Introduction to Calculus Functional Notations

1. LIMITS
Chapter 1
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2. Brief History on Calculus:
• It is a branch of mathematics which
started to develop in the 17th century.
• Sir Isaac Newton and Wilhelm Leibniz
created calculus independently of each
other and about the same time.
CALCULUS 1 Samar College Galina V. Panela

3. Brief History on Calculus:
• The word calculus is derived from the
Latin word for stone or pebble.
• Calculus roughly means a “method of
calculation”
CALCULUS 1 Samar College Galina V. Panela

4. FUNCTIONAL NOTATION
Lesson 1.1
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5. Concept of Function:
• A function is a certain law of
correspondence,
• It is generally associated with a
formula.
• Like the formula of the area of the circle
𝐴 = 𝜋𝑟2
. We say that A is a function of r
and in symbol, we write 𝐴 = 𝑓(𝑟).
CALCULUS 1 Samar College Galina V. Panela

6. Concept of Function:
• In general, if there is a relation
between two variables x and y such
that each value of x corresponds to a
value of y, then we say, y is a function
of x.
𝑦 = 𝑓(𝑥)
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7. Concept of Function:
This concept may be extended to
relations between more than two
variables such that if
𝑧 = 𝑓 𝑥, 𝑦
then the value of z is determined when x
and y are given.
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8. Example:
Express the perimeter P of the
rectangle as a function of the length x
of one side if the area of the rectangle
is 10 square meters.
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9. Exercises:
Evaluate the function, 𝑓 𝑥 = 2𝑥2
− 4𝑥 + 2, if
• 𝑓 −10
• 𝑓 𝑦2 + 8
• 𝑓 𝑥 + ∆𝑥
• 𝑓 𝑥 + 1 − 𝑓(𝑥 − 1)
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10. Exercises:
Express the area of an equilateral triangle
as a function of its side x.
If 𝑓 𝑥 = 𝑥2 + 1, find
𝑓 𝑥+ℎ −𝑓(𝑥)
ℎ
, where h ≠ 0.
If f(𝑥)
4
𝑥+5
and 𝑔 𝑥 = 𝑥2
− 3, find f(g(x)) and
g(f(x)).
CALCULUS 1 Samar College Galina V. Panela