Calculus was developed
by both Leibnitz and Newton as a way of getting information from
graphs that stalled conventional mathematics. Calculus has two
forms: Integration and Differentiation. Basically, differentiation
reduces data to its essentials; integration enhances information
to create new products. Differentiation will reduce a 2-dimensional
plot to a 1-dimensional line whose length and slope reveal performance
data. Integration is used to estimate what data will arise when
a 2-dimensional graph is enhanced to 3-dimensions, or what volume
will result when rotated about its x-axis or y-axis.
For example, using a
graph plotting distance against time, differentiation will find
instantaneous velocity, point of maximum or minimum velocity.
Integration will find total distance covered.
Natural Logarithms and
trig functions and identities seem to give the most pain to students.
I think that is because while the basic trig functions can be
sketched on a graph paper, the concept of multiplying and dividing
trig functions seems ridiculous. Nevertheless, it is possible
and it is logical (well, from a mathematician's perspective anyway).
To visualize these functions, try downloading or buying graphing
software such as Equation
tan = sin/cos (tanning at sun/coast)
sec = 1/cos (my secretary has one
over my cousin)
csc = 1/sin (my cousin's secretary
has won over sin)
cot = cos/sin (I find cotton is cozier
to wear over synthetics)
tan = sin/cos. A well-tanned-but-horny-and-cheap
man, pondering whether to pay for a hooker or save money, chooses
SIN OVER COST.
Trig Function in the
The sign of a trig function
can easily be calculated by knowing that sin=y
and cos=x, and that all sine is positive in the
top half of the circle, all cosine is positive on the right half
of the circle, and that sin + cos combine to define
all trig functions. For example, the upper left quadrant has
sine + and cos -, so only sine and cosecant are positive. In
the lower left of the circle, both sine and cosine are negative
and -- because two negatives make a positive -- cos/sin (cot)
and sin/cos (tan) are positive.
Here are a few mnemonics
to help recall the trig values which are positive in the quadrants
of the unit circle. Starting from the upper right, we have All
Sinning Czechs are on Tan Cots with Costly Secrets, and All
Sing Christmas Carols wearing Tan Cottons with Cozy Secretaries.
I know the x-axis is related to cosine, and the y-axis is related
to sine by thinking of Cossacks in Sinai.(cos-x, sin-y)
Table of Trig Identities
This table shows trig
functions in the centre (center to Americans) column are
differentiated to the left, and integrated to the right. Notice
the integration process always adds the Constant. The
reason for the C is that when a function is differentiated,
the Constant is lost, and when the process is reversed, it must
come back somehow. So to compensate for any lost numbers, always
include the C, even if C = zero.
sec x tan x
-csc x cot x
x + C
sin x + C
-cos x + C
Not applicable. Trig functions need upper boundaries in order
to be integrable.
Trig Derivatives in
a Secluded Cottage
How about a mnemonic
for the trig derivatives? Try: Scot's cozy sinner of a secretary
squarely and secretly co-seeks a cot. Sexy and tanned, cozy secluded
cottage, casual sexual contact twice. The functions are,
in order: derivative of sin = cos, cos =-sin, tan = sec squared,
sec = sec tan, csc = -csc cot, cot = -csc squared. Notice that
all of the co-functions have a negative value. Trying to get
a mnemonic phrase to help recall all of these is difficult; I
usually end up with confusing sentence fragments of secret tanning
spots and cosy secretaries. Personally, I remember tanning
on sun coast as the primary mnemonic, and memorize or know
how to calculate the rest of them.
in the Kingdom
The procedure for graphing
a function is long, but worthwhile. The steps should be done
in sequence, and this is where a lot of students get mixed up.
Briefly, the steps include: find the domain of the function,
the xy intercepts, the asymptotes, the symmetry, and the
first and second derivatives. To help remember the steps (which
can have different terms for the same step) I came up with a
story which applies memory pegs, and
a single sentence based on sex (mneumonic). I couldn't make an
acronym work for this one. First, the story:
In the DOMAIN
of Calculus the Difficult, the King (who leads a graphic lifestyle)
drives a Jensen INTERCEPTOR and greets people with "HA,
VA!". The citizenry are mostly inbred and regard people
having ASYMMETRICAL faces as odd. The King TOOK THE
PRIME Minister and taught him the CRITICAL POINTS
of running a government. It seems the economy was RISING and
FALLING relative to MAXwell House coffee and MINIMUM
wage. He took a SECOND PRIME Minister who had two CONCubines,
one of whom was born on the CUSP between Leo and Virgo.
Of course, "taking
the Prime" is another way of saying "finding the derivative",
and we find the critical points by examining the derivatives
of the function. HA and VA are the horizontal and vertical asymptotes.
The second prime tells us if the curve is concave up or concave
down, and the cusp. The first derivative indicates any maxima
The single sentence
mnemonic: Does Intercourse in a Horizontal position Simply
Drive Critics of Up and Down to take a Second look at Concubines?
Easy to recall now??
Limits -- A Parable
Imagine an action movie
actor who must run down a path dodging bullets, bombs, and debris.
