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Calculus Final Exam Review. By: Bryant Nelson. Common Trigonometric Values. sin(h) h. cos(h) - 1 h. lim h 0. lim h 0. = ?. = ?. Special Trigonometric Limits. 1. 0. f(x+ Δ x) – f(x) Δ x. lim Δ x 0. Differentiation Rules. Definition of a Derivative:. f’(x) = .

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special trigonometric limits

sin(h)

h

cos(h) - 1

h

lim

h0

lim

h0

= ?

= ?

Special Trigonometric Limits

1

0

differentiation rules

f(x+Δx) – f(x)

Δx

lim

Δx0

Differentiation Rules

Definition of a Derivative:

f’(x) =

differentiation rules cont
Differentiation Rules cont.

Product Rule:

(f·g)’ = f’·g + f·g’

Quotient Rule:

(f/g)’ = (f’·g - f·g’)/g2

Natural Log Rule:

d/dx(ln(u)) = (1/u)·du/dx

Exponential Rules:

d/dx(℮u) = (℮u)·du/dx

·

d/dx(bu) = (bu) ·ln(b)·du/dx

differentiation rules cont6
Differentiation Rules cont.

Trigonometric Rules

d/dx(sin(u)) =

(cos(u))du/dx

d/dx(cos(u)) =

(-sin(u))du/dx

d/dx(tan(u)) =

(sec2(u))du/dx

d/dx(cot(u)) =

(-csc2(u))du/dx

d/dx(sec(u)) =

(sec(u)tan(u))du/dx

d/dx(csc(u)) =

(-csc(u)cot(u))du/dx

differentiation rules cont7
Differentiation Rules cont.

Inverse Trigonometric Rules

d/dx(sin-1(u)) =

(1/(√1-u2))du/dx

d/dx(cos-1(u)) =

(-1/(√1-u2))du/dx

d/dx(tan-1(u)) =

(1/(1+u2))du/dx

d/dx(cot-1(u)) =

(-1/(1+u2))du/dx

d/dx(sec-1(u)) =

(1/(|u|·√u2-1))du/dx

d/dx(csc -1(u)) =

(-1/(|u|·√u2-1))du/dx

integration rules
Integration Rules

Power Rules:

∫(un·du) = (un+1)/(n+1) +C; only while n ≠ -1

∫(u-1·du) = ln(|u|) +C

Exponential Rules:

∫(℮u·du) = ℮u +C

∫(bu·du) = bu/ln(b) - u +C, b≠1

Logarithmic Rule:

∫(ln(u)·du) = u·ln(u) - u +C, u>0

integration rules cont
Integration Rules cont.

Trigonometric Rules

∫(sin(u)·du) =

-cos(u) + C

∫(cos(u) ·du) =

sin(u) + C

∫(tan(u) ·du) =

-ln(|cos(u)|) + C

∫(cot(u) ·du) =

ln(|sin(u)|) + C

∫(sec(u) ·du) =

ln(|sec(u) + tan(u)|) + C

∫(csc(u) ·du) =

ln(|csc(u) + cot(u)|) + C

integration rules cont10
Integration Rules cont.

Trigonometric Rules cont.

∫(sec2(u) ·du) =

tan(u) + C

∫(csc2(u) ·du) =

-cot(u) + C

∫(sec(u)tan(u) ·du) =

sec(u) + C

∫(csc(u)cot(u) ·du) =

-csc(u) + C

summation formulas

n

n

n

n

n

n

c·ak =

k2 =

ak

k3 =

k =

1 =

K=1

K=1

K=1

K=1

K=1

K=1

Summation Formulas

n

(n(n+1))/2

(n(n+1)(2n+1))/6

(n2(n+1)2)/4