Finding Points of lnflection
Determine the points of inection and discuss the concavity of the graph of
f(x) = x4 - 41.
Solution Differentiating twice produces the following.
f(x) = x4 - 41' Write original function.
f(x) = 4x3 121'2 Find rst derivative.
f
Application: or Dinrentintion
Applvlng the Flm DetlvntlveTest
Find Lite relative extrema afx I ~ sin I in the interval (0. 21).
Solution Not: that j is cuntinunus on the interval (0, 27). The derivative off is
11x) = mix. To determine the critical numbers
moreover. II HEX) ax = rix) + L, men
Uftx) dx
= for) lltdllllltlllill l\ rlic "Invoua III uncommon.
REMARK The Power Rule for
Integration has the restriction that
n $ l_To evaluatejx" dx, you
must use the natural log rule. (See
Exercise 75.)
These two e
"ive maximum. Derivative Test implies that f(c) is a relative minimum. A proof of the second case is
e 4.30 left to you.
See LarsonCa/culus.com for Bruce Edwards's video of this proof. 3
Tips 7 E x A M P L E 4 Using the Second Derivative Test
:pared to ap
' f(x) <0
* Concave
downward
1 f(x)>0
: Concave
: : upward
.i'
. you can determine
.e graph of f.
2
1!
Inc lCSl llBrVdlS.
I E x A M P LE 1 Determining Concavity
Determine the open intervals on which the graph of
f(X) = (2/2
is concave upward or concave do
Finding Extrema on a Closed Interval
Find the extrema off(x) = 2x - 3.8 on the interval [A I. 3].
(a. 0) Maximum
: - ~-t-t- t Solutlon Begin by differentiating the function,
. f(x) = 2x 3x"J \Jintt- nnytnni ltmttlon.
(L-l) 2
(3.0-33/5) f '(x) = 2
.5 LOHCBVIIY ancl tne DBCDHD UEIIValIVe I851 ll!
Determining Concavitv
Determine the open intervals on which the graph of
is concave upward or concave downward.
" Solution Differentiating twice produces the following.
Concave 2 + I
x) = :1 _ 4 \rl
. f(x)<n
Concave
z downward
f"(x)>0 ! l f"(x)>0
Concave : l Concave
From the sign of f 2 you can determine
the concavity of the graph off.
Figure 4.25
[U apply llluUlElll Iml, lUCillC llIC .xvauucs Ell WHICH] 1 4 U Ufj UUCB IlUl Cxlkl.
Use these x-value
function at the critical numbers and the endpoints is the other part.
Finding Extrema on a Closed Interval
Find the extrema of
f(x) = 3x4 - 4x3
on the interval [ l, 2].
Solution Begin by differentiating the function.
f(x) = 3x4 - 4x3 Write original functi
ECE 45 WINTER QUARTER 2017
ELECTRICAL & COMPUTER ENGINEERING
Course:
ECE 45 Circuits and Systems
URL:
https:/sites.google.com/a/eng.ucsd.edu/ece-45-w2017/
Text:
E. Kudeki, D. C. Munson, Jr., Analog Signals and Systems, Pearson
Instructor:
Robert Lugannani
1.Cells: the basic unit of life
2.Leeuwenhoek : animalcules.
3.Hooke: observed cork cells.
4.Schleiden: all plant are made up of cells.
5.Schwaan: all animals are made up of cell.
6.Pasteur: living things come from other living things.
7.Cell Theory: -cel
1.()Lipids: Composed C,H,O with a H:O ratio greater than 2
2.Fatty acids: Insoluble in water
3.Saturated Fatty Acids: Contains as many H as can possible as room temperature
4.Unsaturated Fatty Acids: does not have as many H bonds as it could bond. Liquid
1.Cells: the basic unit of life
2.Leeuwenhoek : animalcules.
3.Hooke: observed cork cells.
4.Schleiden: all plant are made up of cells.
5.Schwaan: all animals are made up of cell.
6.Pasteur: living things come from other living things.
7.Cell Theory: -cel
1.()Lipids: Composed C,H,O with a H:O ratio greater than 2
2.Fatty acids: Insoluble in water
3.Saturated Fatty Acids: Contains as many H as can possible as room temperature
4.Unsaturated Fatty Acids: does not have as many H bonds as it could bond. Liquid