Chabot Mathematics
§2.1 Basics of
Differentiation
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Chabot College Mathematics
1
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx
Review §
1.6
 Any QUESTIONS About
• §1.6 → OneSided-Limits & Continuity
 Any QUESTIONS About
HomeWork
• §1.6 → HW-06
Chabot College Mathematics
2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx
§2.1 Learning Goals
 Examine slopes of
tangent lines and
rates of change
 Define the derivative,
and study its basic properties
 Compute and interpret a variety of
derivatives using the definition
 Study the relationship between
differentiability and continuity
Chabot College Mathematics
3
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx
Why Calculus?
 Calculus divides into the Solution of TWO
Main Questions/Problems
1. Calculate the SLOPE
of a CURVED-Line
Function-Graph at
any point
2. Find the AREA under
a CURVED-Line
Function-Graph between
any two x-values
Chabot College Mathematics
4
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx
Calculus Pioneers
 Sir Issac Newton Solved the CurvedLine Slope Problem
• See Newton’s MasterWork Philosophiae
Naturalis Principia Mathematica (Principia)
– Read it for FREE:
http://archive.org/download/newtonspmathema0
0newtrich/newtonspmathema00newtrich.pdf
 Gottfried Wilhelm von Leibniz Largely
Solved the Area-Under-the-Curve
Problem
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx
Calculus Pioneers
 Newton (1642-1727)
Chabot College Mathematics
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 Leibniz (1646-1716)
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx
Origin of Calculus
 The word Calculus
comes from the
Greek word for
PEBBLES
 Pebbles were
used for counting
and doing simple
algebra…
Chabot College Mathematics
7
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx
“Calculus” by Google Answers
 “A method of computation or calculation
in a special notation (like logic or
symbolic logic). (You'll see this at the
end of high school or in college.)”
 “The hard deposit of
mineralized plaque
that forms on the
crown and/or root of
the tooth. Also
referred to as tartar.”
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx
“Calculus” by Google Answers
 “The branch of mathematics involving
derivatives and integrals.”
 “The branch of mathematics that is
concerned with
limits and with the
differentiation
and integration
of functions”
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx
“Calculus” by B. Mayer
 Use “Regular” Mathematics (Algebra,
GeoMetry, Trigonometry) and see what
happens to the Dependent quantity
(usually y) when the Independent
quantity (usually x) becomes one of:
• Really, Really TINY
lim
h0
• Really, Really BIG (in Absolute Value)
lim or lim
x  
Chabot College Mathematics
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x  
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx
Calculus Controversy
 Who was first; Leibniz or Newton?
Derivatives
Integrals
 We’ll Do DERIVATIVES First
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx
What is a Derivative?
 A function itself
 A Mathematical Operator (d/dx)
 The rate of change of a function
 The slope of the
line tangent to
the curve
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx
The TANGENT Line
y
single point
of Interest
x
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx
Slope of a Secant (Chord) Line
y
 Slope, m, of Secant Line (− −) = Rise/Run
Rise y2  y1 f x  h   f  x 
m


x  h   x
Run x2  x1
x1, y1 
y1  f x
x
Chabot College Mathematics
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x2 , y2 
y2  f x  h
h
x
xh
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx
Slope of a Closer Secant Line
y
Rise y2  y1 f x  h   f  x 
m


