[email protected] mth15_lec-07_sec_2-1_differeniatation-basics_.pptx 1 bruce mayer, pe chabot...
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[email protected] • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx1
Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Chabot Mathematics
§2.1 Basics of
Differentiation
[email protected] • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx2
Bruce Mayer, PE Chabot College Mathematics
Review §
Any QUESTIONS About• §1.6 → OneSided-Limits & Continuity
Any QUESTIONS About HomeWork• §1.6 → HW-06
1.6
[email protected] • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx3
Bruce Mayer, PE Chabot College Mathematics
§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
[email protected] • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx4
Bruce Mayer, PE Chabot College Mathematics
Why Calculus?
Calculus divides into the Solution of TWO Main Questions/Problems1. 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
[email protected] • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx5
Bruce Mayer, PE Chabot College Mathematics
Calculus Pioneers
Sir Issac Newton Solved the Curved-Line Slope Problem• See Newton’s MasterWork Philosophiae
Naturalis Principia Mathematica (Principia)– Read it for FREE:
http://archive.org/download/newtonspmathema00newtrich/newtonspmathema00newtrich.pdf
Gottfried Wilhelm von Leibniz Largely Solved the Area-Under-the-Curve Problem
[email protected] • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx6
Bruce Mayer, PE Chabot College Mathematics
Calculus Pioneers Newton (1642-1727) Leibniz (1646-1716)
[email protected] • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx7
Bruce Mayer, PE Chabot College Mathematics
Origin of Calculus
The word Calculus comes from the Greek word for PEBBLES
Pebbles were used for counting and doing simple algebra…
[email protected] • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx8
Bruce Mayer, PE Chabot College Mathematics
“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.”
[email protected] • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx9
Bruce Mayer, PE Chabot College Mathematics
“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”
[email protected] • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx10
Bruce Mayer, PE Chabot College Mathematics
“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
• Really, Really BIG (in Absolute Value)0
limh
xxlimorlim
[email protected] • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx11
Bruce Mayer, PE Chabot College Mathematics
Calculus Controversy
Who was first; Leibniz or Newton?
We’ll Do DERIVATIVES First
Derivatives Integrals
[email protected] • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx12
Bruce Mayer, PE Chabot College Mathematics
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
[email protected] • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx13
Bruce Mayer, PE Chabot College Mathematics
The TANGENT Line
single pointof Interest
x
y
[email protected] • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx14
Bruce Mayer, PE Chabot College Mathematics
Slope of a Secant (Chord) Line
x
y
h
Slope, m, of Secant Line (− −) = Rise/Run
xhx
xfhxf
xx
yym
12
12
Run
Rise
x xfy 1
hx
hxfy 2
22 , yx
11, yx
[email protected] • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx15
Bruce Mayer, PE Chabot College Mathematics
Slope of a Closer Secant Line
x
y
h
xhx
xfhxf
xx
yym
12
12
Run
Rise
x xfy 1
hx
xf
[email protected] • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx16
Bruce Mayer, PE Chabot College Mathematics
Move x Closer & Closer
x
y
xhx
Note that distance h is getting Smaller
[email protected] • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx17
Bruce Mayer, PE Chabot College Mathematics
Secant Line for Decreasing h
x
y
The slope of the secant line gets closer and closer to the slope of the tangent line...
[email protected] • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx18
Bruce Mayer, PE Chabot College Mathematics
Limiting Behavior
The slope of the secant lines get closer to the slope of the tangent line...
