Rafia Shabbir

February 25, 2020

Chapters

 

Limit Formulas

 

 

 

 

 

 

 

 

    \[\lim _ {x \rightarrow a} k = k\]

 

    \[\lim _ {x \rightarrow a} [ f (x) \pm g (x) ] = \lim _ { x \rightarrow a} f(x) \pm \lim _ { x \rightarrow a} g(x)\]

 

    \[\lim _ {x \rightarrow a} [ f (x) \cdot g (x) ] = \lim _ { x \rightarrow a} f(x) \cdot \lim _ { x \rightarrow a} g(x)\]

 

    \[\lim _ {x \rightarrow a} \frac { f(x) } { g (x)} = \frac { \lim_ {x \rightarrow a} f(x) } { \lim _{x \rightarrow a} g(x) } if \lim _ {x \rightarrow a} g(x) \neq 0\]

 

    \[\lim_ {x \rightarrow a} [f (x) ^ {g (x)}] = \lim_ {x \rightarrow} [f(x) ] ^ {\lim _ {x \rightarrow a} g(x)} if f(x) >0\]

 

    \[\lim_ {x \rightarrow a} g[f (x)] = g [ \lim _ {x \rightarrow a} f(x) ]\]

g can be a root, a log, sin, cos, tan, etc.

 

    \[\lim_ {x \rightarrow a} \sqrt [n] {f (x)} = \sqrt [n] { \lim_ {x \rightarrow a} f(x)}\]

 

    \[\lim _ {x \rightarrow a} [log _ a f(x)] = log _a [\lim_ {x \rightarrow a} f(x) ] if a >0 and f(x) > 0\]

 

 

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L' Hospital Rule

    \[\lim _ { x \rightarrow a} \frac {f(x)} {g (x)} = \lim _ { x \rightarrow a} \frac {f ' (x)} {g ' (x)}\]

Continuity Formulas

Continuous Function at a Point

    \[\ni f (a)\]

 

    \[\ni \lim _ {x \rightarrow a} f(x) \leftrightarrow \lim _ {x \rightarrow a ^ {-}} f(x) = \lim_ { x \rightarrow a^ {+}} f(x)\]

 

    \[f(a) = \lim {x \rightarrow a} f(x)\]

Directional Continuity

Left-Continuous Function

    \[f(a) = \lim _ {x \rightarrow a ^ {-}} f(x)\]

Right-Continuous Function

    \[f(a) = \lim _ {x \rightarrow a ^ {+}} f(x)\]

 

Discontinuity

Removable Discontinuity

    \[∃ f (a) or f(a) \neq \lim _ {x \rightarrow a} f(x)\]

Jump Discontinuity

    \[\lim _ {x \rightarrow a ^ {-}} f(x) \neq \lim _ {x \rightarrow a ^ {+}} f(x)\]

Essential Discontinuity

 

 

Derivative Formulas

u and v are functions of x.

a, e and k are constants (real numbers)

f(x) = k                                         f ' (x) = 0

 

f(x) = ku                                          f ' (x) = k u'

 

f(x) = x                                           f ' (x) = 1

 

f(x) = u ^ k                                    f ' (x) = k \cdot u ^ {k - 1} \cdot u'

 

f(x) = \sqrt{u}                              f ' (x) = \frac { u'} {2 \cdot \sqrt{u}}

 

f(x) = \sqrt [k] {u}                       f ' (x) = \frac { u '} {k \cdot \sqrt [k] {u ^ {k -1}}}

 

f(x) = u \pm v                               f ' (x) = u ' \pm v'

 

f(x) = u \cdot v                             f ' (x) = u ' \cdot v + u \cdot v'

 

f(x) = \frac {u} {v}                       f ' (x) = \frac {u' v - u v'} {v ^ 2}

 

f(x) = \frac {k } {v}                      f ' (x) = \frac {-k \cdot v ^2} { v ^2}

 

f(x) = \frac {1} {v}                       f ' (x) = \frac {- v '} {v ^2}

 

f(x) = a ^ u                                    f ' (x) = u' \cdot a ^ u\cdot lna

 

f(x) = e ^ u                                    f' (x) = u' \cdot e ^ u

 

f (x) log_a u                                  f ' (x) = \frac {u '} {u \cdot lna} = \frac {u '} {u} \cdot log _a e = \frac {u'}{u} \cdot \frac {1} {lna}

 

f(x) = lnu                                        f ' (x) = \frac {u'} {u}

 

f (x) = sin u                                    f ' (x) = u ' \cdot cos u

 

f(x) = cos u                                    f ' (x) = - u' \cdot sin u

 

f(x) = tan u                                     f ' (x) = \frac {u '} {cos^2 u} = u ' \cdot sec ^2 u = u ' \cdot (1 + tan ^2 u)

