MATH 241, LECTURE 9
1. The Derivative
The derivative of a function f at some point x is the slope of the tangent line to the graph of f at the
point (x, f (x). We can collect these numbers all togethe
MATH 241, LECTURE 10
1. Practice with using rules for derivatives Right now we have three short-cut rules for taking derivatives (we will soon learn more - enough to take the derivative of any functio
MATH 241, LECTURE 11
1. Limits at finite points
Denition 1. (Intuitive) We say the limit of f (x) as x tends to c is L if the values f (x) are always
arbitrarily close to L once x is close enough to c
MATH 241, LECTURE 12
1. Limits and their use for derivatives
Last time we focussed on intuitive methods for nding limits, and introduced a few algebraic methods.
Lets begin by practicing our algebraic
MATH 241, LECTURE 13
Next we continue to learn rules for dierentiation. These are basic skills we must memorize in order to
apply the derivative in dierent settings.
1. The product rule
We have alread
MATH 241, LECTURE 14
0.1. Practice with the quotient rule. Remember our mnemonic, dwarf-friendly form of the quotient
rule as we apply it a few times.
Example 1. Take the derivatives of:
1
x2 .
x
x2
MATH 241, LECTURE 15
1. Lots of practice with the chain rule
Just as synonymous words can be more or less useful in dierent contexts, dierent mathematical
notation for the same concept can be helpful
MATH 241, LECTURE 16
0.1. Partial substitution in applications of the chain rule. When we are using the chain rule to get
a numerical answer, as opposed to a formula for the derivative, we can save a
MATH 241, LECTURE 7
1. Average rate of change
A fundamental philosophical truth is that everything changes. In physics, the change in position is
known as velocity or speed. In economics, the change i
MATH 241, LECTURE 6
1. Logarithmic functions
Many common functions arise through undoing basic functions (more formally, we say they are inverses
of basic functions). For example, subtraction was inve
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MATH 241: CALCULUS FOR BUSINESS AND THE SOCIAL SCIENCES,
LECTURE 1
1. What to expect:
Lecture classes MWF 2, Mackenzie 240A.
Review sections on M or Tu. Quiz most weeks at the beginning of class tim
MATH 251, LECTURE 3
1. The ins and outs of linear functions
There are many ways to describe a given linear function. An important skill to develop is the ability to
translate between the dierent descr
MATH 241, LECTURE 8
1. The derivative
1.1. Slopes of tangent lines.
Denition 1. A tangent line to a curve is a line which intersects the curve at some point, but does not
cross the curve.
Informally,
MATH 241, LECTURE 2
1. Functions: the basic objects of study in calculus
Functions are means for producing one quantity (or set of quantities) from another. More formally we
have the following.
Deniti
MATH 241, LECTURE 4
1. Quadratic functions
After linear functions, the next simplest kinds of functions are quadratic functions. We will review basic
properties of these functions now, so they are fam
MATH 241, LECTURE 17
0.1. Some practice with logarithms, exponential functions, and the chain rule. Derivative rules
are great fun because they can be combined. Taking the derivative of some functions