Introduction to
o Calculuss: The Alggebra of C
Change!
Revised 8//2/2015
“’Tis the Art of numb
bering and m
measuring exxactly a Thingg whose Exisstence cannot be conceived.” ‐‐ Voltaire Ca
alculus is the Latin
L
word foor pebble;
sma
all stones werre used by thee Romans to ddetermine
the equivalent
e
of modern-day
m
ttaxi fares.
So
o when someo
one asks if yo
ou are taking Calculus,
you could say, “No, I don’t
d
need callculus; I take the bus.”
This lab iis a conceptual introducction to the ideas we wiill study in m
much more d
detail as wee go on. It is o
often helpfu
ul to have an
n understand
ding of idea s before seeeing too many equation
ns. Note: Laabs are a disccussion of th
he concepts,, giving exam
mples and po
osing probleems along the way. Som
me guideline
es to help yo
ou read and work througgh the labs:
 When a new W
concept or tterm is intro
oduced, it is usually in eitther bold. A
Anything underlined is of particular importancce. Side notees of interesst (or not) arre usually in ittalics. But yo
ou should re
ead them anyway.  Exxercises for you to workk as part of the lab are inn red; thought questionss (some wou
uld saay they are rrhetorical) appear in blu
ue.  Conjectures aare sprinkled
d throughout the labs; thhese are tho
ought questions which re
equire you to develop a supporting argument or disprove w
with a counteer‐example. Of co
ourse, you’lll often be assked to turn in some of tthis work.  Problem sets that supplement the lab material w
will be assign
ned. Calculus has its rootss in some very old proble
ems: the claassic th
paradoxe
es posed by Zeno in the 5 century BC, the struggless of Pythagoras and Arch
himedes to ccalculate thee areas encclosed by vaarious curved
d shapes. Th
hese mathemaaticians and their successsors wonde
ered how to describe something tthey all knew
w intuitively: As you addd more sides to a regular polygon, it becomes more and m
more circular. Why? The length of its sides shrinkks as we addd more. How does the
e sum of these side lengtths still represen
nt the perime
eter of the ccircle? And w
what keeps tthe circle from just disap
ppearing? https:///xkcd.com/11553/ 1 Fast forward to Enlightenment Europe, when such thinkers as Pierre de Fermat looked for ways to find the maximum and minimum values of functions. But most historians of math (yes, there are such people) will tell you that Calculus was formally invented (or was it discovered?) by Isaac Newton and Gottfried Leibniz, working separately (and very competitively) to describe objects in motion. There is still bitter controversy over whether the glory really belongs to Newton or Leibniz; however, there is no doubt that the Calculus we enjoy today owes its modern form to Augustin‐Louis Cauchy, Joseph‐Louis Lagrange and a few others we’ll soon meet. Just as there are basic operations in Algebra, there are basic operations in Calculus: finding limits, taking derivatives and evaluating integrals. And one thing that makes Calculus different from Algebra is that we finally get to divide by quantities that are essentially 0. The concept of the limit allows us to get away with this wonderful thing – the very thing your Algebra teacher told you was not possible! Review Exercises: If you don’t know where you’ve been, how can you tell where you are going? These are the kind of things you should be able to do by hand, from memory – and quickly. 1. Write the expression for the equation of a line when you know a point (x1, y1) on the line and the slope m of the line at that point. 2. Write the expression for the equation of a line when you know two points (x1, y1) and (x2, y2) on the line. What is the slope of this line in terms of the given points? 3. Write the binomial expansion for the expression (1 + x)2, (1 + x)3 and (1 + x)4. If x << 1, we usually can approximate these expressions by dropping any terms in x2 or higher order. Use the pattern established here to suggest a very useful approximation for the first two terms of a binomial expansion for any polynomial of the form (1 + x)n. Under what circumstances are we justified in ‘dropping’ the terms of higher order than 1? Secants, tangents and limits, oh my! It all starts with Physics – you were expecting maybe Chemistry? Newton sought to describe objects in motion; that seems like a good place for us to start. The one‐dimensional motion of an object may be represented as a graph of position x as a function of time t. 2 Note that the properties of a mathematical function are critical here – objects in our world cannot be in two places (have two distinct x values) at the same time!