...as the values of hget closer to Zero
this Translates to…
[email protected] • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx19
Bruce Mayer, PE Chabot College Mathematics
The Tangent Slope Definition
The Above Equation yields the SLOPE of the CURVE at the Point-of-Interest
With a Tiny bit of Algebra
xhx
xfhxfm
h
0
tan lim
h
xfhxfm
h
0tan lim
[email protected] • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx20
Bruce Mayer, PE Chabot College Mathematics
Example Parabola Slope
want the slopewhere x=2
2xy
[email protected] • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx21
Bruce Mayer, PE Chabot College Mathematics
Example Parabola Slope
Use the Slope-Calc Definition
h
xhx
h
xfhxfm
hh
22
00
)(lim
)()(lim
h
hxh
h
xhxhxhh
)2(lim
2lim
0
222
0
4222)2(lim0
xhxmh
0
0
42222 xxm
[email protected] • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx22
Bruce Mayer, PE Chabot College Mathematics
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/
[email protected] • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx23
Bruce Mayer, PE Chabot College Mathematics
Delta (∆) Notation
Generally in Math the Greek letter ∆ represents a Difference (subtraction)
Recall the Slope Definition
SeeDiagramat Right
x
y
xx
yy
x
ym
Δ
in Change
in Change
Run
Rise
12
12
yin Change
1x 2x
x
y
1y
2y
x
y
[email protected] • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx24
Bruce Mayer, PE Chabot College Mathematics
Delta (∆) Notation From The Diagram
Notice that at Pt-A the Chord Slope, AB, approaches the Tangent Slope, AC, as ∆x gets smaller
Also:
Then →
yin Change
1x 2x
x
y
1y
2y
x
y
xxfxfy
xfy
xxx
122
11
12 11
11
12
12Δ
xxx
xfxxf
xx
yy
x
ymAB
0
[email protected] • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx25
Bruce Mayer, PE Chabot College Mathematics
∆→d Notation Thus as ∆x→0 The
Chord Slope of AB approaches the Tangent slope of AC
Mathematically
Now by Math Notation Convention:
Thus
yin Change
1x 2x
x
y
1y
2y
x
y
x
ymm
xAB
xA
00limlim
x
xfxxfm
xA
11
0lim
xfdx
d
dx
xdf
dx
dy
x
yx
0lim
x
xfxxf
dx
dyx
11
0lim
[email protected] • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx26
Bruce Mayer, PE Chabot College Mathematics
∆→d Notation The Difference
between ∆x & dx:• ∆x ≡ a small but
FINITE, or Calcuable, Difference
• dx ≡ an Infinitesimally small, Incalcuable, Difference
∆x is called a DIFFERENCE
dx is called a Differential
See the Diagram above for the a Geometric Comparison of • ∆x, dx, ∆y, dy
yin Change
1x 2x
x
y
1y
2y
x
y
dy
dxx
[email protected] • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx27
Bruce Mayer, PE Chabot College Mathematics
Derivative is SAME as Slope
From a y = f(x) graph we see that the infinitesimal change in y resulting from an infinitesimal change in x is the Slope at the 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
dx
dym
[email protected] • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx28
Bruce Mayer, PE Chabot College Mathematics
d → Quantity AND Operator Depending on the
Context “d” can connote a quantity or an operator
Recall from before the example y = x2
Recall the Slope Calc
We could also “take the derivative of y = x2 with respect to x using the d/dx OPERATOR
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 60
5
10
15
20
25
30
35
40
x
y =
f(x)
= x
2
MTH15 • Bruce Mayer, PE • dy/dx
XYf cnGraph6x6BlueGreenBkGndTemplate1306.m
dx
dyx
h
xhxm
h
2
)(lim
22
0
xxdx
dxf
dx
dy
dx
d
dx
dy22
[email protected] • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx29
Bruce Mayer, PE Chabot College Mathematics
d → Quantity AND Operator
dy & dx (or d?) Almost Always appears as a Quotient or Ratio
d/dx or (d/d?) acts as an OPERATOR that takes the Base-Function and “operates” on it to produce the Slope-Function; e.g.
