multiple:: � ( ) ( ) = � ( ) ( ) + ( ) � ( ) 3. Prod Product uct:: Mnemonic: If is “hi” and is “ho,” then the product rule is “ho d hi plus hi d ho.” = − 4. Quotien Quotient: t: d dx
d dx
f x
cf x
f x g x
f
f x
g x
Part1: If f (x) is continuous on the interval [a, b], then the area function F (x) = is continuous and differentiable on the interval and F � (x) = f (x) .
f � (x)g (x) f (x)g � (x) (g (x))2
Mnemonic: “Ho d hi minus hi d ho over ho ho.”
then
f (x) dx = F (b)
a
First formulation: (f g ) (x) = f � (g (x)) g � (x)
•
Second formulation:
◦ ◦
1. Left-hand rectangle approximation: n −1 Ln = ∆ x f (xk ) n
3. Midpoint Midpoint Rule:
dy = y dx
� and solve for y � . y ) dy cos y + x d(cos 2y dx = 3 dx , Ex: x cos y y 2 = 3 x . Differentiate to first obtain dx dx dx dx cos y −3 � � � 2yy = 3 . Finally, solve for y = x sin y +2y . and then cos y x(sin y )y
carefully whenever y appears. Then, rewrite
−
−
−
−
M n = ∆ x
d (c ) dx
2. Linear:
d (mx + b ) dx
3. Powers:
d (x n ) dx
4. Polynomials:
d (an x n dx
N B S I
+
= m •
···
+ a 2 x + a 1 x + a 0 ) = a n nx − 1 + n
= e x =
d (sin x ) dx
•
1
•
x
= cos x
Tangent:
d (tan x) dx
= sec 2 x
Secant:
d (sec x) dx
= sec x tan x
− 1 x) = √ 1 1− x −1 x) = 1 1+x
Arcsine:
d (sin dx
•
Arctangent:
d (tan dx
•
Arcsecant:
d (sec dx
•
· · · + 2a2 x + a 1
2
2
−1 x) = √ 1 x
The definite definite integral integral
f x0 ) + 2 f (x1 ) + 2 f (x2 ) +
Arbitrary base:
d (ax ) dx
Arbitrary base:
d (log a x ) dx
•
n
n)
−2 )+ 4f (xn−1 )+ f (xn )
x2
−1
� = ( ) ± ( ) ( )± ( ) ( ) = ( ) Constant multiples: ( ) = − ( ) Definite Definite integrals : reversing the limits: f x
g x
cf x dx
c
dx
f x dx
=
•
Cosine:
d (cos x ) dx
g x dx
•
Cotangent:
d dx
•
Cosecant:
d dx
•
Arccosine:
d (cos dx
−1 x) =
•
Arccotangent:
d (cot dx
− 1 x) =
•
Arccosecant:
d dx
a
f x dx
Definite integrals: concatenation:
f x dx
b
b
f (x) dx +
=
− sin x (cot x ) = − csc 2 x (csc x) = − csc x cot x − √ 1−1 − 1+1 (csc − 1 x) = − √ 1 −1
f (x) dx
≤
b
g (x) dx.
a
� � -substitutions: -substitutions: ( ) ( )
f g x g x dx =
f g x g x dx = F (g (x)) + C if
f (x) dx
a
b
a
2. Substitution Substitution Rule —a.k.a. u •
b
f (x) dx =
p
Definite integrals: comparison: If f (x) g (x) on the interval [a, b], then
� � ( ) ( )
f x dx
≤
1 x ln a
f (u) du
f (x) dx = F (x) + C .
3. I ntegration by Parts Best used to integrate a product when one factor ( u = f (x) ) has a simple derivative and the other factor (dv = g � (x) dx ) is easy to integrate.
x2
x
x
Indefinite Integrals: ( ) � ( ) = ( ) ( ) − � •
2
f x g x dx
x2
f x g x
f � (x)g (x) dx or
b
•
Definite Integrals:
u dv = uv
b
f (x)g (x) dx = f (x)g (x)]a
a
−
−
v du
b
f � (x)g (x) dx
a
4. Trigonomet Trigonometric ric Substitutions: Substitutions: Used to integrate expressions of the form
b
√ 2 2 ±a ± x .
f (x) dx is the signedarea between the function function y = f (x) and the
Expression
x-axis from x = a to x = b .
