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Summary Calculus gr 12 IEB & GDE notes R115,00   Add to cart
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Summary Calculus gr 12 IEB & GDE notes

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Detailed notes of the Grade 12 Calculus section Including: Discontinuities Limits (examples included) Steps on differentiation using first principles Rules for differentiation Tangents to a curve Cubic functions Graph interpretation Curve sketching Optimisation (4 pages of examp...

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  • June 4, 2022
  • 12
  • 2020/2021
  • Summary

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Calculus

, lim


Differential Calculus
x -> 0

lim
h -> 0




Enables us to:
Work out the gradient at a point on a graph Sketch cubic graphs
Discontinuities
lim f(x) exists if xlim f(x) = xlim f(x)
Two types of discontinuities in graphs x -> a -> a- -> a+


Removable discontinuity Jump discontinuity Piecewise/split function




2 3




/
1 2


1 -1 1
-1


lim f (x) = b
lim f(x) = 3 lim- f(x) = 2
x -> a

lim lim the limit as x tends
x -> 1+ f(x) = 2 x -> 1 - f(x) = 2 to a is equal to b x -> 1+ x -> 1


f(1) ≠ 2, the limit exists Limit only exists if lim f(x) ≠ lim f(x)
x -> 1- x -> 1+
lim as x -> a from left & right
the limit does not exist
x -> 1 f(x) = 2 but f(1) ≠1 f (x) tends to same value b


Limits
Determine the following limits: - Sub x value into the function
3x
lim (x + 5) lim lim - if 0/0 factorise and cancel!
x -> -4 x -> 3 (2/3x +3) x -> 0 x
3x
lim - don’t drop lim until sub x in
= -4 + 5 = 2/3(3) + 3 x -> 0 x

=1 =5 =3

x2- 1 x-4
lim lim x2- 4 lim 2x2- x lim
x -> 1 x-1 x -> 2 x2- 3x + 2 x -> 0 x x -> 4 x-2
lim (x - 1) (x + 1) lim (x + 2) (x - 2) lim x(2x -1) lim ( x - 2) ( x + 2)
x -> 1 (x - 1) x -> 2 (x - 1) (x - 2) x -> 0 x x -> 4 ( x - 2)
lim (x + 1) lim (x + 2) lim (2x - 1) lim ( x + 2)
x -> 1 x -> 2 (x - 1) x -> 0 x -> 4


=1+1 = 4/2 = -1 =4
=2 =2

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