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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).
d f (x) ± g(x) = f � (x) ± g � (x) 1. Sum and Difference: dx � � d cf (x) = cf � (x) 2. Scalar Multiple: dx � � d f (x)g(x) = f � (x)g(x) + f (x)g � (x) 3. Product: dx
�
Mnemonic: If f is “hi” and g is “ho,” then the product rule is “ho d hi plus hi d ho.” � � � (x)g � (x) f (x) d = f (x)g(x)−f (g(x)) 2 g(x) dx
4. Quotient:
Mnemonic: “Ho d hi minus hi d ho over ho ho.”
5. The Chain Rule
• First formulation: (f ◦ g)� (x) = f � (g(x)) g � (x) dy du dy = du • Second formulation: dx dx
dy dx
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1. Left-hand rectangle approximation: n−1 � Ln = ∆x f (xk )
dx dx
y) dy = 3 dx − 2y dx cos y + x d(cos , dx dx
and then cos y − x(sin y)y � − 2yy � = 3. Finally, solve for y � =
cos y−3 . x sin y+2y
Mn = ∆x
=0
2. Linear:
d (mx dx
+ b) = m
3. Powers:
d (xn ) dx
= nx n−1 (true for all real n �= 0)
4. Polynomials:
d (an xn dx
+ · · · + a2 x + a1 x + a0 ) = an nx
6. Logarithmic • Base e:
d (ln x) dx
d (sin dx
x
1 x
x) = cos x
d (tan dx
d (sec dx
=
x) = sec 2 x
x) = sec x tan x
n−1
• Arbitrary base:
d (ax ) dx
5. Simpson’s Rule: Sn =
f (x0 )+4f (x1 )+2f (x2 )+· · ·+2f (xn−2 )+4f (xn−1 )+f (xn )
� ∆x 3
�
�
�
�
• Arbitrary base:
d (log a dx
• Cosine: • Cotangent: • Cosecant:
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
1 1+x2
• Arccotangent:
1 x) = − √1−x 2
x) =
d (cot −1 dx
• Arcsecant:
d (sec −1 dx
x) =
√1 x x2 −1
• Arccosecant:
1 x) = − 1+x 2
d (csc −1 dx
x) = − x√x12 −1
INTEGRALS AND INTEGRATION
�
�
b
f (x) dx = −
a
p
f (x) dx + a
• Definite integrals: comparison: If f (x) ≤ g(x) on the interval [a, b], then
1 x ln a
d (tan −1 dx
√ 1 1−x2
• Definite integrals: reversing the limits:
x
d (sin −1 dx
�
�
a
f (x) dx b
b
f (x) dx = p
b a
�
f (x) dx ≤
�
�
b
f (x) dx a
b
g(x) dx. a
� � � 2. Substitution Rule—a.k.a.