1. Not within mindmap
1.1. Format of estimation answer
1.1.1. POSITIVE Infinite Limit
1.1.2. NEGATIVE Infinite Limit
1.1.3. Limits from the LEFT
1.1.4. Limits from the RIGHT
1.2. Transformation of graph
2. Theorems
2.1. ONE SIDED LIMITS IFF lim f(x) = lim f(x) = lim f(x)
2.2. 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)
2.2.1. 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)
2.3. FUNCTIONS HAVE UNIQUE LIMITS If lim f(x) = K AND lim g(x) = L Then, K = L
3. Even and Odd Functions
3.1. Even : f(-x) = f(x)
3.2. Odd : f(-x) = -f(x)
4. Tangent
4.1. To have a tangent at a point of a graph, limit of tangent at that point must exist( Acc. to definition of the tangent to the graph) Therefore, , where the tangent line passes through ( , )
4.2. dy/dx = 0 , gradient = 0
4.2.1. Turnings points of curves
4.2.2. Points of Inflexions - Different from turnings points as the RHS and LHS of turning points goes opp directions ( 1 up 1 down )
5. Limit Laws
5.1. Sums : lim f(x) exists , lim g(x) exists Then, lim( f(x) + g(x) exists and = lim f(x) + lim g(x)
5.2. Products : lim f(x) exists , lim g(x) exists Then, lim f(x)g(x) exists and = lim f(x) * lim g(x)
5.3. Quotients : lim f(x) exists, lim g(x) exists and lim g(x) NOT EQUAL TO 0 Then,
5.4. Simple Functions : a,c ε R 1) lim c = c 2) lim x = a
5.5. Differences : lim f(x) exists, lim g(x) exists Then, lim ( f(x) - g(x) ) exists and = lim f(x) - lim g(x)
5.6. Roots : Let a ε R , a > 0 and n ε N , lim x ^ 1/n = a ^ 1/n
6. Laws
6.1. REPLACEMENT LAW ASSUME that : f(x) = g(x) , x NOT EQUAL TO a Then, 1) lim f(x) = lim g(x) OR 2) no limits exist
7. CONTINUITY
7.1. Definition of Continuous Function
7.1.1. a ε dom(f)
7.1.2. lim f(x) exists
7.1.2.1. Use ONE SIDED LIMITS TO PROVE (RHS = LHS)
7.1.3. lim f(x) = f(a)
7.2. Pass Limits into Continuous Functions (Theorem)
7.2.1. ASSUME that 1) lim g(x) exists
7.2.2. 2) f is continuous at lim g(x)
7.2.3. THEN lim f ( g(x) ) = f ( lim g(x) )
7.3. Compositions of Continuous Functions (Theorem)
7.3.1. ASSUME that 1 ) g(x) is continuous at a
7.3.2. 2) f(x) is continuous at g(a)
7.3.3. THEN : (F o G)(x) is continuous at a
7.4. Continuous on an interval , IVT THEOREM - F IS A FUNCTION - N,a,b ε R with a < b
7.4.1. ASSUME that : f is continuous on a closed interval [ a,b]
7.4.2. f(a) NOT EQUAL TO f(b)
7.4.3. N lies between f(a) and f(b)
7.4.4. THEN : There exists a number c s.t. 1) a < c < b 2) f(c) = N
8. Differentiable/ Derivatives
8.1. Definition of ' Derivative at a '
8.1.1. f'(a) = lim f( a+h) - f(a)
8.1.1.1. 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 '
8.1.1.1.1. Theorem : If a function f is differentiable at point a, it is continuous at a.
8.2. Useful rules
8.2.1. Power Rule
8.2.1.1. Let n ε N , and x,a ε R ,
8.2.1.2. THEN: (x-n)(x ) = x - a
8.2.2. Product Rule
8.2.2.1. ASSUME that : f(x) is differentiable at point a
8.2.2.2. g(x) is differentiable at a
8.2.2.3. THEN: f(x)g(x) is also differentiable at a
8.2.2.3.1. [ f(x)g(x) ] ' = f'(a)g(a) + f(a)g'(a)
9. L' Hopital Rule
9.1. SUPER IMPORTANT!!! ONLY APPLIES ON LIM 0/0
9.1.1. DIFFERENTIATE TOP AND BOTTOM INDIVIDUALLY (leave in the same form)
9.1.2. STATE THAT YOU ARE USING L'HOPITAL RULE
10. ROLLE'S THEOREM Note: - a and b are the EXTREME VALUES
10.1. Within closed interval [ a ,b ]
10.2. Continuous in [ a , b ]
10.2.1. USE ONE SIDED LIMITS TO CHECK, LHS = RHS
10.3. Differentiable in ( a , b )
10.3.1. Check whether dx/dy is defined at every point within interval
10.3.1.1. Basically dy/dx DENOMINATOR NOT EQUAL TO 0