Calculus I
Framework and Overview of Main
Concepts
This document provides an overview of the framework
of Calculus I. You will work through the details in the text, in class notes, and in
communication with the instructor.
If you have already had Calculus I, this document
should provide a brief review of the main topics.
If you are just starting Calculus I, you should
read over this document, though you will probably not understand much of it.
- You should then read over the document one a week.
- You will see your progress in your ability to
understand what is said here, and in your knowledge of the details behind these
statements.
If you are reviewing for a test, this document is a
good place to start.
Topics
The Idea of the Derivative
We know that to find the
average rate of change of a quantity over a time interval we can divide the change in the
quantity by the duration of the time interval.
- When the quantity is given by a function y(t), the
change in the value of the function between clock times t and t + Dt
is y(t + Dt) - y(t) and the duration of the time
interval is Dt, so that the average rate
of change between these clock times is [ y(t + Dt)
- y(t) ] / Dt.
- The limiting value of this average rate of change,
as Dt -> 0, is called the derivative of
the function y with respect to t, denoted dy / dt or y ' (
t ) .
If y(t) is a linear function y(t) = m t + b
of clock time, then dy / dt = lim {t -> 0} [ y(t + Dt)
- y(t) ] / Dt is easily simplified to give us dy
/ dt = m. That is, the derivative of a linear function is constant, and
is equal to the slope of the graph of that function.
- If y(t) is a quadratic function y(t) = a t^2
+ b t + c, then dy / dt = lim {t -> 0} [ y(t + Dt)
- y(t) ] / Dt can be simplified to give dy
/ dt = 2 a t + b. Thus the derivative of a quadratic function is a
linear function.
- Since a linear function changes at a constant rate,
and a linear function is the rate at which a quadratic function changes, a quadratic
function has the distinction that its rate of change changes at a constant rate.
- The value dy / dt of the derivative of this
quadratic function at a given clock time is equal to the slope of the parabolic
graph of the function at that clock time.
- Formulas for the derivatives of power functions,
exponential and logarithmic functions, and trigonometric functions, as well as rules for
calculating derivatives of combined functions, will be developed during the first semester
of this course.
If we interpret y(t) as the depth of water
in a uniform cylinder as water flows from a hole at the bottom of the cylinder,
then as we have observed through a modeling exercise y(t) is very closely modeled
by a quadratic function of clock time.
- The rate at which water depth changes, as
calculated from a table of observed y vs. t values, appears to be a linear
function of clock time.
- The rate at which water depth changes is
approximated by the derivative dy / dt of the quadratic model y(t). As we have seen
this derivative is a linear function of clock time, which agrees with our observation that
the rate is linear in time.
- For more detailed information on the quadratic model
of the flow, you may if you wish consult the Precalculus I homepage. If
you are interested in the physics of the situation you may consult the Physics I
homepage.
The Idea of the Integral
We know that to find the change in a
quantity over a time interval we can multiply the rate of change in the
quantity by the duration of the time interval.
- Given that the rate of change of some quantity is
given by a function r(t), then the approximate average value of this rate of change
between clock times t1 and t2 is [ r(t1) + r(t2) ] / 2.
- It follows that the change in the quantity
is approximately equal to the product
[ r(t1) + r(t2) ] / 2 * (t2 - t1)
- of the approximate average rate of
change and the duration of the time interval.
- This process is equivalent to finding the area
of the trapezoid beneath the line segment connecting the points (t1,
r(t1)) and (t2, r(t2)).
- The average altitude of this
trapezoid represents the approximate average rate at which the function
changes and the width of the trapezoid represents the duration of
the time interval.
- If we directly observe the average rate of change of
a quantity at a series of clock times, we can find the approximate
change in the quantity between each successive pair of clock times.
- We can then add the approximate changes to
get the total approximate change.
- The smaller the time interval Dt = t2 - t1, then the closer the graph of the
trapezoid will tend to be to the graph of the actual function,
and the closer the approximate change thus calculated will be to the actual
change.
- For any reasonably well-behaved rate function, the accuracy
of the approximation is approximately proportional to the square of Dt, which means that the approximation improves
very rapidly as Dt decreases.
- The process of finding the change in a
quantity from its rate function is called integration.
- The specific process outlined above is a form of approximate
integration.
- Using approximate integration we
can find for a given rate function r(t) the approximate change in a quantity from some
clock time t = a to another clock time t = b.
- The precise change in the quantity would
be called the definite integral of the rate function r(t) between clock
times t = a and t = b.
Connection between Integral and Derivative
If a function y(t) represents some quantity
vs. clock time, then its derivative dy / dt represents the time rate of
change of that quantity.
- We can therefore write r(t) = dy / dt.