Formal definition: Let n be an integer and ∆x = b−na . For each k = 0, 2 , . . . , n 1 , n− 1 f (x∗k ) pick point x∗k in the interval [a + k ∆x, a + ( k + 1)∆ x] . The expression ∆x
− − ( ∗ ). ( ) is a Riemannsum. The definite integral is defined as lim ∆ →∞ Antiderivative: The function ( ) is an antiderivative of ( ) if �( ) = ( ) .
a2
− x2
x2
− a2
k =0
b
n
f x dx
n
a
1
x
k =0
f xk
Trig substitution
Expression becomes
x = a sin θ
Range of θ π
a cos θ
dx = a cos θ dθ x = a sec θ
F x
f x
Indefin Indefinite ite integra integral: l: The indefinite integral
F x
f x
x2 + a 2
f (x) dx represents a family of
π
π
a tan θ
π
dx = a sec θ tan θ dθ x = a tan θ dx = a sec2 θ dθ
Pythagorean identity used
− 2 ≤ θ ≤ 2 (−a ≤ x ≤ a ) 0≤θ< 2 π ≤ θ < 32
INDEFINITE INTEGRAL
•
n
· · · +2 f (x
a
= a x ln a
a
•
· · · + 2 f (x −1) + f (x
a
DEFINITE INTEGRAL
�(
p
INTEGRALS AND INTEGRATION
N
2
b
•
2
8. Inverse Trigonome Trigonometric tric
A . k C C r a L 5 L m 9 s e . e d 5 t a n 1 $ o r o i N t t k a 2 5 r d e i c a r 9 p e l . b 3 3 S t s u i y g P 4 $ b . e . e 3 d r C l 5 0 e a L L b v A 0 r o i s N 6 S 2 e s e U s s t © e t o & 7 e t r r a N s h h t s h k e 8 t g r n i d C h r g k a r e y i r p a 9 t r a S B p l n i o l r p f C A S o A 0 1 P
xk + x k+1
�
•
d (ln x) dx
•
f
5. Simpson’ Simpson’ss Rule: Rule: S n = ∆3x f (x0 )+ 4f (x1 )+ 2f (x2 )+
�
6. Logarithmic Base e : •
∆x 2
4. Trapezoid Trapezoidal al Rule: Rule: T n =
= nx n−1 (true for all real n = 0 )
d ( ex ) dx
•
9
f (xk )
TECHNIQUES OF INTEGRATION
=0
5. Exponen Exponentia tiall Base e : •
•
6 8 5 1 8 7
1
− k =0
1. Properties of Integrals • Sums and differences:
1. Consta Constants nts::
7. Trigonometric Sine: •
2 6 9 8 3 6
n
Rn = ∆ x
of x. Differentiate both sides of the equation with respect to x. Use the chain rule
COMMON DERIVATIVES
3 6 9 8 3 6 6 8 5 1
2. Right-hand rectangle approximation:
k =1
6. Implicit Implicit differentiatio differentiation: n: Used for curves when it is difficult to express y as a function
5 9 3 0 5
− F (a) .
k =0
dy dy du = du dx dx
f (t) dt
a
APP RO XIM ATI NG DE FI NI TE IN TE GR AL S
�
•
x
b
5. The Chain Chain Rule Rule
Part 2: If f (x) is continuous on the interval [a, b] and F (x) is any antiderivative of f (x),
f x g x
g
f (x) g (x)
d dx
g x
cf x
f x g x
f (x) dx = F (x) + C if F � (x) = f (x) .
FUNDAMENTAL THEOREM OF CALCULUS
DERIVATIVE RULES
1. Sum and and Diffe Differenc rence: e:
antiderivatives:
π
a sec θ
π
− 2 < θ < 2
1
− sin2 θ = cos2 θ
sec 2 θ
− 1 = tan2 θ
1 + tan2 θ = sec 2 θ
A P P L I C AT I O N S GEOMETRY
Volume of revolved solid (shell method):
b
Area:
�
f (x) if f
a
f (x)
− g (x) dx is the area bounded by y = f (x), y = g (x) , x = a and x = b
≥ g(x) on [a, b].
b
Volume of revolved solid (disk method): π
� ( ) f x
2
dx is the volume of the solid swept
a
out by the curve y = f (x) as it revolves around the x-axis on the interval [a, b]. b
Volume of revolved solid (washer method): π
� � ( ) − ( ) f x
2
g x
2
dx is the volume of
a
the solid swept out between y = f (x) and y = g (x) as they revolve around the x-axis on the interval [a, b] if f (x)
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. b
Arc length:
� � 1+ ( ) f x
2
dx is the length of the curve y = f (x) from x = a
a
to x = b .
b
Surface Surface area: area:
a
2πf (x)
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 .
