Unformatted text preview: ti v e (>0).
increasing
positiv
Conversely : if y ’(x) is positi v e then y(x) is incr easing
positiv
increasing
easing.
minimum.
Let f(x) be a function and b the point where f(x) is minimum
At a minimum point b the value of the function is lesser than
the values around it. Let δ x be a small change in x.
b
b
Then f(b  δ x) > f(b) and f(b + δ x) > f(b)
b
b
At a minimum point f ’(x) is
neg tiv befor zer positiv after.
ne g a ti v e bef or e , z er o , positi v e after
What can you say about f ”(b) ?
b
114 Exercise 1: Find an instant where f(x) = x 2 + x h as an extreme value
(maximum or minimum).
Exercise 2: Find an instant where f(x) =  x 2 + x h as an extreme value
(maximum or minimum).
Exer
ercise
Ex er cise 3: Draw the curves of the functions below and answer the following :
a) f(x) = x 2 for x < 0 . The shape of the curve is concave upwards
and falling.
b) f(x) =  x 2 f or x > 0 . The shape of the cur ve is concave downwards
and falling.
c) f(x) = x 2 for x > 0 . The shape of the curve is concave upwards
and rising.
d) f(x) =  x 2 for x < 0 . The shape of the cur ve concave downwards
and rising.
i) What kind of angle does the tangent to the cur ve make ?
(acute, obtuse) . What is the sign of the tangent ?
ii) What is the sign of the first derivative of the function ?
(positive, negative)
iii) Is the function increasing or decreasing ?
Exercise 4: Repeat exercise 3 for the functions below and describe the shape of
the curve.
a) f(x) = x 3 f or x < 0 .
b) f(x) = x 3 f or x > 0 .
c) f(x) =  x 3 f or x < 0 .
d) f(x) =  x 3 f or x > 0 .
115 2 3. MAXIMA and MINIMA Given a function f(x) how can we find the instants where f(x) has maximum
or minimum values ?
If the point a is an extr eme then the first derivative must equal zero.
xtreme
Solve the equation f ’(a) = 0 to find a .
a
This information is necessary but not sufficient to know whether the point a
is a maximum point or a minimum point. There are three methods.
Method 1 : We need to fur ther check around a if...
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This note was uploaded on 11/29/2012 for the course PHYSICS 105 taught by Professor Tamerdoğan during the Fall '09 term at Middle East Technical University.
 Fall '09
 TAMERDOğAN

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