x  h   x
Run x2  x1
f x 
y1  f x
x
Chabot College Mathematics
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h
x
xh
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx
Move x Closer & Closer
 Note that distance h is getting Smaller
y
x
x
Chabot College Mathematics
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xh
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx
Secant Line for Decreasing h
 The slope of the secant line gets closer and
closer to the slope of the tangent line...
y
x
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx
Limiting Behavior
 The slope of the secant lines get
closer to the slope of the tangent
line...
...as the values of h
get closer to Zero
 this Translates to…
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx
The Tangent Slope Definition
mtan
f x  h   f x 
 lim
h 0
x  h   x
 The Above Equation yields the SLOPE
of the CURVE at the Point-of-Interest
 With a Tiny bit of Algebra
mtan
f x  h   f x 
 lim
h 0
h
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx
Example  Parabola Slope
yx
2
want the slope
where x=2
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx
Example  Parabola Slope
 Use the Slope-Calc Definition
f ( x  h)  f ( x )
( x  h) 2  x 2
m  lim
 lim
h 0
h 0
h
h
0
x  2 xh  h  x
h( 2 x  h)
 lim
 lim
h 0
h 0
h
h
2
2
2
m  lim (2 x  h)  2 x  2  2  4
0
h 0
mx  2  2x  2  2  4
Chabot College Mathematics
21
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx
SlopeCalc ≡ DerivativeCalc
 The derivative IS the slope of the line
tangent to the curve (evaluated at a
given point)
 The Derivative (or Slope) is a LIMIT
 Once you learn the rules of derivatives,
you WILL forget these limit definitions
 A cool site for additional explanation:
• http://archives.math.utk.edu/visual.calculus/2/
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx
Delta (∆) Notation
 Generally in Math the Greek letter ∆
represents a Difference (subtraction)
 Recall the
Rise Change in y y2  y1 Δy
Slope Definition m  Run  Change in x  x2  x1  x
y
 See
Diagram
y
y
at
y
Right
2
Change in y
1
x
Chabot College Mathematics
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x1
x2
x Mayer, PE
Bruce
BMayer@ChabotCollege.edu • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx
Delta (∆) Notation
y
 From The Diagram
Notice that at Pt-A
the Chord Slope,
AB, approaches the
Tangent Slope, AC,
as ∆x gets smaller
y2
y
x
 Also:
x1
x2  x1  x
y1  f x1 
y2  f x2   f x1  x 
Change in y
y1
m AB
Δy y2  y1 f  x1  x   f  x1 



x x2  x1
x1  x  x1
0
 Then →
Chabot College Mathematics
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x
x2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx
∆→d Notation
 Thus as ∆x→0 The
Chord Slope of AB
approaches the
Tangent slope of AC
 Mathematically
m A  lim m AB
x 0
y
 lim
x 0 x
y
y2
y
Change in y
y1
x
x1
f  x1  x   f  x1 
m A  lim
x 0
x
x2
x
y dy df  x  d
lim



f x 
x 0 x
dx
dx
dx
 Thus
dy
f  x1  x   f  x1 
 Now by Math
 lim
x 0
dx
x
Notation Convention:
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx
∆→d Notation
y
 The Difference
between ∆x & dx:
y2
dy
y
• ∆x ≡ a small but
FINITE, or Calcuable,
Difference
xxdx
• dx ≡ an Infinitesimally
x1
x
small, Incalcuable,
 See the Diagram
Difference
y1
Change in y
2
 ∆x is called a
DIFFERENCE
 dx is called a
Differential
Chabot College Mathematics
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above for the a
Geometric
Comparison of
• ∆x, dx, ∆y, dy
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx
x
Derivative is SAME as Slope
 From a y = f(x) graph we see that the
infinitesimal change in y resulting from
an infinitesimal change
dy
in x is the Slope at the
m
dx
point of interest. Generally:
 The Quotient dy/dx is read as:
“The DERIVATIVE of y with respect to x”
 Thus “Derivative” and “Slope” are
Synonymous
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx
d → Quantity AND Operator
 Depending on the
Context “d” can
connote a quantity
or an operator
 Recall from before
the example y = x2
MTH15 • Bruce Mayer, PE • dy/dx
40
35
( x  h) 2  x 2
dy
m  lim
 2x 
h 0
h
dx
 We could also “take
the derivative of y = x2
with respect to x using
the d/dx OPERATOR
 
30
y = f(x) = x2
 Recall the Slope Calc
dy d
d
d 2

y
f x  
x  2x
dx dx
dx
dx
25
20
15
10
5
0
-6
XY f cnGraph6x6BlueGreenBkGndTemplate1306.m
-5
-4
-3
-2
Chabot College Mathematics
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-1
0
x
1
2
3
4
5
6
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx
d → Quantity AND Operator
 dy & dx (or d?) Almost
y dy
Always appears as a lim

x  0  x
dx
Quotient or Ratio
 d/dx or (d/d?) acts as
d 2
an OPERATOR that
x  2x
takes the Base-Function dx
and “operates” on it to
produce the
Slope-Function; e.g.
 
