This document provides an overview of calculus concepts including derivatives, integrals, and their applications in physics. It defines key terms like singularity, differentiation, integration, and discusses notation for derivatives. It also covers derivatives and integrals of basic functions, applications to physics concepts like velocity, acceleration, and Newton's Second Law, as well as examples of calculating derivatives and integrals.

1. Beginning Direct3D Game Programming:
Mathematics 4
Calculus
jintaeks@gmail.com
Division of Digital Contents, DongSeo University.
March 2016

2. Singularity
 In mathematics,
a singularity is in general a
point at which a given
mathematical object is not
defined, or a point of an
exceptional set where it fails
to be well-behaved in some
particular way, such
as differentiability.
 For example, f(x) is singular
when x≡0.
2

3.  The function f(x) = |x| also has a singularity at x= 0, since it is
not differentiable there.
3

4.  The graph defined by y2 = x also has a singularity at (0,0), this
time because it has a "corner" (vertical tangent) at that point.
4

5. Differentiation
5
f(x)=x2

6. Differentiation
 The derivative of a function of a real variable measures the
sensitivity to change of a quantity (a function value
or dependent variable) which is determined by another
quantity (the independent variable).
 Derivatives are a fundamental tool of calculus.
 For example, the derivative of the position of a moving object
with respect to time is the object's velocity: this measures how
quickly the position of the object changes when time is
advanced.
6

7.  The slope m of the secant line is the difference between
the y values of these points divided by the difference between
the x values, that is:
7

8. 8

9. 9

10. Practice
 Write a function that calculate the derivative of x2.
 For example, FPrime( float x ) returns a f'(x2).
 Print the tangent line at (3.0, f(3.0)).
10

11. Get the slope of (3,9) of y=x2
11

12. 𝑦 = 𝑥2
(9 + 𝑑𝑞) = (3 + 𝑑𝑝)2
9 + 𝑑𝑞 = 9 + 6𝑑𝑝 + 𝑑2
𝑝2
𝑑𝑞 = 6𝑑𝑝 + 𝑑𝑑𝑝2
𝑞 = 6𝑝 + 𝑑𝑝2
12

13. 𝑞 = 6𝑝 + 𝑑𝑝2
𝑞/𝑝 = 6 + 𝑑𝑝
(dp is infinitely small, so)
𝑞/𝑝 = 6
13

14. Get the slope of (a,a2) of y=x2
14

15. 𝑦 = 𝑥2
(𝑎2
+ 𝑑𝑞) = (𝑎 + 𝑑𝑝)2
𝑎2 + 𝑑𝑞 = 𝑎2 + 2𝑎𝑑𝑝 + 𝑑2 𝑝2
𝑑𝑞 = 2𝑎𝑑𝑝 + 𝑑𝑑𝑝2
𝑞 = 2𝑎𝑝 + 𝑑𝑝2
15

16. derivative of f(y)=x2
𝑞 = 2𝑎𝑝 + 𝑑𝑝2
𝑞/𝑝 = 2𝑎 + 𝑑𝑝
(dp is infinitely small, so)
𝑞/𝑝 = 2𝑎
16
dy/dx=F'(x2)=2x

17. Rules for basic functions
Derivatives of powers:
if
where r is any real number, then
wherever this function is defined. For example, if
then
17

18. Remember
18
𝑓′
𝑥 𝑛
= 𝑛𝑥 𝑛−1

19. Leibniz's notation
 The notation for derivatives introduced by Gottfried Leibniz is
one of the earliest.
 It is still commonly used when the equation y = f(x) is viewed
as a functional relationship between dependent and
independent variables. Then the first derivative is denoted by
 Higher derivatives are expressed using the notation
 for the nth derivative of y = f(x) (with respect to x).
19

20. Lagrange's notation
 Sometimes referred to as prime notation, one of the most
common modern notation for differentiation is due to Joseph-
Louis Lagrange and uses the prime mark, so that the
derivative of a function f(x) is denoted f′(x) or simply f′.
 Similarly, the second and third derivatives are denoted like
below.
20

21. Newton's notation
 Newton's notation for differentiation, also called the dot
notation, places a dot over the function name to represent a
time derivative. If y = f(t), then below expression denote the
first derivatives of y with respect to t.
𝑦
 The second derivatives of y with respect to t can be denoted
like this.
𝑦
21

22. Application: Tangent vector
 A Tangent space is a real vector space that intuitively
contains the possible "directions" at which one can
tangentially pass through x.
 The elements of the tangent space are called tangent
vectors at x.
22

23. 23
 A pictorial representation of the tangent space of a single
point, x, on a sphere.

24. Normal mapping
 To calculate the Lambertian (diffuse) lighting of a surface, the
unit vector from the shading point to the light source(L)
is dotted with the unit vector normal(N) to that surface, and
the result is the intensity of the light on that surface.
• Example of a normal map (center) with the scene it was
calculated from (left) and the result when applied to a flat
surface (right).
24

25. Gradient: ∇(read as del or gradient)
 In mathematics, the gradient is a generalization of the usual
concept of derivative of a function in one dimension to a
function in several dimensions. If f(x1, ..., xn) is
differentiable, scalar-valued function of standard Cartesian
coordinates in Euclidean space, its gradient is
the vector whose components are the n partial
derivatives of f.
25
The Derivation can be
extended to more higher
dimensions!

