The Tangent Problem. Determine the line tangent to a given curve
at a given point. (Also, 1a: Define precisely the concept of "tangent.")
For example, determine the line in the plane tangent to the circle with
equation X 2 + Y 2 = 25 at the point (-4,3).
The Velocity Problem. Determine the instantaneous velocity of a
moving object. (Also, 2a: Define precisely the concept of "instantaneous
velocity.") For example, determine the instantaneous velocity of an object
moving on a straight line at time t = 5 seconds when its position at time
t seconds is t 2 + 6t meters from a given point P.
The Extremum Problems. Determine the maximum and minimum values
of a variable quantity that is dependent upon another variable quantity.
For example, suppose the variable Y depends on the variable X with Y =
X 2 - 6X. Determine any maximum and minimum values for the dependent variable
Y when the variable X is allowed to vary over real numbers between 0 and
The Area Problem. Determine the area of a planar region enclosed
by a suitably defined curve. For example, determine the area of the planar
region enclosed by the X - axis, the lines X = 2, X = 5, and the
parabola with equation Y = X 2 - 6X.
The Arc Length Problem. Determine the length of a suitably defined
curve. For example, determine the length of the parabola with equation
Y = X 2 - 6X between the points (0,0) and (6,0).
The Tangent Curve Problem Reversed. Determine a curve so that the
tangent to the curve at any point on the curve is predetermined by some
description depending on the point's position in the plane. For example,
determine a curve in a coordinate plane so that the slope of the tangent
to the curve at the point (a,b) is a + b.
The Position Problem. (The Velocity Problem Reversed.) Determine
the position of an object moving on a straight line from knowledge of its
initial position and instantaneous velocity at every instant. For example,
determine the position of an object moving on a straight line at
time t = 5 seconds knowing its initial position is P on the line and its
instantaneous velocity at time t is precisely t 2 - 6t meters per second.
The Growth/Decay Problem. Given the size of a population or populations
and the rates of growth or decay of the population(s) as functions of the
time and current population sizes, describe the size of the population(s)
in the future or in the past. For example suppose a population is measured
by its biomass and it currently has a biomass of 1000 kilograms. Suppose
the population is growing at a rate that is proportional to its current
size and that one hour ago its biomass was 950 kilograms. Find
the biomass in 10 hours. Find how long it will take before the biomass
will be 2000 kilograms, 5,000 kilograms, and 10,000 kilograms.
The Probability Expected Value - Mean Problem. In an experiment
where we measure the outcome of a variable X, suppose the probability that
X< A is given by some function F(A). What is the expected value of X?
that is, what number in theory would be close to the average or mean of
the values for X if the experiment is repeated a large number of times.
For example, suppose we throw a magnetic dart at a circular dart board
of radius 2 feet and we measure X as the distance from center of the board
to where the dart lands. Suppose that the probability that X< A is given
by the function F(A)= A2/4 for 0< A < 2; 0 for A < 0; and 1 for
2 < A. What is the expected value of X? that is, what is the expected
distance of the dart from the center of the target?
Precalculus Mathematics: So now that you have a better idea
of what mathematical problems are connected to the "calculus," what
is "precalculus" mathematics? Here the simple answer we gave earlier
can be expanded. Precalculus mathematics is mathematics that gives
background for the mathematical concepts, problems, issues and techniques
that appear in the calculus course. Certainly one key background tool for
the calculus is the function concept. Being familiar with function concepts
and specific functions provides an important foundation and language for
To understand calculus you should have a background that allows you
So what is the sensible approach to prepare you for calculus? In
the sensible precalculus approach we will
numbers and variables in the context of algebra,
equations and functions both algebraically and visually, and
"real world" applications that use functions to relate the quantities
Review and renew your understanding of numbers and variables as used in
Review and renew your understanding of equations both algebraically and
Review, renew, and expand your understanding of functions both algebraically
Connect "real world" applications to the equations and functions we introduce
Introduce problems of the type encountered in calculus when precalculus
techniques can be used for solution.
Introduce methods from current technology that will make precalculus analysis
easier and form a foundation for later calculus use of technology.