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx
Prime Notation
 Writing dy/dx takes too much work;
need a Shorthand notation
 By Mathematical Convention define the
“Prime” Notation as
f ( x  h)  f ( x )
f ' ( x)  lim
h 0
h
y
 lim
 y'
x 0 x
• The “Prime” Notation is more compact
• The “d” Notation is more mathematically
Versatile
– I almost always recommend the “d” form
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx
Average Rate of Change
 The average rate of
change of function f on
the interval [a,b] is given by
f (b) - f (a)
b-a
 Note that this is simply the Secant, or
Chord, slope of a function between two
points (x1,y1) = (a,f(a)) & (x2,y2) = (b,f(b))
Chabot College Mathematics
31
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx
Example  Avg Rate-of-Change
40
35
30
y = f(x) = x2
 For f(x) = y = x2 find
the average rate of
change between x = 3
(Pt-a) and x = 5 (Pt-b)
 By the Chord Slope
MTH15 • Avg Rate-of-Change
45
25
y
20
15
10
x
5
0
XY f cnGraph6x6BlueGreenBkGndTemplate1306.m
0
1
2
3
4
x
Chabot College Mathematics
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6
f (b)  f (a ) 5  3 16 y




8
ba
53
2 x
2
mavg
5
2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx
7
Example  Avg Rate-of-Change
MTH15 • Avg Rate-of-Change
45
40
MTH15 • Avg Rate-of-Change
35
25
25
y
20
15
10
x
5
0
XY f cnGraph6x6BlueGreenBkGndTemplate1306.m
0
1
2
3
4
x
5
6
7
y = f(x) = x2
y = f(x) = x2
30
20
Chord
Slope
y
15
10
3
x
XY f cnGraph6x6BlueGreenBkGndTemplate1306.m
4
x
Chabot College Mathematics
33
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx
5
Chabot College Mathematics
34
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx
MATLAB Code
% Bruce Mayer, PE
% MTH-15 • 01Jul13
% XY_fcn_Graph_BlueGreenBkGnd_Solid_Marker_Template1306.m
%
% The Limits
xmin = -3; xmax1 = 1; xmin2 = xmax1; xmax = 3; ymin = -4; ymax = 10;
% The FUNCTION
x1 = linspace(xmin,xmax1,500); y1 = 1-x1.^2;
x2 = linspace(xmin2,xmax,500); y2 = 3*x2+1;
% The Total Function by appending
x = [x1, x2]; y = [y1, y2];
%
% The ZERO Lines
zxh = [xmin xmax]; zyh = [0 0]; zxv = [0 0]; zyv = [ymin ymax];
%
% the 6x6 Plot
axes; set(gca,'FontSize',12);
whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Green
plot(x1,y1,'b', x2,y2,'b', zxv,zyv, 'k', zxh,zyh, 'k', x1(end),y1(end),
'ob', 'MarkerSize', 12, 'MarkerFaceColor', 'b',...
'LineWidth', 3),axis([xmin xmax ymin ymax]),...
grid, xlabel('\fontsize{14}x'), ylabel('\fontsize{14}f(x) \rightarrow
PieceWise'),...
title(['\fontsize{14}MTH15 • Bruce Mayer, PE • 2-Sided Limit',]),...
annotation('textbox',[.51 .05 .0 .1], 'FitBoxToText', 'on', 'EdgeColor',
'none', 'String',
'XYfcnGraphBlueGreenBkGndSolidMarkerTemplate1306.m','FontSize',7)
hold on
plot(x2(1),y2(1), 'ob', 'MarkerSize', 12, 'MarkerFaceColor', [0.8 1 1],
'LineWidth', 3)
set(gca,'XTick',[xmin:1:xmax]); set(gca,'YTick',[ymin:1:ymax])
hold off
Slope vs. Rate-of-Change
 In general the Rateof-Change (RoC) is
simply the Ratio, or
Quotient, of Two
quantities. Some
Examples:
 A Slope is a
SPECIAL RoC
where the UNITS of
the Dividend and
Divisor are the
SAME. Example
• Pay Rate → $/hr
• Speed → miles/hr
• Fuel Use → miles/gal
• Paper Use →
words/page
Chabot College Mathematics
35
• Road Grade →
Feet-rise/Feet-run
• Tax Rate →
$-Paid/$-Earned
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx
Example  Rice is Nice
 The demand for rice
in the USA in 2009
approximately
followed the function
100
D( p) 
p
 Use this Function to:
a) Find and interpret
D' 500
b) Find the equation of
the tangent line to
D at p = 500.
• Where
– p ≡ Rice Price in
$/Ton
– D ≡ Rice Demand in
MegaTons
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx
Example  Rice is Nice
 SOLUTION
a) Using the definition
of the derivative:
dD
D ( p  h)  D ( p )
 lim
dP h0
h
100
100
p+h
p
= lim
h®0
h
Chabot College Mathematics
37
 Clear fractions by
multiplying by p × p + h.
100
100

