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Module 1 Examples
CS 546
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Outline
One-to-one functions
Limits
Continuous functions Derivatives
Differentials
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definition: one-to-one function
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example
1
0
-1
-2
2
0
1
4
A B
Both 1 and -1 map to 1 and both 2 and -2 map to 4. Since some elements in A map to the sameelement in B, this is not a one-to-one function
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horizontal line test
one-to-one not one-to-one
1 1 23
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example
1 2
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example
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example
Every customer should have a unique Social Security number. No customer
should have more than one Social Security number.
In this case, it is possible that more than one item in the department store has the
same price.
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recipe for proving afunction is one-to-one
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example
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recipe for finding the limit ofa fraction as the variable
approaches infinity
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example
)(xf
x
Step 1: The highest power
polynomial is in the denominator.
Step 2: Since the highest powerpolynomial is in the denominator the
limit is equal to 0.
As x approaches infinity, f(x)
get closer and closer to 0.
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example
)(xf
x
Step 1: The highest polynomial is in the
denominator.
Step 2: Since the highest polynomial is in the
denominator, the limit is equal to 0.
As x approaches infinity, f(x) getscloser and closer to 0.
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example
)(xf
x
Step 1: The highest polynomial is located
both in the numerator and the
denominator.
Step 2: Since the highest polynomial is
located in the numerator and the
denominator, we take the ratio of thepolynomials = 2x3/x3= 2.
As x approaches infinity, f(x) gets closer and closer to 2.
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example
)(xf
x
Step 1: The highest polynomial is in the
numerator.
Step 2: Since the highest polynomial is in
the numerator, the limit is infinity.
From the plot
we see that as
x increases,f(x)
approaches
inifinity.
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definition : continuous(at a point)
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)( 0xfx
)(xf
0x
1
x
)()( 00 xfxxf
)()(lim 000 xfxxfx
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)( 0xfx
)(xf
0x
3
x
)()( 00 xfxxf
)()(lim 000 xfxxfx
As we continue to decrease
x, f(x0+x) f(x0) gets
closer to 0.
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)( 0xfx
)(xf
0x
4
x
)()( 00 xfxxf
)()(lim 000 xfxxfx
As we continue to decrease
x, f(x0+x) f(x0) gets
closer to 0.
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)( 0xfx
)(xf
0x
5
x
)()( 00 xfxxf
)()(lim 000 xfxxfx
As we continue to decrease
x, f(x0+x) f(x0) gets
closer to 0.
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)( 0xfx
)(xf
0x
6
x
)()( 00 xfxxf
)()(lim 000 xfxxfx
As we continue to decrease
x, f(x0+x) f(x0) gets
closer to 0.
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)( 0xfx
)(xf
0x
7
x
)()(00
xfxxf
)()(lim 000 xfxxfx
As we continue to decrease
x, f(x0+x) f(x0) gets
closer to 0.
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)( 0xfx
)(xf
0x
8
x
)()( 00 xfxxf
)()(lim 000 xfxxfx
As we continue to decrease
x, f(x0+x) f(x0) gets
closer to 0.
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)( 0xfx
)(xf
0x
9
)()(lim 000 xfxxfx 0x
)()( 00 xfxxf
As we continue to decrease
x, f(x0+x) f(x0) gets
closer to 0.
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example
)()(lim 000 xfxxfx
)0()0(lim 0 fxfx
x
0)(lim 0 xfx 0
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example
)()(lim 000 xfxxfx
)0()0(lim 0 fxfx
x
1)(lim 0 xfx 0
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example
1x
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continuity check
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definition: continuous function(contd)
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recipe for determining if afunction is continuous
recipe for proving afunction is continuous
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example
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example
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example
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rules of differentiation(derivative of a sum)
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example2)( xxxf
x
)(xf
xxg )(2
)( xxh
),()()( xhxgxf
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example
x
)(xf
1)(' xgxxh 2)('
)(')(')(' xhxgxf
xxf 21)('
2)( xxxf
xxg )(2
)( xxh
),()()( xhxgxf
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rules of differentiation(derivative involving a constant)
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rules of differentiation(derivative of a product)
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example2)( xxf
x
)(xf
xxg )(
xxh )(
),()()( xhxgxf
1)(' xg
1)(' xh
xxxxxxf 211)('
)(')()()(')(' xhxgxhxgxf
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n 1n
rules of differentiation(derivative of a power)
xxf )( nn'
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)(' xgc
)(xgc
xxf 2)(
example52)( xxf
5 154xxf 2)( 55'
5
410x
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example
f(x) = 3x4+ x3+ 3x2+ 6
f(x) = 3*4*x4-1+ 3*x3-1+ 3*2x2-1+ 0
f(x) = 12*x3+ 3*x2+ 6x
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rules of differentiation(derivatives of exponentials and logarithms)
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example
xexf 5)(
)()( xgcxf
)(')(' xgcxf
x
exf
5)('
xexg )( 5c
xexg )('
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)()()( xhxgxf
example
xxexf )(
)(')()()(')(' xhxgxhxgxf
xx
exexf 1)('
xxg )(
1)(' xg
xexh )(
xexh )('
1
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)()()( xhxgxf
example for you to try
xexxf 2)(
)(')()()(')(' xhxgxhxgxf
xx
exexxf
2
2)('
2)( xxg
xxg 2)('
xexh )(
xexh )('
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rules of differentiation(derivatives of composite functions)
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)),(()( xhgxf
example
2
)( xexf
)('))((')(' xhxhgxf
xexf x 2)('2
xexg )(
2)( xxh
)(')(' )( xhexf xh
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)),(()( xhgxf
example
2
)( xexf
)('))((')(' xhxhgxf
xexf x 2)('2
xexg )(
2)( xxh
)(')(' )( xhexf xh
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definition: differential
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recipe for finding a differential ofa function
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example
Find the differential:
1. f(x) = x2+6
Step 1: Find the derivative
f(x) = 2*x2-1+ 0 = 2x
Step 2: Multiply the derivative by dx
f(x)*dx = 2x*dx