dx
dy
x
yx
0lim
xxdx
d22
[email protected] • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx30
Bruce Mayer, PE Chabot College Mathematics
Prime Notation
Writing dy/dx takes too much work; need a Shorthand notation
By Mathematical Convention define the “Prime” Notation as
• The “Prime” Notation is more compact• The “d” Notation is more mathematically
Versatile– I almost always recommend the “d” form
'lim)()(
lim)('00
yx
y
h
xfhxfxf
xh
[email protected] • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx31
Bruce Mayer, PE Chabot College Mathematics
Average Rate of Change
The average rate of change of function f on the interval [a,b] is given by
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))
[email protected] • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx32
Bruce Mayer, PE Chabot College Mathematics
Example Avg Rate-of-Change
For f(x) = y = x2 find the average rate of change between x = 3 (Pt-a) and x = 5 (Pt-b)
By the Chord Slope0 1 2 3 4 5 6 7
0
5
10
15
20
25
30
35
40
45
x
y =
f(x)
= x
2
MTH15 • Avg Rate-of-Change
0 1 2 3 4 5 6 70
5
10
15
20
25
30
35
40
45
x
y =
f(x)
= x
2
MTH15 • Avg Rate-of-Change
XYf cnGraph6x6BlueGreenBkGndTemplate1306.mXYf cnGraph6x6BlueGreenBkGndTemplate1306.m
8
2
16
35
35)()( 22
x
y
ab
afbfmavg
y
x
[email protected] • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx33
Bruce Mayer, PE Chabot College Mathematics
Example Avg Rate-of-Change
3 4 5
10
15
20
25
x
y =
f(x)
= x
2
MTH15 • Avg Rate-of-Change
XYf cnGraph6x6BlueGreenBkGndTemplate1306.m
0 1 2 3 4 5 6 70
5
10
15
20
25
30
35
40
45
x
y =
f(x)
= x
2
MTH15 • Avg Rate-of-Change
0 1 2 3 4 5 6 70
5
10
15
20
25
30
35
40
45
x
y =
f(x)
= x
2
MTH15 • Avg Rate-of-Change
XYf cnGraph6x6BlueGreenBkGndTemplate1306.mXYf cnGraph6x6BlueGreenBkGndTemplate1306.m
y
xy
x
ChordSlope
[email protected] • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx34
Bruce Mayer, PE Chabot College Mathematics
MA
TL
AB
Co
de
% Bruce Mayer, PE% MTH-15 • 01Jul13% XY_fcn_Graph_BlueGreenBkGnd_Solid_Marker_Template1306.m%% The Limitsxmin = -3; xmax1 = 1; xmin2 = xmax1; xmax = 3; ymin = -4; ymax = 10;% The FUNCTIONx1 = linspace(xmin,xmax1,500); y1 = 1-x1.^2;x2 = linspace(xmin2,xmax,500); y2 = 3*x2+1;% The Total Function by appendingx = [x1, x2]; y = [y1, y2];% % The ZERO Lineszxh = [xmin xmax]; zyh = [0 0]; zxv = [0 0]; zyv = [ymin ymax];%% the 6x6 Plotaxes; set(gca,'FontSize',12);whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Greenplot(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 onplot(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
[email protected] • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx35
Bruce Mayer, PE Chabot College Mathematics
Slope vs. Rate-of-Change In general the Rate-
of-Change (RoC) is simply the Ratio, or Quotient, of Two quantities. Some Examples:• Pay Rate → $/hr• Speed → miles/hr• Fuel Use → miles/gal• Paper Use →
words/page
A Slope is a SPECIAL RoC where the UNITS of the Dividend and Divisor are the SAME. Example• Road Grade →
Feet-rise/Feet-run• Tax Rate →
$-Paid/$-Earned
[email protected] • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx36
Bruce Mayer, PE Chabot College Mathematics
Example Rice is Nice The demand for rice
in the USA in 2009 approximately followed the function
• Where– p ≡ Rice Price in
$/Ton– D ≡ Rice Demand in
MegaTons
Use this Function to:a) Find and interpret
b) Find the equation of the tangent line to D at p = 500.p
pD100
)(
500'D
[email protected] • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx37
Bruce Mayer, PE Chabot College Mathematics
Example Rice is Nice SOLUTION
a) Using the definition of the derivative:
Clear fractions by multiplying by
Simplifying
• Note the Limit is Undefined at h = 0
h
pDhpD
dP
dDh
)()(lim
0
hpp
hpp
h
php
dp
dDh
100100
lim0
hpph
hpp
dP
dDh
0lim100
[email protected] • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx38
Bruce Mayer, PE Chabot College Mathematics
Example Rice is Nice Remove the UNdefinition by multiplying
by the Radical Conjugate of the Numerator: hpp
hpp
hpp
hpph
hpp
dp
dDpD
h
0
lim100'
hpphpph
hphpphppp
dp
dDh
)(lim100
0
hpphpph
hpD
h
0
lim100'
[email protected] • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx39
Bruce Mayer, PE Chabot College Mathematics
Example Rice is Nice
Continue the Limit Evaluation
hpphppdp
dDh
1lim100
0
2/350' ppD
[email protected] • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx40
Bruce Mayer, PE Chabot College Mathematics
Example Rice is Nice
Run-Numbers to Find the Change in DEMAND with respect to PRICE
Unit analysis for dD/dp
Finally State: for when p = 500 the Rate of Change of Rice Demand in the USA:
2/350' ppD
$
Ton10
$
Ton
1
Ton10
Ton$
Ton10
Ton$
MTon 2666
dp
dD
.00447.050050500' 2/3 D
$
Ton 4470
$
Ton 1000447.0500'
226
D
[email protected] • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx41
Bruce Mayer, PE Chabot College Mathematics
Example Rice is Nice
Thus The RoC for D w.r.t. p at p = 500:
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 $1/Ton INCREASE in unit price.
