Math Review

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Math Review

• Graphing
• Algebraic Equations (quadratic equation)
• Powers and Logarithms
• Basic Calculus Derivatives, Integrals

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Graphing
Graph - a visual representation of the
relationship between two or more variables
Descriptive Title
y
Dependent variable (units)

• Elements of a graph
• axes (x, y)
• origin
• axis labels and units
• data (y vs x)
• title

x
Origin (0,0)
Independent variable (units)
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Graphing a straight line

• y mx b

m slope b y-intercept
Chapter 18 Kinetics The Arrhenius relationship
(ln k vs 1/T)
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Algebraic Equations

• Solve a quadratic relationship
• ax2 bx c 0
• using the quadratic equation

Example Solving an equilibrium expression x2
3x 7 0 x H30
Solution x 4.5414, -1.5414
H30 cannot have a negative value, therefore x
4.5414
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Solving Algebraic Equations by Approximation
Time Saving Technique Faster and Easier than
solving exactly
Assumption y ltlt 0.100
Always Check the Assumption y ltlt 0.100 ( y lt
0.05 (0.100)
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Solving Algebraic Equations by Approximation
Time Saving Technique Faster and Easier than
solving exactly Especially for higher order
equations
Approximation x ltlt 2.00, 3.00
Always Check the Assumption x ltlt 2.00,3.00
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Solving Algebraic Equations by Approximation

• What if the approximation is not valid?

Approximation x ltlt 2.00, 3.00
x is smaller than 2.00, but not so small it can
be ignored!
Solve by iteration!!!
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Solving Algebraic Equations by Iteration
Solution x1 0.122
Substitute x0 0
x1 0.122
x2 0.116
x2 0.116
x3 0.117
x4 0.117
x3 0.117
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Powers and Logarithms
100 1 101 10 102 100 etc..
Ba n logB n a
Base Power Inverse 10 10x n log10 n
x e 2.7182818 ex n loge n ln n
x ln natural log
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Properties of Powers and Logs
Addition/Subtraction of Logs Example
log 5 log 3 log 15
log m log n log (m x n)
log 6 - log 3 log 2
log m - log n log (m/n)
Multiplication/ Division of Powers
10m x 10n 10mn
1021 x 106 1027
10m / 10n 10m-n
1021 / 106 1015
Logs of Powers
log 32 2 log 3
log nx x log n
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Significant figures of logarithms

• log (7310) 3.863917377
• 104.34892 22331.608
• How many figures are significant?

log (7310.) log (7.310 x 103)
log (7.310 x 103) log 7.310 log 103
log 7.310 log 103 0.8639 3 3.8639
3.8639
mantissa
characteristic
The number of significant figures corresponds to
the number of digits in the mantissa
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Derivative the slope of a curve
The slope of a line is a constant ? m Dy/Dx The
slope of a curve is a function ? consider smaller
changes, dy/dx
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Derivatives you need to know!
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Integral the Area under a Curve

• The area under a line is calculated by simple
  geometry
• Area (x2-x1) a
• Area average height x base
• Area ½(mx1 b mx2 b)(x2 x1)

The area under a curve is calculated by geometry
as well The area is approximated with a series
of small rectangles of width Dx and heights
dependant on the function.
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Integral the Area under a Curve
Area f(x1)Dx f(x2)Dx .. Area ?
f(xi)Dx As Dx ? 0, the area is defined by the
definite integral
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Integrals you should know!
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Practice Problems

• To practice any of these types of problems, try
  the problems in the Appendix, p A34 A35