Calculus I Syllabus
We will follow the order of topics in the text, Salas, Helle, & Etgen's Calculus,
but we'll stress some topics and pass over some others.
Review and Preview
Preview: Calculus is about the relation between a quantity and its rate of change.
For an example, if the quantity is the distance travelled at a given time, then its rate
of change is velocity. If the velocity is constant, then calculus is not required: the
distance travelled is the product of the elapsed time and the velocity. But when the
velocity is not constant, then this formula doesn't apply. Nonetheless, the distance
and velocity are intimately related. If the distance travelled at all times is known,
then the velocity at any given time can be determined; and if the velocity at all times
is known, then the distance travelled at any given time can be determined. These two
operations are called differentiation and integration.
Much of calculus involves analyzing and developing these concepts and their
applications.
Review: The review only contains things that you should already know, but
it helps to see them again just before you use them. Concepts under review are real
numbers, functions, intervals and inequalities, graphs of functions, absolute value,
piecewise defined functions, symmetry and even and odd functions, operations on graphs,
and composition of functions. A variety of functions are reviewed including linear
functions (and with them slopes of lines), power functions, polynomials, rational
functions, trigonometric, exponential, and logarithmic functions. Notations under
review include functional notation and substitution, interval notation, unions,
intersections, set notation, and algebra of functions.
These are discussed in chapter 1 of the text, but we will not go over the chapter
section by section.
Limits and Continuity
We first must clarify the concept of derivative. In some ways it is intuitively clear
that a travelling body has a velocity, or more generally, any changing quantity has a
rate of change. But just what is the rate of change? The answer is the rate of
change at an instant is the limit of the average rates of change near that instant.
The concept of limit is much more subtle than it first appears. We will discuss it
in some detail and develop a formal defintion of a limit and a formal notation to go
along with it.
Key concepts associated to the concept of limit are tangent lines, limit laws,
continuity, the pinching theorem, left- and right-limits, trigonometric limits,
the intermediate value theorem (IVT), and the exterme value theorem (EVT).
Chapter 2 Limits and Continuity
2.1 The Idea of Limit
2.2 Definition of Limit
2.3 Some Limit Theorems, Additional information on Infinite Limits
2.4 Continuity
2.5 The Pinching Theorem; Trigonometric Limits
2.6 Two Basic Properties of Continuous Functions
Derivatives
With a solid defintion of limit, we can proceed to define a derivative (instantaneous
rate of change) as the limit of average rates of change, and then develop the
properties of derivatives. There are a number of rules for differentiation (finding
derivatives), mostly easily learned, although the chain rule, for some reason, seems
to be more difficult to master. There are a couple of different notations for
derivatives that everyone uses.
It is assumed that you know the trig functions, sine, cosine, etc., and we will
find and use their derivatives.
Further topics in differentiation include implicit differentiation, and higher
derivatives.
Chapter 3 Differentiation
3.1 The Derivative
3.2 Some Differentiation Formulas
3.3 The d/dx Notation; Derivatives of Higher Order
3.4 The Derivative as a Rate of Change
3.5 The Chain Rule
3.6 Differentiating the Trigonometric Functions
3.7 Implicit Differentiation; Rational Powers
The Mean Value Theorem and Curve Sketching
The purpose of curve sketching is no so much to draw the graph of the
function, but to get a better understanding of the relation between a
function and its derivative. For instance, if the derivative is positive,
then the function is increasing; at a maximum or a minimum of a function,
the derivative is zero. We will prove these (obvious) statements using a
theorem called the mean value theorem. We'll also see what second derivatives
have to do with the graph of a function.
The applications of derivatives are numerous. Besides classical applications in
physics and the natural sciences, there are applications in the social sciences, for
instance, marginal profits are just derivatives of profits.
Chapter 4 The Mean-Value Theorem and Applications
4.1 The Mean-Value Theorem
4.2 Increasing and Decreasing Functions
4.3 Local Extreme Values
4.4 Endpoint and Absolute Extreme Values
4.5 Some Max-Min Problems
4.6 Concavity and Points of Inflection
4.7 Vertical and Horizontal Asymptotes
4.8 Curve Sketching
4.9 Velocity and Acceleration; Speed
4.10 Related Rates of change per Unit Time
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