# Calculus

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Calculus

## 2. CONTINUITY

### 2.1. Definition of Continuous Function

2.1.1. a ε dom(f)

2.1.2. lim f(x) exists

2.1.2.1. Use ONE SIDED LIMITS TO PROVE (RHS = LHS)

2.1.3. lim f(x) = f(a)

### 2.2. Pass Limits into Continuous Functions  (Theorem)

2.2.1. ASSUME that 1) lim g(x) exists

2.2.2. 2)  f is continuous at lim g(x)

2.2.3. THEN lim f ( g(x) ) = f ( lim g(x) )

### 2.3. Compositions of Continuous Functions  (Theorem)

2.3.1. ASSUME that 1 ) g(x) is continuous at a

2.3.2. 2) f(x) is continuous at g(a)

2.3.3. THEN : (F o G)(x) is continuous at a

### 2.4. Continuous on an interval , IVT THEOREM           - F IS A FUNCTION                                         - N,a,b ε R with a < b

2.4.1. ASSUME that : f is continuous on a closed interval [ a,b]

2.4.2. f(a) NOT EQUAL TO f(b)

2.4.3. N lies between f(a) and f(b)

2.4.4. THEN : There exists a number c s.t.                                                                                                           1) a < c < b                                                     2) f(c) = N

## 3. Differentiable/ Derivatives

### 3.1. Definition of ' Derivative at a '

3.1.1. f'(a) =  lim f( a+h) - f(a)

3.1.1.1. IF LIMIT EXISTS, f(a) IS DIFFERENTIABLE AT a, if limits DNE, DERIVATIVE IS NOT DEFINED AT a       NOTE:                                                          THIS ACTUALLY ALSO MAKES UP THE DEFINITION OF ' THE DERIVATIVE FUNCTION '

3.1.1.1.1. Theorem : If a function f is differentiable at point a, it is continuous at a.

### 3.2. Useful rules

3.2.1. Power Rule

3.2.1.1. Let n ε N , and x,a ε R ,

3.2.1.2. THEN:                                                              (x-n)(x                             ) = x   - a

3.2.2. Product Rule

3.2.2.1. ASSUME that :  f(x) is differentiable at point a

3.2.2.2. g(x) is differentiable at a

3.2.2.3. THEN: f(x)g(x) is also differentiable at a

3.2.2.3.1. [ f(x)g(x) ] ' = f'(a)g(a) + f(a)g'(a)

## 4. L' Hopital Rule

### 4.1. SUPER IMPORTANT!!! ONLY APPLIES ON LIM 0/0

4.1.1. DIFFERENTIATE TOP AND BOTTOM INDIVIDUALLY (leave in the same form)

4.1.2. STATE THAT YOU ARE USING L'HOPITAL RULE

## 5. ROLLE'S THEOREM                                                                                                            Note:                                                          - a and b are the EXTREME VALUES

### 5.2. Continuous in [ a , b ]

5.2.1. USE ONE SIDED LIMITS TO CHECK, LHS = RHS

### 5.3. Differentiable in ( a , b )

5.3.1. Check whether dx/dy is defined at every point within interval

5.3.1.1. Basically dy/dx DENOMINATOR NOT EQUAL TO 0

## 7. Tangent

### 7.2. dy/dx = 0 , gradient = 0

7.2.1. Turnings points of curves

7.2.2. Points of Inflexions                                         - Different from turnings points as the RHS and LHS of turning points goes opp directions ( 1 up 1 down )

## 10. Theorems

### 10.2. COMPARISON THEOREM                                                                                                               lim f(x) exists, lim g(x) exists ,                             f(x) < g(x) for all x in the neighbourhood of a except for a itself                                                                                                                        Then,    lim f(x) < lim g(x)

10.2.1. SQUEEZE THEOREM                                                                                                                                                                                                                                                                                       lim g1(x) exists , lim g2(x) exists,                                                                                                           lim g1(x) = g2(x)                                                                                                                              g1(x) < f(x) < g2(x) for all x in the neighbourhood of a except for a itself                                                                                                                                                                                                     Then, 1) limf(x) exists                                                                                                                                      2) lim f(x) = lim g1(x) = lim g2(x)

## 11. Not within mindmap

### 11.1. Format of estimation answer

11.1.1. POSITIVE Infinite Limit

11.1.2. NEGATIVE  Infinite Limit

11.1.3. Limits from the LEFT

11.1.4. Limits from the RIGHT