limits and derivatives
TRANSCRIPT
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Limits and Derivatives
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Concept of a Function
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FUNCTIONS
• “FUNCTION” indicates a relationship among objects.
• A FUNCTION provides a model to describe a system.
• A FUNCTION expresses the relationship of one variable or a group of variables (called the domain) with another variables( called the range) by associating every member in the domain to a unique member in range.
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TYPES OF FUNCTIONS
• LINEAR FUNCTIONS
• INVERSE FUNCTIONS
• EXPONENTIAL FUNCTIONS
• LOGARITHMIC FUNCTIONS
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y is a function of x, and the relation y = x2 describes a function. We notice that with such a relation, every value of x corresponds to one (and only one) value of y.
y = x2
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Since the value of y depends on a given value of x, we call y the dependent variable and x the independent variable and of the function y = x2.
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Notation for a Function : f(x)
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The Idea of Limits
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Consider the function
The Idea of Limits
2
4)(
2
x
xxf
x 1.9 1.99 1.999 1.9999 2 2.0001 2.001 2.01 2.1
f(x)
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Consider the function
The Idea of Limits
2
4)(
2
x
xxf
x 1.9 1.99 1.999 1.9999 2 2.0001 2.001 2.01 2.1
f(x) 3.9 3.99 3.999 3.9999 un-defined
4.0001 4.001 4.01 4.1
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Consider the function
The Idea of Limits 2)( xxg
x 1.9 1.99 1.999 1.9999 2 2.0001 2.001 2.01 2.1
g(x) 3.9 3.99 3.999 3.9999 4 4.0001 4.001 4.01 4.1
2)( xxg
x
y
O
2
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If a function f(x) is a continuous at x0,
then . )()(lim 00
xfxfxx
4)(lim2
xfx
4)(lim2
xgx
approaches to, but not equal to
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Consider the function
The Idea of Limits
x
xxh )(
x -4 -3 -2 -1 0 1 2 3 4
g(x)
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Consider the function
The Idea of Limits
x
xxh )(
x -4 -3 -2 -1 0 1 2 3 4
h(x) -1 -1 -1 -1 un-defined
1 2 3 4
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1)(lim0
xhx
1)(lim0
xhx
)(lim0
xhx does not
exist.
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A function f(x) has limit l at x0 if f(x) can be made as close to l as we please by taking x sufficiently close to (but not equal to) x0. We write
lxfxx
)(lim0
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Theorems On Limits
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Theorems On Limits
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Theorems On Limits
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Theorems On Limits
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Limits at Infinity
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Limits at Infinity
Consider1
1)(
2
xxf
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Generalized, if
)(lim xfx
then
0)(
lim xf
kx
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Theorems of Limits at Infinity
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Theorems of Limits at Infinity
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Theorems of Limits at Infinity
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Theorems of Limits at Infinity
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The Slope of the Tangent to a Curve
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The Slope of the Tangent to a Curve
The slope of the tangent to a curve y = f(x) with respect to x is defined as
provided that the limit exists.
x
xfxxf
x
yAT
xx
)()(limlim of Slope
00
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Increments
The increment △x of a variable is the change in x from a fixed value x = x0 to another value x = x1.
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For any function y = f(x), if the variable x is given an increment △x from x = x0, then the value of y would change to f(x0 + △x) accordingly. Hence thee is a corresponding increment of y(△y) such that △y = f(x0 + △x) –
f(x0).
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Derivatives(A) Definition of Derivative.
The derivative of a function y = f(x) with respect to x is defined as
provided that the limit exists.
x
xfxxf
x
yxx
)()(limlim
00
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The derivative of a function y = f(x) with respect to x is usually denoted by
,dx
dy),(xf
dx
d ,'y ).(' xf
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The process of finding the derivative of a function is called differentiation. A function y = f(x) is said to be differentiable with respect to x at x = x0 if the derivative of the function with respect to x exists at x = x0.
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The value of the derivative of y = f(x) with respect to x at x = x0 is denoted
by or .0xxdx
dy
)(' 0xf
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To obtain the derivative of a function by its definition is called differentiation of the function from first principles.