We can find the change in y(t) between two given
clock times either by evaluating y(t) at these clock times, or by integrating r(t) between
these clock times.
- If we integrate this derivative function
r(t) between clock times t = a and t = b, we will obtain the change in
the quantity between these clock times.
- We can also find the change in the quantity by evaluating
y(t) at t = a and at t = b, obtaining quantities y(a) and y(b). The
change in the quantity is thus y(b) - y(a).
We thus see that the definite integral of
r(t) = dy / dt between t = a and t = b is simply equal to y(b) - y(a).
A more familiar way of stating this is to say that
is the following:
- If f(t) and F(t) are
functions such that f(t) = dF(t) / dt (i.e., f(t) is the derivative of F(t)),
then the definite integral of f(t) between t = a and t = b is simply equal to F(b)
- F(a).
The above statement is
the First Fundamental Theorem of Calculus.
This Theorem tells us that to
integrate a function f(t), which for now we can think of as a rate function, we
need only find some function F(t) whose derivative is equal to f(t).
- In the context of rates, F(t)
will be a quantity function whose rate function is f(t).
- The function F(t) is called an antiderivative
of f(t).
For a linear rate of depth
change function r(t) = m t + b, a corresponding quantity function
is the quadratic function y(t) = 1/2 m t^2 + b t (it is easy to verify
that the derivative of y(t) is r(t), so y(t) is an antiderivative of r(t)).
- Other quantity functions
are possible.
- For example the derivative of 1/ 2 m
t^2 + b t + 12 is also equal to m t + b.
- In fact any function of
the form 1/2 m t^2 + b t + c, where c can be any constant number, is an antiderivative
of r(t) = m t + b.
- It doesn't matter which
antiderivative we use to determine the change in depth between to clock times,
since the constant c will cancel when we calculate the change y(b) -
y(a).
- This means that when we integrate a
rate function, we do not find the quantity function, we rather find a quantity
function.
- Any quantity function we find for a
given rate function will differ from any other quantity function by at most a constant.
More generally, we can say that if F(t) is
an antiderivative of a function f(t), then so is F(t) + c, where c is any
constant number. We can furthermore say that the set of all such functions F(t) +
c, where c can be any constant number, includes all possible
antiderivatives of f(t).
The graphs of a function f(t) and an
antiderivative function F(t) are related.
- The 'heights' of the f(t)
graph are the slopes of the F(t) graph.
- To obtain the f(t) graph from the
given F(t) graph we plot the slopes of the given graph as the
heights of our f(t) graph.
- To obtain an F(t) graph from the
given f(t) graph we start at an arbitrary point and begin constructing slopes which
are equal to the heights of the f(t) graph.
- Since the starting point for
constructing the F(t) graph is arbitrary, we see that there are infinitely
many possible F(t) graphs. However, since the slopes of these graphs are all
determined by f(t), the F(t) graphs will all be congruent in that one
graph will differ from the others only by its vertical location.
We can construct an approximation to the graph of a
given function by partitioning its domain into trapezoids and labelling information
relevant to the interpretation of the graph.
If the function represents a quantity Q(t) which depends on clock time t, then
- the width Dt of the trapezoid represents the
duration of the time interval associated with trapezoid, and
- the rise associated with the trapezoid represents the change DQ
in the quantity Q over that time interval, so that
- the slope of the trapezoid is slope = rise / run = DQ
/ Dt, which represents the average rate
at which quantity Q is changing for that time interval.
If the function represents the rate dQ / dt at which some quantity Q changes with
respect to clock time, then
- the average altitude of a trapezoid represents the approximate average rate rateAve =
(dQ / dt)Ave and
- the width of the trapezoid represents the duration Dt
of the time interval.
- The area of the trapezoid therefore represents the approximate
change DQ in the quantity.
- The run Dt associated with the trapezoid
represents the duration of the time interval while the rise D
(dQ / dt) represents the change in the rate dQ / dt.
- The slope associated with the trapezoid is therefore slope = rise / run
= D (dQ / dt) / Dt
and represents the average rate at which the rate dQ / dt changes for the time
interval.
If we know the value of a function y(t) at t = t0 and also the value y ' (t) of its
derivative at t = t0, then we can estimate y(t) at any t value which is close to t0.
- The change in the value of y corresponding to the change Dt
from t0 to t + Dt is Dy
= y ' (t0) * Dt.
- So y (t0 + Dt) is approximately equal
to y(t0) + y'(t0) * Dt.
This is easily understood from a graph of y(t) vs. t in the vicinity of t = t0.
- The graph point ( t0, y(t0) ) is
easily plotted.
- Since the slope of the graph of y
vs. t at this point must be equal to the derivative y ' (t0), we construct the straight
line through (t0, y(t0) ) with slope y ' (t0).
