Page 33 - Focus SPM 2022 - Additional Mathematics

Additional Mathematics SPM Chapter 2 Differentiation

2.1 Limit and its Relation to (b) Constructing Table
Differentiation x –0.1 –0.01 –0.001 0 0.001 0.01 0.1
f(x) –2.99 –2.9999 –2.999999 –3 –2.999999 –2.9999 –2.99
A Investigating and determining the
value of limit of a function when its The table above shows that when x → 0, f(x) → –3.
variable approaches zero Therefore 3
lim
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1. The limit value a of a function f(x) when the x → 0 1 x – 3x 2 = –3
x
variable x approaches zero is written as
Investigation through graph
lim f(x) = a x – 3x
3
2
x → 0 f(x) = = x – 3
x y
2. The investigation and determination of the limit
value is done through graph and by constructing Observe that when 0 x
a table. x → 0, f(x) → –3. Therefore
3
lim

3. The determination of limit value can also be x → 0 1 x – 3x 2 = –3 –3
x
done through direct substitution such that the
a
product of the value , (a = constant), which is Direct substitution
0
2
3
lim
lim

not defined cannot be accepted and the function x → 0 1 x – 3x 2 = x → 0 1 x(x – 3) 2
x
x
needs to be simplified to get the correct value. lim
= x → 0 (x – 3)
2
2
1 = (0) – 3
= –3
lim
Find the value of x → 0 f(x) if
(a) f(x) = x + 3, REMEMBER!
x – 3x
3
(b) f(x) = .
x Simplify the function first before doing direct substitution
Solution because lim x – 3x 0 – 3(0)
3
3
(a) Constructing Table x → 0 1 x 2 = 0
x –0.1 –0.01 –0.001 0 0.001 0.01 0.1 produces result which is not defined, that is 0 0 and this
is not the correct solution.
f(x) 2.9 2.99 2.999 3 3.001 3.01 3.1
Consider the values of f(x) when x approaching Try Questions 1 – 3 in ‘Try This! 2.1’
zero either from the left (negative value) or from
the right (positive value).
B Determining the first derivative
The table above shows that when x → 0, x + 3 → 3 of a function f(x) by using the first
either from the left (→) or from the right ( ). principle
Therefore,
1. Differentiation is about the changes of a variable
Form 5
lim with respect to the changes of another variable
x → 0 (x + 3) = 3
(normally time). Generally, differentiation is
Investigation through graph y the process of determining the instantaneous
Observe that when changes happen to a given variable at a particular
x → 0, f(x) → 3. Therefore, 3 moment.
lim
3
x → 0 (x + 3) = 3 x 2. Graphically, assuming that the graph f(x) = x
–3 0 representing the motion of an object, therefore
Direct substitution the gradient on the curve at a particular moment
lim
x → 0 (x + 3) = 0 + 3 t and t can be determined through the process
1
2
= 3 of diferentiation.
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