≥ g (x ) . CONTINUED ON OTHER SIDE
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MOTION 1. Position s (t) vs. time t graph: • The slope of the graph is the velocity: s (t) = v (t ) . • The concavity of the graph is the acceleration: s (t) = a (t) .
N O I T I S O P
�
��
2. Velocity v (t) vs. time t graph: • The slope of the graph is the accleration: v (t ) = a (t) . • The (signed) area under the graph gives the displacement displacement (change in position):
+
�
s( t )
− s(0) =
Y T I C O L E V
v (τ ) dτ
0
N + O I T A R E L E C C A
0
–
PROBABILITY AND STATISTICS
•
= Average value of f (x) between a and b is f =
b
b
1
−a
f (x) dx .
CONTINUOUS DISTRIBUTION FORMULAS
∞
2.
−∞
≥ 0 for all x;
f (x) dx = 1 .
b a
f (x) dx xf (x) dx Expected value (a.k.a. expectation or mean) of X X : E (X ) = µ = � � x − E (x) f (x) dx = E (X ) − E (X ) Variance: Var( Variance: Var(X ) = σ = Standard deviation: Var(X ) = σ f (x) dx = f (x) dx = . Median m satisfies Cumulative density function (F (x) is the probability that X is at most x): ≤ x) = f (y ) dy F (x) = P (X ≤ Joint probability density function g (x, y ) chronicles distribution of X and Y . Then g(x, y) dy . f (x) = �x − E (X )�y − E (Y ) ) f (x, y) dxdy ) = σ = Covariance: Cov( Covariance: Cov(X, Y )
Probability that X is between a and b : P (a
•
≤ X ≤ ≤ b) =
X
2
∞
2
•
X
−∞
2
2
L = 2
X
m
•
∞
x
∞
−∞
∞
•
Correlation: ρ (X, Y ) =
∞
−∞
σXY Cov(X, Y ) = σX σY Var(X )Var( )Var(Y )
COMMON DISTRIBUTIONS 1. Normal distribution distribution (or Bell curve ) with mean µ and (x µ )2 1 variance σ : f (x) = e 2σ 2 σ 2π P (µ σ X µ σ ) = 68.3% • −
√
•
95%
• •
− ≤ ≤ ≤ − ≤ µ + 2 σ) = 95.5% P (µ − 2σ ≤ X ≤
•
2. χ-square distribution: with mean ν and variance 2ν : ν x 1 f (x) = ν x 2 1e 2 2 2 Γ ν 2 −
•
−
�
Gamma function: Γ( x) =
∞
x −1
t 0
dx .
σ
-1
0
1
2
MICROECONOMICS COST • Cost function C (x ) : cost of producing x units. • Marginal cost: C (x ) C (x ) : cost per unit when x units produced. • Average cost function C (x ) = x • Marginal average cost: C (x ) If the average cost is minimized, then average cost = marginal cost. • If C (x) > 0 , then to find the number of units (x ) that minimizes average cost, solve C x for x in (x ) = C (x) . �
�
��
�
REVENUE, PROFIT • Demand (or price ) function p (x ): price charged per unit if x units sold. • Revenue (or sales ) function: R (x ) = xp (x ) • Marginal revenue: R (x ) • Profit function: P (x ) = R (x ) − C (x ) • Marginal profit function: P (x ) If profit is maximal, then marginal revenue = marginal cost. • The number of units x maximizes profit if R (x ) = C (x ) and R (x ) < C (x ) . �
�
�
1
∂g 0 . ∂p >
P (t): the amount after t years. P 0 = P (0) : the original amount invested (the principal ). r : the yearly interest rate (the yearly percentage is 100r %).