- If we move through a 'run' Dt we must move along this slope through a 'rise' of Dy = slope * Dt
= y ' (t0) * Dt, and
- our new y coordinate will
be y(t0) + y ' (t0) * Dt, which as long as Dt is small and y ' doesn't vary too drastically will
be a good approximation of y(t0 + Dt).
For a function f(x) which is continuous and increasing on an interval a <=
f(t) <= b we form a partition a = t(0) < t(1) < t(2) < . . . < t(n) = b
with t(i+1) - t(i) = `dt and approximate the integral int( f(t), t, a, b) by a lower sum
and an upper sum.
The actual integral lies between the lower sum and the upper sum, and as `dt
-> 0 the difference between these sums approaches zero.
The integral is therefore equal to the limiting value of either the lower or the
upper sum, and must be equal to F(b) - F(a) for any function F(t) which is an
antiderivative of f(t).
- Since the function is increasing the lower sum is f(t(0)) * `dt + f(t(1)) * `dt +
. . . + f(t(n-1)) * `dt while the upper sum is f(t(1)) * `dt + f(t(2)) * `dt + . . . +
f(t(n)) * `dt.
- The difference between the lower and upper sum on any interval is therefore
f(t(n)) * `dt - f(t(0)) * `dt = f(b) `dt - f(a) `dt = ( f(b) - f(a) ) `dt.
- Since f(b) and f(a) are constant and finite we see that the difference approaches
zero as `dt -> 0.
The derivatives of the basic functions are as follows:
- d / dx ( x^n) = n x^(n-1) for positive integers n, obtained using the Binomial
Theorem.
- d / dx ( e^x ) = e^x, obtained from the definition of the derivative and the fact
that as x -> 0, the difference between e^x and x approaches 0 faster than does x.
- d / dx (sin(x)) = cos(x), obtained by a geometrical argument showing that as x
-> 0 the ratio between the arc length on the unit circle corresponding to angle x and
the y coordinate of the unit circle point defining sin(x) approaches 1.
- d / dx ( cos(x) ) = - sin(x), obtained by a geometric argument similar to that
used for sin(x) (also derivable using the chain rule (see below) and the Pythagorean
identity).
The derivative of f(g(x)) is the product of the rate at which g(x) changes at x,
and the rate at which f(z) changes at z = g(x).
- When x changes it causes a change in the value of z = g(x). A change in z
results in a change in the value of f(z).
- If x changes by `dx then z = g(x) changes by approximately `dz = g ' (x) `dx.
- If z changes by `dz then f(z) changes by approximately `df = f ' (z) `dz.
- If `dz = g ' (x) `dx then `df = f ' (z) * [ g ' (x) `dx ], approximately.
- Thus in the limit df / dx = f ' (z) * g ' (x) = g ' (x) * f ' (g(x)).
If a function f(z) depends on the value of z(w), which in turn depends on the
value of w(v), which in turn depends on v(x), then a change in x causes a change in v
which causes a change in w which causes a change in z which causes a change in f.
- The changes in successive variables form a 'chain', so that we have the
approximations (which become precise as `dx -> 0):
- `dv = v ' (x) * `dx,
- `dw = w ' (v) * `dv = w ' (v) * v ' (x) * `dx, or `dw / `dx = (`dw / `dv) * (`dv
/ `dx)
- `dz = z ' (w) * `dw = z ' (w) * w ' (v) * v ' (x) * 'dx, or `dz / `dx = (`dz /
`dw ) * (`dw / `dv) * (`dv / `dx).
- The chain keeps growing until we have the approximation `df / `dx = ( `df /
`dz) * (`dz / `dw ) * (`dw / `dv) * (`dv / `dx).
- In the limit as `dx -> 0 we have df / dx = df / dz * dz / dw * dw / dv * dv /
dx.
If g(x) = f^-1(x), then f(g(x)) = x and f ' (g(x)) = 1. However, we also
have f ' (g(x)) = g ' (x) * f ' (g(x)), so that
- g ' (x) = 1 / [ f ' ( g ( x) ) ].
Provided we can obtain an expression for f ' (g(x) ) we can thus find the
formula for the derivative g ' (x) of the inverse function g(x) = f^-1(x).
Using this technique we find that
- d / dx ( x^p) = p x^(p-1) for any rational number p not equal to -1.
- d / dx (ln(x)) = 1 / x, using f(x) = e^x and g(x) = ln(x).
- d / dx (arcsin(x)) = 1 / `sqrt(1-x^2), using f(x) = sin(x) and g(x) = arcsin(x)
- d / dx (arctan(x) ) = 1 / (x^2 + 1), using f(x) = tan(x) and g(x) = arctan(x).