If we know two points on the position‐time curve, positions x1 and x2 separated by an interval of time delta t (t = t2 ‐ t1), connecting the points with a straight line represents the object’s average velocity over the time interval. These lines are known as secants, as they intersect the position‐time curve at two points. We refer to the slope of this line as the object’s average velocity: change in position divided by x
change in time, v 
. Slope is also known colloquially as ‘rise over run.’ How does this idea t
apply to a graph of position vs. time? Note that velocity has a direction (here, it can be positive – the direction of increasing x, negative (decreasing x) or 0 – no change in x). In one dimensional motion, the algebraic sign, + or –, indicates direction. It is presumed that time always ‘moves’ forward, ie, t is non‐negative. Displacement refers to the change in position x (a signed quantity); distance traveled is how far you moved during the trip, ignoring the distinction between positive and negative. If position is constant, what is the slope of the position time graph, ie, the average velocity? The magnitude of velocity – number without a direction, which is always treated as if it is positive – is known as speed. If an object’s average velocity changes from one time interval to another, it may be speeding up, slowing down or changing direction; all of these changes in v
velocity are collectively known as acceleration, a 
is the average acceleration during the t
time interval t. It should then be clear that for every position‐time graph, there is an associated velocity‐time graph; if velocity is constant, what is the slope of the velocity‐time graph? How does the idea of ‘rise over run’ apply to a velocity‐time graph? Exercises a. If a car travels in one direction along a straight road a distance of 50 miles in the first hour and 30 miles in the second hour, what is its average velocity over the entire trip? b. Suppose the car traveled 50 miles one way in the first hour and 30 miles in the opposite direction during the second hour. What is its average velocity over this two‐hour trip? What is its displacement from its point of origin? How far did it travel? Why are these last two different? 3 Velocity at a specific time It should be clear from the exercises above (and a glance at your car’s speedometer) that the average velocity is not necessarily an object’s velocity at any given time. If we want to know velocity at a specific instant in time (which we call the instantaneous velocity), we just reduce the size of the time interval t so that the two points approach one another as closely as possible. This raises the question suggested by Voltaire’s clever observation at the top of this lab: If the two points are so close together, what does it even mean to speak of the line connecting them? How can an object have any change in position (displacement) x during an infinitesimal time interval t? How do we know that the tangent line to a function at a point is unique? So, if we want to study objects in motion (or if we just want to study rates of change in general), we need to find a way to find the slope of this very short line: We need the concept of the limit of a function. The graph at right is a position‐time graph; it has position x on the vertical axis and time t on the horizontal axis. t 2 - 0.3 t - 5
4
2
-3
-2
-1
1
2
3
-2
-4
The figure illustrates the distinction between the average velocity (slope of the blue secant line) over the interval between red and blue points and the instantaneous velocity (slope of the tangent green line) at the red point. We learned how to draw secants at an early age (hopefully you are familiar with “connect the dots”), but most people can’t draw tangents very accurately. And what we really want a formula for the tangent that makes mathematical sense. You should already have an expression for the slope of the secant line from Review Exercise 2 above. Be sure your expression matches what you see in the figure! 4 t
Recall that a secant line to a curve is a line that intersects the curve in two points in a given neighborhood, whereas a tangent line is one that just touches ‐ a curve at only one point in a given neighborhood. However, the tangent may intersect the curve again at some unrelated point farther away. Comparing the figure above to the next one suggests that our secant line gets closer to the tangent line as we shrink the distance between the two points, ie, reduce the size of the interval delta t towards 0. t 2 - 0.3 t - 5
4
2
-3
-2
-1
1
2
3
1
2
3
t
-2
-4
t 2 - 0.3 t - 5
The secant about to become tangent. 4
2
-3
-2
-1
-2
-4
5 t
Limits These examples illustrate that our position‐time function x(t) has average velocities that approach the instantaneous velocity at a specific point as delta t approaches 0. In general, we can say that the limit of the slope of secant lines approaches the slope of a single tangent line at a point as the two points defining the secant approach one another, ie, the interval between them approaches 0. For a given function, the operation of evaluating a limit is little more than finding the value of the function as the independent variable t approaches a specific value. If we're lucky, this limit exists and we have to do nothing more than plugging in the specified value of t and finding the corresponding x. If we're not so lucky, some tricks and techniques will be needed ‐ and even then, the limit still might not exist. While you will soon have to learn how to do these things by hand, right now we'll let Wolfram alpha do the work. Enter the following expressions into Walpha. Observe the proper notation for limits and make sure you can see how these straightforward limits are calculated.
a. limit t^2 as t 0
b. limit t^2 / t as t  0
c. limit t^2 / t as t  -1
d. limit t^2 / (t+1) as t  -1
This last example illustrates an important rule: if a function doesn't have the same limiting value as you approach from the left and from the right, the limit does not exist. Try some other combinations ‐ be sure you can think through the answer provided. Conjecture: The limit of a function at a point, if that limit exists is the value of the function at that point. Support or deny, using examples you create with Wa. Slope of the tangent line Now apply the limit operation to find the slope of the tangent line. Construct the slope of a secant, as we've done above in the average velocity discussion, and then set up this very specific limit: lim
t  0
f (t  t )  f (t )
t
This limit is the definition of the derivative of a function. When that limit exists (and we will dive into the existence question later on), we can find the slope of a tangent line at a point. The 6 derivativve of a functiion is also kn
nown as its rrate of changge (which is why we calll Calculus the Algebra o
of change). Note that most peop
ple assume that t is possitive in the aabove expreession. A lesss frequentlyy f (t  t )  f (t )
seen, butt equally valid expressio
on for the de
erivative is liim
. t  0
t
Applyingg the definitiion of derivaative What is tthe derivativve of a consttant? Without doing anyy mathemattical work, w
we can invokee an importan
nt life rule: ""Constants d
don't change." Since thee derivative is a rate of change, this answer sshould alwayys be 0 ‐ and if you plug values of a cconstant fun
nction, say f((t) = 5, into tthe 55
numerator of our definition of th
he derivative
e, you do inddeed get 0 (aactually you get lim
, t  0  t
which in this case is 0
0).
What is tthe derivativve of a straigght line, y = m
mx + b? Se
et up the lim
mit ssimplify befo
ore taking the limit and e
evaluate. Any surprise t
A
there? , Summaryy The derivvative of a co
onstant is __
_________; the derivativve of a straigght line is ________________. The derivvative is the slope of ___
__________
________________. If the fun
nction is position as a fun
nction of tim
me, the first derivative w
with respect to time represen
nts ________
_________ aand the seco
ond derivativve representts _______________________. The derivvative of a fu
unction reprresents the _
___________________________ rate of change of the function.. 7