26. • In the above two images, the values of the function are represented in
black and white, black representing higher values, and its corresponding
gradient is represented by blue arrows.
26

27. • The gradient of the function f(x,y) = −(cos2x + cos2y)2 depicted as a
projected vector field on the bottom plane.
27

28.  The gradient (or gradient vector field) of a scalar
function f(x1, x2, x3, ..., xn) is denoted ∇f or where ∇ denotes
the vector differential operator, del. The notation "grad(f)" is
also commonly used for the gradient.
 In a rectangular coordinate system, the gradient is the vector
field whose components are the partial derivatives of f:
28

29.  In the three-dimensional Cartesian coordinate system, this is
given by
 where i, j, k are the standard unit vectors.
 For example, the gradient of the function
 is:
29

30. Integral
 A definite integral of a function can be represented as the
signed area of the region bounded by its graph.
30

31. Integral
 In mathematics, an integral assigns numbers to functions in a
way that can describe displacement, area, volume, and other
concepts that arise by combining infinitesimal data.
 Integration is one of the two main operations of calculus, with
its inverse, differentiation, being the other.
 Given a function f of a real variable x and an interval [a, b] of
the real line, the definite integral
 is defined informally as the signed area of the region in the
xy-plane that is bounded by the graph of f, the x-axis and the
vertical lines x = a and x = b.
31
𝑎
𝑏
𝑓 𝑥 𝑑𝑥

32. Observation
 A plane is the collection of infinite lines.
 Similarly, a cube is the collection of infinite planes.
 Cavalieri's principle
32

33. Practice
33
 Write a function that calculate the integration 0
2
𝑓 𝑥2
𝑑𝑥
 Approximations to integral of
√x from 0 to 1, with
5 ■ (yellow) right endpoint
partitions and 12 ■ (green)
left endpoint partitions

34. Remember
34
𝑎
𝑏
𝑥 𝑛
𝑑𝑥 =
1
𝑛 + 1
𝑥 𝑛+1
][ a
b

35. Calculus
 The operation of integration is the reverse of differentiation.
 For this reason, the term integral may also refer to the related
notion of the antiderivative, a function F whose derivative is
the given function f.
 In this case, it is called an indefinite integral and is written:
35
constant of integration
𝑥 𝑛
𝑑𝑥 =
1
𝑛 + 1
𝑥 𝑛+1
+ 𝐶

36. Calculus
36
constant of integration
𝑥 𝑛 𝑑𝑥 =
1
𝑛 + 1
𝑥 𝑛+1 + 𝐶

37. Calculus
 The fundamental theorem of calculus that connects
differentiation with the definite integral: if f is a continuous
real-valued function defined on a closed interval [a, b], then,
once an antiderivative F of f is known, the definite integral
of f over that interval is given by
37

38. 38
F'(x2)=2x
2𝑥 𝑑𝑥 = 𝑥2
+ 𝐶

39. 39
𝑛𝑥 𝑛−1
𝑑𝑥 = 𝑥 𝑛
+ 𝐶
𝑓′
𝑥 𝑛
= 𝑛𝑥 𝑛−1

40. 40
𝑛𝑥 𝑛−1
𝑥 𝑛
Differentiation Integration

41. Physics
 The velocity of an object is the rate of change of
its position with respect to a frame of reference, and is a
function of time.
41
• Kinematic quantities of a classical particle: mass m,
position r, velocity v.

42. Instantaneous velocity
 Velocity is defined as the rate of change of position with
respect to time, which may also be referred to as the
instantaneous velocity to emphasize the distinction from the
average velocity.
42
S(t) = S0 + v0t
S=vt
S0 v0
v0t

43. Instantaneous velocity
 If we consider v as velocity and x as the displacement (change
in position) vector, then we can express the (instantaneous)
velocity of a particle or object, at any particular time t, as
the derivative of the position with respect to time:
43

44. Acceleration
 Acceleration, in physics, is the rate of change of velocity of an
object.
 An object's acceleration is the net result of any and
all forces acting on the object, as described by Newton's
Second Law.
44
v(t) = v0 + at
v=at
v0
v1
a=(v1 – v0)/t

45. Relationship between velocity and acceleration
 An object's instantaneous acceleration at a point in time is
the slope of the line tangent to the curve of a v vs. t graph
at that point.
 In other words, acceleration is defined as the derivative of
velocity with respect to time:
45

46.  Example of a velocity vs. time graph, and the relationship
between velocity v on the y-axis, acceleration a (the three
green tangent lines represent the values for acceleration at
different points along the curve) and displacement s (the
yellow area under the curve.)
46

47. Distance, Velocity and Acceleration
 From bottom to top:
• an acceleration
function a(t);
• the integral of the
acceleration is the velocity
function v(t);
• and the integral of the
velocity is the distance
function s(t).
47

48. Remember
48

49. Newtos's second law, F=ma
 In classical mechanics, for a body with constant mass, the
(vector) acceleration of the body's center of mass is
proportional to the net force vector (i.e. sum of all forces)
acting on it (Newton's second law):
49

50. Uniform acceleration
 Calculation of the velocity difference for a uniform
acceleration.
50

51. 51
Integration

52. Practice
 Write a Win32 bouncing ball program.
 Initial position is (100,100) and initial velocity is (10,-10).
 Initial acceleration is (0,0).
 The ball must bounced in the boundaries of client area.
52

53. Practice2
 Add external acceleration when direction keys are pressing.
if( ::GetAsyncKeyState( VK_LEFT ) ) {
acceleration = acceleration + KVector( -100, 0, 0 );
bAccelModified = true;
}//if
53

54. References
 https://en.wikipedia.org/wiki/Derivative
 https://en.wikipedia.org/wiki/Gradient
 https://en.wikipedia.org/wiki/Integral
 https://en.wikipedia.org/wiki/Acceleration
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