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Differentiation Rules
1. 0)( cdx
d
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Differentiation Rules
1. 0)( cdx
d
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Differentiation Rules
2. dx
dv
dx
duvu
dx
d )(
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Differentiation Rules
2. dx
dv
dx
duvu
dx
d )(
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Differentiation Rules
2. dx
dv
dx
duvu
dx
d )(
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Differentiation Rules
3. dx
duccu
dx
d)(
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Differentiation Rules
3. dx
duccu
dx
d)(
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Differentiation Rules
4. 1)( nn nxxdx
d for any positive integer n
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Differentiation Rules
4. 1)( nn nxxdx
d for any positive integer n
Binominal Theorem
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Differentiation Rules
5. dx
duv
dx
dvuuv
dx
d)( product rule
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Differentiation Rules
5. dx
duv
dx
dvuuv
dx
d)( product rule
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Differentiation Rules
6.
2)(
vdxdv
udxdu
v
v
u
dx
d
where v ≠ 0
quotient rule
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Differentiation Rules
6.
2)(
vdxdv
udxdu
v
v
u
dx
d
where v ≠ 0
quotient rule
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Differentiation Rules
7. 1)( nn nxxdx
d for any integer n
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DIFFERENTIATION RULES
• y,u and v are functions of x. a,b,c, and n are constants (numbers).
The derivative of a constant is zero. Duh! If everything is constant, that means its rate, its derivative, will be zero. The graph of a constant, a number is a horizontal line. y=c. The slope is zero.
The derivative of x is 1. Yes. The graph of x is a line. The slope of y = x is 1. If the graph of y = cx, then the slope, the derivative is c.
1xdx
d
0cdx
d
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MORE RULES
• When you take the derivative of x raised to a power (integer or fractional), you multiply expression by the exponent and subtract one from the exponent to form the new exponent.
1 nn nxxdx
d
23 3xxdx
d
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OPERATIONS OF DERIVATIVES• The derivative of the sum or
difference of the functions is merely the derivative of the first plus/minus the derivative of the second.
dx
duv
dx
dvuuv
dx
d
dx
dv
dx
duvu
dx
d
• The derivative of a product is simply the first times the derivative of the second plus second times the derivative of the first.
2vdxdv
udxdu
v
v
u
dx
d
• The derivative of a quotient is the bottom times the derivative of the top, minus top times the derivative of the bottom….. All over bottom square..
• TRICK: LO-DEHI – HI-DELO
• LO2
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JUST GENERAL RULES
• If you have constant multiplying a function, then the derivative is the constant times the derivative. See example below:
• The coefficient of the x6 term is 5 (original constant) times 7 (power rule.)
67 355 xxdx
d
dx
dvccv
dx
d
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SECOND DERIVATIVES
• You can take derivatives of the derivative. Given function f(x), the first derivative is f’(x). The second derivative is f’’(x), and so on and so forth.
• Using Leibniz notation of dy/dx
2
2
dx
yd
dx
dy
dx
d
For math ponders, if you are interesting in the Leibniz notation of derivatives further, please see my article on that. Thank you. Hare Krishna >=) –Krsna Dhenu
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EXAMPLE 4:
• Find the derivative:
• Use the power rule and the rule of adding derivatives.
• Note 3/2 – 1 = ½. x½ is the square root of x.
• Easy eh??
22
35 22 xxxy
xxxy 435 4
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EXAMPLE 5
• Find the equation of the line tangent to y = x3 +5x2 –x + 3 at x=0.
• First find the (x,y) coordinates when x = 0. When you plug 0 in for x, you will see that y = 3. (0,3) is the point at x=0.
• Now, get the derivative of the function. Notice how the power rule works. Notice the addition and subtraction of derivative. Notice that the derivative of x is 1, and the derivative of 3, a constant, is zero.
35 23 xxxy
1103 2 xxdx
dy
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EX 5 (continued)
• Now find the slope at x=0, by plugging in 0 for the x in the derivative expression. The slope is -1 since f’(0) = -1.
• Now apply it to the equation of a line.
10
xdx
dy
)( 00 xxmyy
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EX 5. (continued)
• Now, plug the x and y coordinate for x0 and y0 respectively. Plug the slope found in for m.
• And simplify
• On the AP, you can leave your answer as the first form. (point-slope form)
)0(13 xy
3 xy
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EXAMPLE 6
• Find all the derivatives of y = 8x5.
• Just use the power rule over and over again until you get the derivative to be zero.
• See how the power rule and derivative notation works?
0
960
960
480
160
40
8
6
6
5
5
4
4
23
3
32
2
4
5
dx
yd
dx
yd
xdx
yd
xdx
yd
xdx
yd
xdx
dy
xy