and with Excel =COS(130*PI()/180) we find that Therefore,
a2 = b2+c2–2bccosA
a2 = 4.22+2.72–2 4.2 2.7× × × cos130° = 58 42 0.707– × = 28.306 130°
cos = –0.643
a2 = 17.64 7.29 14.58+ + = 39.51 a = 6.28
Chapter 5
Fundamentals of Calculus
his chapter introduces the basics of differential and integral calculus. It is intended for readers who need an introduction or an accelerated review of this topic. Readers with a strong mathematical background may skip this chapter. Others may find it useful, as well as a convenient source for review.
5.1 Introduction
Calculus is the branch of mathematics that is concerned with concepts such as the rate of change, the slope of a curve at a particular point, and the calculation of an area bounded by curves. Many principles governing physical processes are formulated in terms of rates of change. Calculus is also widely used in the study of statistics and probability.
The fundamental concept of calculus is the theory of limits of functions. A function is a defined relationship between two or more variables. One variable, called dependent variable, approaches a limit as another variable, called independent variable, approaches a number or becomes infinite. In calculus, we are interested in related variables. For instance, the amount of postage required to mail a package is related to its weight. Here, the weight of the package is considered the indepen- dent variable, and the amount of postage is considered the dependent variable.
The two branches into which elementary calculus is usually divided, are the differential calculus, based on the limits of ratios, and the integral calculus, based on the limits of sums.
5.2 Differential Calculus
Suppose that the dependent variable, denoted as , is a function of the independent variable, denoted as . This relationship is written as . If the variable changes by a particular amount , the variable will also change by a predictable amount . The ratio of the two amounts of change is called a difference quotient. If the rate of change differs over time, this quotient indicates the average rate of change of in a particular amount of time.
If the ratio has a limit as approaches , this limit is called the derivative of . The deriva- tive of may be interpreted as the slope of the curve graphed by the equation , measured at a particular point. It may also be interpreted as the instantaneous rate of change of . The pro- cess of finding a derivative is called differentiation.
If the derivative of is found for all applicable values of , a new function is obtained. If , the new function, which we call the first derivative, is written as , or , or . If the first derivative is itself a function of , its derivative can also be found; this is called the second
T
y
x y = f x( ) x
h y k
k h⁄
y = f x( )
k h⁄ h 0 y
y y = f x( )
y
y x y = f x( )
y' dy dx⁄ df x( )⁄dx
y' x
Chapter 5 Fundamentals of Calculus
derivative of and it is written as or or . Third and higher derivatives also exist and they are written similarly.
Example 5.1
Suppose that the distance between City A and City B is miles. Distance is usually denoted with the letter s, so for this example miles. This distance is fixed; it does not change with time; therefore, we say that distance is independent of time because whether we decide to drive from City A to City B today or tomorrow, we need to drive a distance of miles. What is more important to us, is the time we need to spend driving. This depends on the velocity (speed)* which, in turn, depends on traffic conditions, and our driving habits. Thus, if we start driving from City A towards City B at a constant speed of miles per hour, we will spend ten hours driving.
As we drive, the distance changes, that is, it is being reduced. For instance, after driving for one hour, we have covered miles and the remaining distance has been reduced miles. Like- wise, if our speed during the second hour is miles per hour, the remaining distance is further reduced to miles.
We denote velocity with the letter and time with . Then,
(5.1) that is, velocity is the distance divided per unit of time; in this case, miles divided by hours.
In reality, it is impossible to maintain exactly the same speed per unit of time, that is, when we say that during the first hour we drove at a velocity of miles per hour, this is an average velocity since the velocity may vary from to , or from to miles per hour. Therefore, the for- mula of (5.1) is valid for average velocities only.
The instantaneous velocity, that is, at any instant, the velocity is expressed as
† (5.2)
* Velocity is speed with direction assigned to it.
† The development of this expression follows. Let be the distance at time and the distance at time . Next, let the difference between and be denoted with the Greek letter D (delta), that is, and likewise, let the differ- ence between and be denoted as . Then, . Now, let ; this notation says that we let become so small that it approaches zero in the limit but never becomes exactly zero. Obviously, as becomes smaller and smaller, so does . Then,
y y'' d2y dx⁄ 2 d2f x( )⁄dx2
500 s = 500
500
50
50 450
65 385
v t
v = s t⁄
50
49 51 48 52
v t( ) ds ---dt
=
s1 t1 s2 t2
s2 s1 Δs = s2–s1
t2 t1 Δt = t2–t1 v t( ) s2–s1 t2–t1 --- Δs
---Δt
= = Δt→0
Δt Δt
Δs v t( ) Δs Δt ---
Δtlim→0 ds ---dt
= =