If y is a function of x and g is a function of y,
then dg / dx is the derivative with respect to x of the composite
function g ( y(x) ).
This derivative, by the Chain rule, is dg / dx = y ' (x) * g ' ( y (x) ).
- g ( y(x) ) is often written just g(y) and we have to remember that if the
derivative is with respect to x, this is a composite.
- Instead of writing y ' (x) and g ' ( y(x) ) we might just write y and g ' (y), or
even y and dg / dy. When we do so we have to remember that y is a function
of x and g ' means dg / dy.
- Thus we might write dg / dx = y ' dg / dy.
A product function of the form f(x) * g(y) can be differentiated
with respect to x (recalling that dg / dx = dg / dy * dy / dx, or dg / dx = dg /
dy * y ' ) using the product rule.
- Thus d / dx ( f(x) * g(y) ) = df / dx + dg / dx = df /
dx + dg / dy * y ', where y ' = dy / dx.
- For example
- d / dx ( x^2 y^3) = 2x y^3 + x^2 * 3y^2 y '
- d / dx ( sin (x^2 y^3) ) = [ 2x y^3 + x^2 * 3y^2 y' ] cos (x^2 y^3)
(note the use of the Chain Rule)
An equation f(x, y) = 0 will often be satisfied by an infinite
number of order pairs or points (x, y).
- These points will typically form a curve or a set of curves in
the x-y plane.
- Usually it is impossible to find the equation of the curve(s) because
we can't solve the equation explicitly for y.
- However, the curve still exists and at a point such a curve will
typically have a slope.
- If we know the coordinates of a point on the curve and have a
formula for y ' in terms of x and y we can simply insert the coordinates into the formula
to find the slope at that point. Given the point and the slope we
can find the equation of the tangent line at the point.
Given an equation of the form f(x, y) = 0, where f(x, y)
denotes any expression involving x and y, we can differentiate the equation with
respect to x. The resulting equation will involve the variables x
and y and the derivative y ' = dy / dx. The equation can
often be solved for y ' in terms of x and y, especially when y ' appears
only as a linear factor of one or more terms of the equation.
- Having obtained the solution y ' = dy / dx in terms of x
and y we can find y ' at any point (x, y) for which the original
equation f(x, y) = 0 is true.
- Thus given any point (x, y) on the curve implicitly defined by f(x, y) = 0 we can
find the equation of the tangent line to the curve at that point.
Often we need to evaluate the limiting value of an expression of the form f(x)
/ g(x) where both f(x) and g(x) have limiting values 0.
- Since 0 / 0 is undefined (i.e., it could mean 0, it could mean
an infinite quantity, it could mean any finite quantity), we can't get the desired limit
from the quotient of the limits.
- However if f(x) and g(x) have tangent lines at
the limit point, f(x) / g(x) will at that point approach the
ratio of the slopes of these tangent lines.
- Thus we see that
- If lim {x -> a} f(x) = 0 and lim {x -> b} g(x) = 0, then near x = a
we have the approximations
- f(x) = f(a) + f ' (a) * ( x - a ) = 0 + f ' (a) * (x - a) = f ' (a) * (x
- a)
- g(x) = g(a) + g ' (a) * ( x - a ) = 0 + g ' (a) * (x - a) = g ' (a) * (x
- a)
- both of which become accurate as x -> a. Thus
- the approximation f(x) / g(x) = f ' (a) * (x - a) / [ g ' (a) * (x - a) ]
= f ' (a) / g ' (a) becomes accurate in the limit as x -> a.
Thus
- when f(x) and g(x) both approach 0 as x -> a, and when f(x) and g(x)
have derivatives at x = a, lim{x -> a} [ f(x) / g(x) ] = lim {x -> a} f(x) / [
lim {x -> a} g(x) ] .
This is known as l'Hopital's Rule.
This rule is easily adapted to the case where f(x) and g(x)
both have infinite limits at x = a (instead of f(x) / g(x) we use [ 1 /
f(x) ] / [ 1 / g(x) ], where the numerator and denominator both have limit 0. It is
also easy to adapt to the case where the limits of the two functions are both
zero as x -> infinity.
A function f(x) which is continuous and twice-differentiable near
x = a will have a maximum at x = a under the following conditions:
- f ' (a) = 0 and f ' (x) changes from positive to negative at x = a (the first-derivative
test),
or
- f ' (a) = 0 and f '' (a) is negative (the second-derivative test).
A function f(x) which is continuous and twice-differentiable near x = a will
have a minimum at x = a under the following conditions:
- f ' (a) = 0 and f ' (x) changes from negative to positive at x = a (the first-derivative
test),
or
- f ' (a) = 0 and f '' (a) is positive (the second-derivative test).