ph
p
dD
 lim

dp h0
h
p ph
p ph
 Simplifying
p  ph
dD
 100  lim
h0 h p 
dP
ph
• Note the Limit is
Undefined at h = 0
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx
Example  Rice is Nice
 Remove the UNdefinition by multiplying
by the Radical Conjugate of the
Numerator: p  p  h
p  ph
p  ph
dD
D'  p  
 100  lim

h0 h p 
dp
ph
p  ph
p  p p  h  p p  h  ( p  h)
dD
 100  lim
h0
dp
h p ph p  ph


h
D'  p   100  lim
h0 h p 
ph p  ph

Chabot College Mathematics
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
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx
Example  Rice is Nice
 Continue the Limit Evaluation
dD
 100  lim
h0
dp
=100
= 100
1
p ph p  ph

-1
p × p+0
(
-1
p 2 p
(
p + p+0

)
)
D'  p   50 p 3 / 2
Chabot College Mathematics
39
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx
Example  Rice is Nice
 Run-Numbers to Find the Change in
DEMAND with respect to PRICE
D'  p   50 p 3 / 2
 D' 500   50500 
3 / 2
 0.00447.
 Unit analysis for dD/dp
dD
dp

MTon
$ Ton
106 Ton

$ Ton
106 Ton Ton


1
$
106 Ton 2

$
 Finally State: for when p = 500 the Rate
of Change of Rice Demand in the USA:
106 Ton 2
Ton 2
D' 500  0.00447 
 4470
$
$
Chabot College Mathematics
40
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx
Example  Rice is Nice
 Thus The RoC for D w.r.t. p at p = 500:
2
Ton
Ton
D' 500  4470
 4470
$
$ Ton
 Negative Derivative???!!!
• What does this mean in the context?
 Because the derivative is negative, at a
unit price of $500 per ton, demand is
decreasing by about 4,470 tons per
$1/Ton INCREASE in unit price.
Chabot College Mathematics
41
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx
Example  Rice is Nice
 SOLUTION
b) Find the equation of the tangent line to
D at p = 500
 The tangent line to a function f is
defined to be the line passing through
the point and having a slope equal to
the derivative at that point.
Chabot College Mathematics
42
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx
Example  Rice is Nice
 First, find the value
100
D(500) 
 4.47 MegaTons
500
of D at p = 500:
 So we know that the tangent line
passes through the point (500, 4.47)
 Next, use the derivative of D for the
slope of the tangent line:
dD
dp p 500  50500 
Chabot College Mathematics
43
3 / 2
 0.00447
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx
Example  Rice is Nice
 Finally, we use the point-slope formula
for the Eqn of a Line and simplify:
y - y1 = m(x - x1 )
y - 4.47 = -0.00447(x - 500)
y = -0.00447x + 6.707
 The Graph of
D(p) and the
Tangent Line
at p = 500 on
the Same Plot:
Chabot College Mathematics
44
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx
Operation vs Ratio
 In the Rice Problem we could easily
write D’(500) as indication we were
EVALUATING the derivative at p = 500
 The d notation is not so
? dD
? dD500 
dD
ClearCut. Are these
500 

