CALCULUS I - MAS 101
(DRAFT: ALL PICTURES AND SOME DETAIL STEPS MAY BE
OMITTED. IT MAY CONTAIN TYPOS!)
Contents
Part 1. Derivatives
1. Week 1 [ref. Sec 3.1, Lect 2, Textbook]
1.1. Introduction
1.2. Geometric interpretation of derivative
1.3. Physical interpretation of derivative: Rate of change
1.4. Limit and Continuity
Date: September 1, 2026.
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CALCULUS I
Part 1. Derivatives
1. Week 1 [ref. Sec 3.1, Lect 2, Textbook]
1.1. Introduction. What is a derivative?
• Geometrical interpretation
• Physical interpretation
• Important for any measurement (physics, ecomonic, ...etc)
How can we differentiate any function we possible know, e.g.,
d x tan−1 x
(e
)
dx
or
d x
(x ).
dx
1.2. Geometric interpretation of derivative.
Question 1.1. Find a tangent line to the graph of y = f (x) at the point P =
(x0 , y0 )?
The equation of tangent line is
y − y0 = m(x − x0 ),
where m is the slope of the line.
We have the geometric definition of the derivative as follows.
Definition 1.2. The derivative f ′ (x0 ) of f at x0 is the slope of the tangent line to
y = f (x) at P = (x0 , y0 ).
Remain: What is f ′ (x0 ) analytically and how to compute f ′ (x0 )?
Criterion of the tangent line:
(a) Not just a line meet the graph at one point.
(b) It is the limit of of the secant line P Q as Q → P (Q belongs to the graph!)
Pick another point Q = (x, y) = (x0 + ∆x, f (x0 , x0 + ∆x)) on the graph. The
slope of the secant line P Q is
∆f
f (x0 + ∆x) − f (x0 )
=
.
∆x
∆x
Now Q → P if and only if ∆x → 0 and the slope of P Q goes to the slope m = f ′ (x0 )
of the tangent line. That is
(1.1)
∆f
f (x0 + ∆x) − f (x0 )
= lim
.
∆x→0 ∆x
∆x→0
∆x
f ′ (x0 ) = lim
Remark 1.3. The process in (1.1) contains two steps.
1o ) compute the different quotient
∆f
,
∆x
this is the algebra part.
2o ) Find the limit of this quotient as ∆x → 0, this is the calculus part.
Example 1.4. Find the derivative of f (x) = x1 at a point x0 .
(1.2)
1
f ′ (x0 ) = − 2
x0
CALCULUS I
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Example 1.5. Show that the area of the triangle enclosed by the axes and the
tangent line to y = x1 in the first quadrant is a constant.
Notations. As y = f (x) we often write ∆y = ∆f = f (x0 + ∆x) − f (x0 ). We
use several notation for the derivative of f depending on the contexts.
• (Leibniz’s notation)
∆y
∆x
∆x→0
dy
.
dx
∆y
∆x
∆x→0
−→
f ′ (x).
f ′,
Df.
−→
• (Newton’s notation)
• (others)
df
,
dx
d
Example 1.6. Compute dx
(xn ), where n = 1, 2, ....
d n
x = nxn−1 for n = 1, 2, ...
dx
Automatically, we can find the derivative of all polynomials.
(1.3)
1.3. Physical interpretation of derivative: Rate of change. Many of quantities in real life is a function y = f (t) in time t. The different quotient
∆y
∆t
tells us about the average rate of change in ∆t, while the derivative f ′ (t0 ) tells us
the instantaneous rate at a given time t0 .
Example 1.7.
(a) y = s(t) the distance, then ds
dt is the speed.
(b) y = q(t) is the electric charge, then dq
dt is the current.
(c) y = T (t) the temperature, T ′ (t) the rate of change of the temperature.
(d) Many other such as population, stock market price, ...
To see the difference between the average rate of change and the instantaneous
rate we consider
Example 1.8. an object falls from the top of the building to the ground. H = 80m.
1.4. Limit and Continuity. The simple process of taking limit in Example 1.5
is as follows:
−1
−1
= 2
lim
∆x→0 x(x + ∆x)
x
which means that we just simply ”plugging” ∆x = 0 in the formula. Similarly,
x+3
7
lim
=
.
x→4 x2 + 1
17
The above two examples are ”easy limits”. However, in derivative formula
f (x0 + ∆x) − f (x0 )
∆x→0
∆x
lim
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CALCULUS I
we cannot do that because the undefined form 00 will appear. This is a ”hard limit”
one. Most of the time, we find the way to converse the hard limit become the easy
one as we already saw in Example 1.6.
The abstract writing for the limit is
(1.4)
lim f (x) = L.
x→c
Equivalently saying, the value of f (x) approaches the limit L as x approaches c.
Example 1.9. Consider the function
(
x+1
f (x) =
−x + 2
What is the limit limx→0 f (x) =?
for x > 0,
for x < 0.