INTEREST t Simple interest: P (t) = P 0 (1 + r ) • Compound interest Interest compounded m times a year: P (t) = P 0 1 + • Interest compounded continuously: P (t) = P 0 ert •
�
�
x
0
− f (x)
•
-2
e−t dt
x
FINANCE
68%
−
y = L( L( x) x) completely equitable distribution
X and Y are two commodities with unit price p and q , respectively. The amount of X demanded is given by f ( p, q ) . • The amount of Y demanded is given by g ( p, q ) . • 0 and 1. X and Y are substitute commodites (Ex: pet mice and pet rats) if ∂f ∂q > 2. X and Y are complementary commodities ( Ex: pet mice and mouse feed) ∂g if ∂f < 0 and ∂p < 0 . ∂q
•
−∞
x
x
SUBSTITUTE AND COMPLEMENTA COMPLEMENTATRY TRY COMMODITIES
−∞
XY
p = D( D( x) x)
The quantity L is between 0 and 1 . The closer L is to 1, the more equitable the income distribution.
•
•
� 0
1 2
m
−∞
( x, p x, p)
1
−∞
•
p = S( S( x) x)
≥
≤
∞
•
y 1
��
PRICE ELASTICITY OF DEMAND • Demand curve: x = x ( p ) is the number of units demanded at price p .
��
S T R A H C K R A P S
M T
LORENTZ CURVE The Lorentz Curve L(x) is the fraction of income received by the poorest x fraction of the population. 1. Domain Domain and range of L(x) is the interval [0, 1] . 2. Endpoints: L (0) = 0 and L(1) = 1 3. Curve is nondecreasing: L� (x) 0 for all x 4. L(x) x for all x Coefficient of Inequality (a.k.a. Gini Index ): •
Probability density function f (x) of the random v ariable X satisfies: 1. f (x)
−
a
X and Y are random variables. •
�
CONSUMER AND PRODUCER SURPLUS Demand function: p = D (x) gives price per unit • p ( p ) when x units demanded. • Supply function: p = S (x ) gives price per unit ( p ) when x units available. consumer ¯ units at price p¯. is x • Market equilibrium surplus p producer (¯x) .) (So p¯ = D (¯ x) = S (¯ surplus • Consumer surplus: x x ¯ ¯ ¯ ¯ CS = 0 D (x) dx − p ¯x¯ = 0 ( D (x) − p ¯) dx 0 • Producer surplus: ¯ ¯ ¯ ¯ x x PS = p ¯x¯ − 0 S (x) dx = 0 ( p¯ − S (x)) dx
–
−
−
�
t
3. Acceleration a (t) vs. time t graph: • The (signed) area under the graph gives the change t in velocity: v (t) v (0) = a(τ ) dτ
p x� ( p)
Price elasticity of demand: E ( p) = x ( p) Demand is elastic if E ( p) > 1 . Percentage change in p leads to larger percentage • change in x( p). Increasing p leads to decrease in revenue. Demand is unitary if E ( p) = 1 . Percentage change in p leads to similar percentage • change in x( p). Small change in p will not change revenue. Demand is inelastic if E ( p) < 1 . Percentage change in p leads to smaller • percentage change in x( p). Increasing p leads to increase in revenue. Formula relating elasticity and revenue: R ( p) = x ( p) 1 E ( p) • •
r mt m
EFFECTIVE INTEREST RATES The effective (or true) interest rate, reff , is a rate which, if applied simply (without compounding) to a principal, will yield the same end amount after the same amount of time. r m 1 Interest compounded m times a year: reff = 1 + m • r 1 Interest compounded continuously: reff = e •
�
−
−
PRESENT VALUE OF FUTURE AMOUNT The present value (PV ) of an amount (A) t years in the future is the amount of principal that, if invested at r yearly interest, will yield A after t years. mt r Interest compounded m times a year: PV = A 1 + m • rt PV = Ae Interest compounded continuously: •
�
−
−
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 yearly P 1 − (1+1r )t • Annuity paid for t years: PV = r Perpetuity: PV = P • r 2. Interest compounded continuously P (1 − e rt ) = erP 1 (1 − e rt ) • Annuity paid for t years: PV = reff P P Perpetuity: PV = reff = er 1 •
−
−
−
−
s m a i l l i W . O n g r y a e k D , b s v s l s d e o e i m i r d a e i n F v l l a h d i D a r e W t t a M . a S O : a M r n n : o n a n t i A D o i d
: t E : n a s r g r e t e i t i s i s r r e u l e l W D I S
N A C 5 9 . 5 $ 5 9 . 3 $ 3
3 6 9 8 3 3 9 5 0 2
7