dp dp
dp
things the SAME?
 Generally They are NOT
• The d/dx Operator Produces the Slope
Function, not a NUMBER
• Find dy/dx at x = c DOES make a Number
Chabot College Mathematics
45
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx
“Evaluated at” Notation
 The d/dx operator produces the Slope
Function dy/dx or df/dx; e.g.:
y  f x   x 2  7 x 


d
 y  f x   dy  df  d x 2  7 x  2 x  7
dx
dx dx dx
 2x+7 is the Slope Function. It can be
used to find the slope at, say, x = −5 & 4
• y’(−5) = 2(−5) + 7 = −10 + 7 = −3
• y’(4) = 2(4) + 7 = 8 + 7 = 5
 Use Eval-At Bar to Clarify a NumberSlope when using the “d” notation
Chabot College Mathematics
46
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx
Eval-At BAR
 To EVALUATE a derivative a specific
value of the Indepent Variable Use the
“Evaluated-At” Vertical BAR.
 Eval-At BAR Usage → Find the value of
the derivative (the slope) at x = c (c is a
df
NUMBER): dy  y' C 
 f ' C 
dx
 Often the “x =”
is Omitted
Chabot College Mathematics
47
dx
x c
dy
 y' C 
dx c
x c
df
 f ' C 
dx c
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx
Example: Eval-At bar
 Consider the Previous
2
y  f x   x  7 x
f(x) Example:
 Using the d notation to find the Slope
(Derivative) for x = −5 & 4


d
d 2
dy
y
x  7x  2x  7 
dx
dx
dx
dy
dx
 2 5  7  3
x  5
Chabot College Mathematics
48
dy
 24  7  15
dx 4
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx
Continuity & Smoothness
 We can now define a “smoothly” varying
Function
 A function f is differentiable at x=a if
f’(a) is defined.
• e.g.; no div by zero, no sqrt of neg No.s
 IF a function is differentiable at a point,
then it IS continuous at that point.
• Note that being continuous at a point does
NOT guarantee that the function is
differentiable there.
Chabot College Mathematics
49
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx
Continuity & Smoothness
 A function, f(x), is SMOOTHLY Varying
at a given point, c, If and Only If df/dx
Exists and:
df
lim
x c dx
x c
df
 K  lim
x c dx
x c
• That is, the Slopes
are the SAME when
approached from
EITHER side
Chabot College Mathematics
50
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx
WhiteBoard Work
 Problem From §2.1
MTH15 • P2.1-46
• P46 → Declining
Marginal
Productivity
250
Q (k-Units)
200
150
100
50
0
XY f cnGraph6x6BlueGreenBkGndTemplate1306.m
0
1
2
3
4
L (k-WorkerHours)
Chabot College Mathematics
51
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx
5
6
All Done for Today
A Different
Type of
Derivative
Chabot College Mathematics
52
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx
Chabot Mathematics
Appendix
r  s  r  s r  s 
2
2
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
–
Chabot College Mathematics
53
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx
Chabot College Mathematics
54
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx
Chabot College Mathematics
55
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx
Chabot College Mathematics
56
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx
P2.1-46
Chabot College Mathematics
57
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx