Tangent Lines and Derivatives
There is a beautiful inter pretation of the difference quotient as an
approximation to the slope of the tangent line to the graph of a
function, and the derivative as the actual slope of the tangent line to
the graph at a given point.
The main aspect of this interpretation is illustrated in
the figure
above. Note that the line L intersecting the graph of
y = F(t) has slope (F(t + Δ t) - F(t))/Δ t = Δ F(t)/Δ t. As
Δ t
approaches zero the line L approaches the tangent line to the graph
at the point (t, F(t)) and the difference quotient approaches
dF(t)/dt. Thus the derivative of the function F(t) at t is the
slope
of the tangent line to its graph at that point.
Discussion
As you can see, the study of calculus will involve a combination of
algebra, geometry and the art of taking limits. In this way studying
calculus involves all the mathematics that you have learned up to
this point.
Gottfried Wilhelm Leibbniz
(b.1646,d.1716) was a German philosopher, mathematician ,and logician
who wrote about mathematics, logic, science, history, law and
theology .)
Isaac newton
Sir Isaac Newton at 46 in Godfrey Kneller's 1689 portrait
Born 4 January 1643 [ OS:25 December 1642]
Woolsthorpe - by- Colsterworth, Lincolnshire
England
Died 31 March 1727 [ OS:20 mARCH 1727]
kensington, London, England
Occupation Physicist, mathematician, astronomer,
alchemist, and natural phiosopher
Newton discovered the calculus as an aid for doing
dynamical
physics to describe gravity and the motions of the planets. Leibniz
discovered the calculus in the course of mathematical and
philosophical investigations. The notation dF/dt is due to Leibniz.
Leibniz dreamed of a universal calculus that could be used to settle
all problems involving reasoning. His dream is un realized to this
day , but the evolution of calculus, mathematical logic, computer
programming and computer science is, from our modern point of
view, the beginning of a possible realization of that dream.
Integ ration
One of the great results of the calculus (due to Newton and Leibniz)
is its application to the problem of finding areas and volumes.
Consider the fol lowing problem : Find the area between the graph of
a function y = F(t) and the t-axis between the values a and
t.
Lets call this area A(t) = A(F, a, t).
Note that the area A(t) under the curve from a
to
t is a function of
t. We can ask what is the rate of change of this area as a function of
t? In other words, what is dA(t)/dt?
The remarkable answer is the
THE FUNDAMENTAL THEOREM OF CALCULUS:
dA(F,a,t)/dt = F(t).
Because we shall develop lots of techniques for differentiation we
will be able to use this theorem to calculate all sorts of areas.
The art of finding areas using calculus is called integral calculus
It is not hard to see pictorially why the fundamental
theorem of
calculus is true. Consider the difference
ΔA= (A(t + Δ t) - A(t)).
This is the area of a small strip between t and t + Δt, and is
well-approximated
by (base x height) = Δ t F(t). Therefore the
difference quotient ΔA/Δ t is approximately equal to F(t), and
this
approximation gets better and better as Δt goes to zero.
Integrals are Limits of Sums.
One way to think about the area under a curve is to cut the area up
into many vertical strips and take the sum of the areas of the strips.
The base of a given strip has length Δt. If Δt
is very small, then the
area of the strip will be approximately F(t)Δt. Thus the integral of
y = F(t) from t=a to t=b is approximately the sum of the
numbers
F(t)Δt as t ranges over a finite but ever-larger sub division of the
interval from a to b. We have illustrated such a subdivision in
the
figure above.
Notation for the Integral
The Leibniz notation for the area A(F,a,b) (the area under the
graph y = f(t) between t = a and t = b ) is
The notation is de signed to remind us that the area can be
approximated by dividing it into many small vertical strips with
height F(t) and base Δ t. We than take the limit of a such a
finite
sum to get the actual area. The Leibniz notation for the integral
reminds us of this by replacing Δt by dt and the summation by the
long S, 
that represents the limit of the
sum.
We can then restate the fundamental theorem of calculus with the
formula
There is much more to be said about the relationship
between
differentiation and integration. We will be covering much of this as
the course goes along. For example, we would like to find the area
from t = a to t = b under the parabola y = t2 . This is
represented
by the integral 
. The fundamental theorem of
calculus
tells us
It also turns out that d(x3)/dx = 3x2
.
You can work this derivative out for yourself in the same way that
we worked out the derivative of x2. (Note that we have changed
from t to x here.) This implies that d(x3/3)/dx = x2
. So the
function x3/3 has the same derivative as the area function for
the
parabola. A little more work shows that one has the area formula
This is an example of how calculus can solve area problems
that
involve curved and complex boundaries .
Discussion
We have now covered the main ideas. These ideas are the
key to
learning differential and integral calculus.
Advanced Problems
1. Recall our early problem about average velocity. Now suppose
that we know that the positions of our car at the times
are
respectively.
The total distance travelled is 
.
With this notation, we can think about lots of examples.
We will take the velocity of the car in the time interval
to be
We let 
. Here k runs from 0 to
n-1.
Since the car has velocity 
in the time
interval
Δk, the average
velocity of the car over the time interval [a,b] is equal to
AvgVelocity =
where 
is the fraction of the time the car
spent at the velocity .
(a) Work out an example of your choice for this problem using
n = 4, with numbers for the times and the positions. Check that the
average velocity does work out to be equal to X/T where X is the
total distance travelled, and T is the total time for the journey.
(b) Show in general, by using algebra, that
the average velocity is equal to X/T where X is the total distance
travelled, and T is the total time for the journey.
where X is the total distance travelled and T is the total time
for the
trip.
2. (a) Convince yourself by multiplying out the algebra that
(a + b)3 = a3 + b3 + 3a2 b + 3ab2
.
Given that we have the two -dimensional geometric interpretation of
(a+b)2 = a2 + b2 + 2ab as shown below,
find a three-dimensional geometric interpretation for the
formula
for (a + b)3 .
(b) Use the results of part (a) of this problem to show that
d ( t 3 )/dt = 3t2 .
(c) Generalize everything and show that d(tn)/dt = (n-1)tn-1.