To make the action look real, the director has to have the projectiles
come as close to the actor as possible without touching him,
for as soon as the position of the bullet equals the position
of actor, he dies and the whole process halts. Similarly, in
calculus, the Limit x --> 0 expression means that the
value for the denominator can come as close to zero as possible
without actually being zero and killing the equation. Just as
the movie director overcomes this problem by manipulating the
images so the action appears real but the actor lives, the math
student manipulates the factors to remove the formula killer
and to see what happens at that point. Usually, this involves
finding the zero factor hiding in the denominator and removing
it so we can see what happens when x = 0. We can:
- factor when we see
a perfect square or cube in the denominator
- multiply by conjugate
when we have a root in the denominator
- divide by highest exponent
when Limit --> infinity
My prof used to say
"This formula has a disease which makes it unable to
function. Mathematicians are surgeons who must find this disease
and remove it by cutting away misleading fat and tissue. The
disease is a zero virus which is always taking new shape and
hiding better." This is illustrated below, where you
can see a straight substitution of x=2 will result in
zero/zero. This zero virus is cured by a
dose of factoring medicine to make the function healthy.
Most problems in calculus
can be solved surprisingly easy -- if you follow the techniques.
I solve most calculus problems by taking 4 or 5 steps from this
list: find domain, find critical values, find formula, take the
derivative / anti-derivative, plug in the values, plot on a number
line, check your answer against the question, check your answer
against the domain, and factor the numerator and denominator.
In addition to these
nine, here are some sub-steps you may have to invoke to smooth
over bumps in the calculation process: simplify, use the quadratic
formula, factor top and bottom to remove common factors from
the numerator and denominator, know when to apply the Chain Rule,
U-substitution, look for horizontal or vertical asymptotes, find
where f'=0, find where f''=0, know the trig identities,
know how to factor the difference of squares/cubes, remember
to use absolute values, and did I mention Simplify?
Oil Spill Problem
Sometimes a story is
better way of remembering the procedure for solving a problem
in math. Here is the famous Oil Spill Rate problem paraphrased.
I saw a BSA motorcycle
parked outside of the Pies Are Square Bakery and Computer
Shoppe. As usual, there was a lot of oil leaking from the
motorcycle and I saw the puddle's radius was increasing at a
constant rate. Inside, the clerk greets me. "I'll take the
PRIME of your PIE selection, the RADIUS
monitor, and I want to see your RATES." I asked the man
with SQUARE FEET for the time. He replied in seconds.
At that moment, the oil spill stopped.
The steps: sketch, find
the two relevant formulae and merge them, simplify with respect
to the variable, differentiate and solve for radius.
Newton's Dirty Quotient
Finding the slope of
the tangent line gives us rise/run, a relationship used heavily
in physics. When used with Limit h--> 0 it gives us
instantaneous velocity. Here's a cute story to help remember
I was working on MY
TAN when I saw this gorgeous woman jogging. Yes, I got a RISE
over her RUN and felt embarrassed, so I used NEWTON's Cream to
"relieve my sexual anxiety". Time was running out as
it APPROACHED THE ZERO hour, but I found the VELOCITY of my hand
was part of the solution.
The power rule is a
shortcut to finding the derivative of X. Instead of applying
Newton's Quotient, simply multiply the factor of X by the exponent,
then subtract 1 from that exponent.
Observe the animation
just above. See how the area of a square equals the square of
x? Now double the area, and the length of the side increases
by only the square root of two. Increase area to three times
the original size, and the length of a side increases by only
the square root of three, and so on. Differentiation (and its
reversed cousin, Integration) are based upon this phenomenon.
Thus, differentiation allows you to calculate the changing area
with respect to the changing sides of a square. This applies
not only to squares, but to rectangles, triangles, cones, spheres,
or just about all geometric figures.
of a curve is a technique with many practical applications, including
cost analysis, finding the most efficient way of building something,
or even streamlining a process.
One application of
the min/max problem is to calculate the maximum volume of a box
made from a sheet of paper from which the four corners are excised
and the sides folded. Let the dimensions of the paper be 5 by
3 feet, and the size of the corner we cut be x. Obviously,
if x=0, the volume is zero because we still have a flat
piece of paper. And if the 2 corners are just as big as the width
of the paper, then we're back to zero volume. Therefore, a graph
showing volume versus x will peak somewhere between 0
and 1.5 feet. The four steps to find where the graph peaks are:
FORMULA, DOMAIN, DERIVATIVE, MAXIMUM.
- Formula: l x w x
h is not 3 x 5 x X, for the length is 5-2x;
width is 3-2x; and height is x. Volume is (5-2x)
- Find the domain by
finding the critical points of the volume formula, that is, when
3-2x=0, when 5-2x=0, and when x=0. The domain
is (1.5, 2.5, 0). Remove 2.5 from the list, as the domain ends
when we reach 1.5.
- You must multiply out
the volume formula before taking the derivative. (5-2x)(3-2x)x
= 15x-16x2=4x3. The derivative is 15-32x+12x squared.
- To find the critical
points, we usually factor the formula to make it easy. In this
case, there is no clear factor so we apply the quadratic formula.
Remember: a=12, b=32, c=15. The quadratic
gives two solutions: approximately 2.06 and 0.61. Which one falls
within the domain? The solution: maximum volume is achieved when
x = 0.61 feet.