Ton$
Ton4470
$
Ton 4470500'
2
D
[email protected] • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx42
Bruce Mayer, PE Chabot College Mathematics
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.
[email protected] • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx43
Bruce Mayer, PE Chabot College Mathematics
Example Rice is Nice
First, find the value 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:
MegaTons 47.4500
100)500( D
00447.050050 2/3500
pdpdD
[email protected] • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx44
Bruce Mayer, PE Chabot College Mathematics
Example Rice is Nice
Finally, we use the point-slope formula for the Eqn of a Line and simplify:
The Graph ofD(p) and theTangent Lineat p = 500 on the Same Plot:
[email protected] • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx45
Bruce Mayer, PE Chabot College Mathematics
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 ClearCut. Are these 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
dp
dD
dp
dD
dp
dD 500500
??
[email protected] • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx46
Bruce Mayer, PE Chabot College Mathematics
“Evaluated at” Notation
The d/dx operator produces the Slope Function dy/dx or df/dx; e.g.:
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 Number-Slope when using the “d” notation
7277 22 xxxdx
d
dx
df
dx
dyxfy
dx
dxxxfy
[email protected] • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx47
Bruce Mayer, PE Chabot College Mathematics
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 NUMBER):
Often the “x =” is Omitted
Cfdx
dfCy
dx
dy
cxcx
''
Cfdx
dfCy
dx
dy
cc
''
[email protected] • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx48
Bruce Mayer, PE Chabot College Mathematics
Example: Eval-At bar
Consider the Previous f(x) Example:
Using the d notation to find the Slope (Derivative) for x = −5 & 4
xxxfy 72
dx
dyxxx
dx
dy
dx
d 7272
15742375245
dx
dy
dx
dy
x
[email protected] • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx49
Bruce Mayer, PE Chabot College Mathematics
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.
.
[email protected] • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx50
Bruce Mayer, PE Chabot College Mathematics
Continuity & Smoothness
A function, f(x), is SMOOTHLY Varying at a given point, c, If and Only If df/dx Exists and:
• That is, the Slopesare the SAME whenapproached fromEITHER side
cxcx
cxcx dx
dfK
dx
df
limlim
[email protected] • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx51
Bruce Mayer, PE Chabot College Mathematics
WhiteBoard Work
Problem From §2.1• P46 → Declining
MarginalProductivity
0 1 2 3 4 5 60
50
100
150
200
250
L (k-WorkerHours)
Q (
k-U
nits
)
MTH15 • P2.1-46
XYf cnGraph6x6BlueGreenBkGndTemplate1306.m
[email protected] • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx52
Bruce Mayer, PE Chabot College Mathematics
All Done for Today
A DifferentType of
Derivative
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Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Chabot Mathematics
Appendix
–
srsrsr 22
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Bruce Mayer, PE Chabot College Mathematics
[email protected] • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx55
Bruce Mayer, PE Chabot College Mathematics
[email protected] • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx56
Bruce Mayer, PE Chabot College Mathematics
[email protected] • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx57
Bruce Mayer, PE Chabot College Mathematics
P2.1-46