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Differentiation
We define the derivative of a continuous function as
Notice that this is a point-wise definition. The derivative is evaluated at each
and is said to exist at each iff the limit above exists.
Another equivalent definition is
The standard rules of differentiation are all derived from these limit definitions.
At any point where the derivative exists, the function must be continuous. However, the converse is not true: a function may be continuous
at a point but not differentiable. The following examples are illustrative.
Green Question: Find where
(A)
(B)
(C)
Derivative formulas
Skill Set
Basic Diff Formulas
No. 1
STEP 2005, Math II, #1: Find the three values of
for which the derivative of is zero.
Red Question: Suppose you are given distinct positive numbers,
and . Find a polynomial such that the derivative of is
zero for
, and , but for no other values of .
STEP 2006, Math I, #4: By sketching on the same axes the graphs of and , show that, for ;
(i)
(ii)
for small .
A regular polygon has
sides, and perimeter . Show that the area of the polygon is
.
Show by differentiation (treating
as a continuous variable) that the area of the polygon increases as increases with fixed.
Show also for large
, the ratio of the area of the polygon to the area of the smallest circle which can be drawn around the polygon is approximately 1.
A complete solution to this problem is posted below.
STEP 2011, Math II, #3: In this question, you may assume without proof that any function
for which is increasing,
that is
if .
(i) (a) Let
. Show that is increasing for and deduce that for .
(b) Given that for , show that
for
(c) Let
for . Show that is increasing and deduce that
for
(ii) Given that for show by considering the function that
for
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