An introduction to calculus…• In his Principia Mathematica, Sir Isaac

Newton devised a new study of mathematics in order to describe dynamics systems that could not be explained by algebra alone. We call this mathematics calculus.

• The physics C examination requires the use of calculus both in conceptual studies as well as in solving problems.

The Limit• A limit in calculus is simply a value that is

neared and sometimes reached as an independent value changes.

• The classic example of a limit is the doorway problem…

• You are standing in the middle of a room. There is one exit immediately in front of you. Each time you take a step towards the doorway you cover one-half the distance between yourself and the door. The next step takes you one half of the remainder and so on. Do you ever reach the door?

YES!

• Although it seems to defy common sense, you do, in fact, reach the door. In this case, the door represents a limit. If you take an infinite number of steps (impossible, of course), no matter how small the distance becomes you are guaranteed to eventually reach the door.

• I like to think of this in terms of real life. You have to reach the limit because eventually the distance between you and the door is smaller than your foot. Hence you cannot take another step without touching the threshold.

Limit notation

• We typically write a limit in mathematical notation in terms of the variable of a function. For example:

• Let’s give f(x) a real function. Let f(x) = x-1

valuesome)(lim

xfx

? lim 1

x

x

• As x (the independent variable in this case) becomes larger and larger, the value of y, or rather f(x), approaches zero. Thus the limit of this function as x approaches infinity is zero.

• In calculus you will do more work specifically with limits. We use limits in physics to define another tool, the instantaneous rate of change or derivative.

0 lim 1

x

x

The derivative• Consider the function y = mx + bThis function is a straight line. In order to

find the rate of change of that function (how much y changes as x changes) we perform the following operation on two distinct ordered pairs (x1,y1) & (x2,y2)

where m is the slope of the line created by the function.

mxx

yy

)(

)(

12

12

Change in y

Change in x

Slope (rate of change)

• The derivative is the slope of a function at a single point. This is difficult mathematically as a point is only a single ordered pair (x’,y’). If we simply perform a slope calculation we get an undefined quantity.

• Classical mathematics does not allow this operation. Enter the derivative…

“Instantaneous Slope”

hell!math to tripa0

0

''

''

xx

yy

• Look at the positive x portion of the function y = x2

• Select any two points (x1,y1) & (x2,y2)

• It is easy to find the average slope of the function between these points.

x1

y1

x2

y2

avemxx

yy

)(

)(

12

12

• In fact, the average slop of this function between any two points (x1,y1) & (x2,y2) is the slope of the secant line connecting those points. This is true for any continuous function.

x1

y1

x2

y2

mave

Now let’s define x as simply the denominator of the slope operation, x= (x2-x1)

• To find the instantaneous rate of change of the function (i.e., the slope at a point, let’s use (x1,y1)) we need to have x2 approach x1

x1

y1

x2

y2

As the interval between x2 & x1 approaches zero the slope of the secant line approaches the slope of the line tangent to the point (x1,y1)

x1

y1

minstant

The derivative

• The derivative is the instantaneous rate of change, or slope, of a function at a point. We define it as follows:

dx

dy

x

y

xx

yyxx

012

12

0lim

)(

)(lim

dx

dyis the derivative of y with respect to x, or the

instantaneous slope of the function “y” at a point “x”

Finding derivatives• We have discussed the concept of the

derivative. Now we need to look at finding a derivative given a specific function. Fortunately there are a number of easy rules to allow us to do this.

• You will practice and research these rules extensively in calculus. In this class you simply need to be able to use them and as such, I will allow you to use the reference sheet you have been given on exams and quizzes (except for next class!)

The Power Rule• We will discuss one very important rule of

differentiation in class as it forms the basis for our next topic (integration).

• Any simple polynomial involving separate terms of any order can be differentiated (that is to say that the slope or derivative can be found) by means of the power rule.

CBxAxy 2

Where A,B, and C are constants

BAxdx

dy2

CBxAxy 2

For any function of the form y = Axn

1 nnAxdx

dy

So…

2

3

3xdx

dy

xy

xxdx

dy

xy

22

2

1

11 0

xdx

dy

xy

0

Constant

dx

dy

y

2

3

12

4

xdx

dy

xy

Examples:

7

8

16

2

xdx

dy

xy

Cdx

dy

Cxy

Other rules for differentiating more complex functions are in your handout.

“Prime” notation

• The addition of a superscripted tick mark, verbally called a “prime” to functional notation means that the expression is a derivative.

)('' xfydx

dy

The Product Rule

• F(x) = AB

• F’(x) = (A’B) + (B’A)

The Chain Rule• y = f(g(x))

• y’ = [f’(g(x))] (g’(x))

• e.g., y = (3x2) =(3x2)1/2

y’ = [½(3x2)-1/2](6x)

A note about notation…

• Sometimes a derivative operator may be written as

where f(x) is the function. This simply means take the derivative of the function f(x). You could re-write this expression as

)(xfdx

d

)( where OR )(

xfydx

dy

dx

xdf

Integration

• Integration is the opposite of differentiation. In terms of mathematics the integral is actually the “anti-derivative”, but it has a further meaning: the infinite sum.

• You have likely seen the symbol used to find the sum of a finite (ending) series. The integral is the sum of an infinite, and potentially un-ending series. The integral of a function defines the area between the function and the x-axis. Area above the x-axis is positive, area below the axis is considered negative.

The power rule for integration

• The power rule for integration is exactly opposite the rule for differentiation. In a nutshell:

constant a is C where,1

1

Cn

AxdxAx

nn

Cxdx 44 CxCx

xdx 22

22

44

Cxdxx 32

3

44

Examples:

representing the vertical shift of the function

Limits of integration• As indicated, the integrals on the last slide are

called indefinite integrals. That means they measure the area “under” a curve from -∞ to +∞.

• Adding limits to an integral indicates that only the area within that domain is desired.

3

2)( dxxf

Solving a definite integral

66.179

)3(3

1)8(

3

1

3

1 338

3

3

8

3

2

x

dxxSolve the integral.

Notice no constantis necessary fordefinite, or defined integrals

Plug in the limits of integration and subtract the initial limit solution from the final limit solution

Why use calculus in physics?• Recall that velocity and acceleration are rates of change.

We can use derivatives to more precisely define and examine these quantities.

• Power is the time rate of energy consumption. Work is energy used. As you will see in this course we will often use integration to calculate energy use.

• When a system has a non-constant force (i.e., a force which has a constantly changing value) classical kinematics cannot describe the behavior of the system. Integration can be used to look at the diverse values of force as the system changes.

• We can also use integration and differentiation extensively in studying electricity and magnetism. Most of the laws that govern E & M are mathematical in nature, relying on infinite sums and instantaneous changes.