 

f(x) = cot u                                    f ' (x) = - \frac {u '} {sen ^2 u } = - u' \cdot c sc ^2 u = - u' \cdot (1 + cot ^2 u)

 

f(x) = sec u                                   f ' (x) = \frac {u ' \cdot sin u} { cos ^2 u} = u ' \cdot sec u \cdot tan u

 

f(x) = csc u                                    f ' (x) = - \frac {u ' \cdot cos u} {sin ^2 u} = - u' \cdot csc u \cdot cot u

 

f(x) = arcsin u                              f ' (x) = \frac { u'} { \sqrt {1 - u ^2}}

 

f(x) = arccos u                             f ' (x) = - \frac {u '} { \sqrt {1 - u ^2}}

 

f(x) = arctan u                             f ' (x) = \frac { u '} { 1 = u ^2}

 

f (x) = arc cot u                            f ' (x) = - \frac {u '} { 1 + u ^2}

 

f(x) = arc sec u                          f ' (x) = \frac { u '} {u \cdot \sqrt {u ^2 - 1}}

 

f(x) = arocsc u                            f ' (x) = - \frac {u '} {u \cdot \sqrt {u ^2 - 1}}

 

f(x) = u ^v                                    f ' (x) = v \cdot u ^ {v - 1} \cdot u ' + u ^v \cdot v' \cdot ln u

 

Chain Rule

(g o f) ' (x) = g ' [f(x)] \cdot f ' (x)

Derivative of an Implicit Function

y' = \frac { F ' _x} {F ' _y}

Integrals Formulas

\int dx = x + C

 

\int k dx = k \cdot x + C

 

\int u ^n \cdot u ' dx = \frac { u ^ {n + 1}} {n + 1} + C                n \neq -1

 

\int \frac {u '} {u} dx = ln u + C

 

\int a ^ u \cdot u ' dx = \frac { a ^ u} { ln a} + C

 

\int e ^ u \cdot u' dx = e ^ u + C

 

\int sin u\cdot u ' dx = -cos u + C

 

\int cos u \cdot u ' dx = sin u + C

 

\int \frac { u ' } { cos ^2 u } dx = \int sec ^2 u \cdot u ' dx = \int (1 + tan^2 u) \cdot u ' dx = tan u + C

 

\int \frac { u '} {sen ^2 u} dx = \int cosec ^ 2 u \cdot u ' dx = \int (1 + cot g ^2 u) \cdot u ' dx = -cotg u + C

 

\int \frac { u ' } { \sqrt { 1 - u ^2}} dx = arcsin u + C

 

\int \frac {u ' } { 1 + u ^2} dx = arc tan u + C

 

Integration by Parts

\int u \cdot v ' dx = u \cdot v - \int u ' \cdot v dx

 

Integration by Substitution

\int f ' (u) \cdot u' dx = F (u) + C

 

Change of Variables

\int R (x , \sqrt {a ^2 - x ^2}) dx                                                    x =a sin t

\int R (x , \sqrt {a ^2 + x ^2}) dx                                                   x =a tan t

\int R (x , \sqrt {x ^2 - a ^2}) dx                                                    x =a sec t

\int R ( x , \sqrt [n] {\frac {ax + b} {cx + d}}) dx                         t = \frac {ax + b} {cx - d}

If R( sinx , cos x) is even

Change                    t = tan x

sin x                         \frac {t} {\sqrt {1 + t ^2}}

cos x                         \frac {1} {\sqrt {1 + t ^2}}

tan x                               t

dx                              \frac {dt} {\sqrt {1 + t ^2}}

 

If R( sinx , cos x) is not even:

Change                      tg\frac {x}{2}

sen x                           \frac {2t} {1 + t ^2}

cos x                            \frac {1 - t ^2} {1 + t ^2}

tgx                                \frac {2t} {1 - t ^2}

dx                                 \frac {2dt} {1 + t ^2}

Definite Integral

\int _{a} ^ {b} f(x) = [F (x) ]_{a} ^{b} = F (b) - F(a)

 

\int _ {a} ^ {b} f(x) dx = - \int _{b} ^ {a} f(x) dx

 

\int _ {a} ^ {a} f(x) dx = 0

 

\int _ {b} {a} f(x) dx = \int_ {c} ^ {a} f(x) dx + \int _ {c} ^ {b} f(x) dx

 

\int _{a} ^ {b} [ f (x) + g(x) ^ 1 dx = \int _ {a} {b} f(x) dx + \int _ {a} ^ {b} g(x) dx

 

\int _ {a} ^ {b} \cdot f(x) dx = k \cdot \int _ {a} ^ {b} f(x) dx

 

\int _ {a} ^ {b} f(x) dx = (b - a) \cdot f(c)

 

A = \int_ {a} ^ {b{ f(x) dx

 

V = \pi \int_ {a} ^ {b} [ f (x) ] ^2 dx

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Rafia Shabbir