derivative power point
TRANSCRIPT
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I’m going nutsover derivatives!!!
2.1The Derivative and theTangent Line Problem
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Calculus grew out of 4 major problems that European mathematicians were working onin the seventeenth century.
1. The tangent line problem
2. The velocity and acceleration problem
3. The minimum and maximum problem
4. The area problem
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The tangent line problem
(c, f(c))
secant line
f(c+ ) – f(c)x(c, f(c)) is the point of tangency and is a second point on the graph of f.
)()( cfxcf
x
)(, xcfxc
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The slope between these two points is
cxccfxcfm
)()(
sec
xcfxcf
)()(
Definition of Tangent Line with Slope m
mx
cfxcfx
)()(lim0
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Find the slope of the graph of f(x) = x2 +1 at the point (-1,2). Then, find the equation of the tangent line.
(-1,2)
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xxfxxf
x
)()(lim0
x
xxxx
11)(lim22
0
x
xxxxxx
112lim222
0
xxxx
x
)2(lim0
x2
Therefore, the slopeat any point (x, f(x))is given by m = 2x
What is the slope at the point (-1,2)?
m = -2
The equation of the tangent line is y – 2 = -2(x + 1)
f(x) = x2 + 1
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The limit used to define the slope of a tangentline is also used to define one of the two funda-mental operations of calculus --- differentiation
Definition of the Derivative of a Function
xxfxxfxf
x
)()(lim)('0
f’(x) is read “f prime of x” Other notations besides f’(x) include:
][)],([,', yDxfdxdy
dxdy
x
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Find f’(x) for f(x) = and use the result to findthe slope of the graph of f at the points (1,1) & (4,2). What happens at the point (0,0)?
,x
xxfxxfxf
x
)()(lim)('0
xxxxxf
x
0lim)('
xxxxxx
xxxxxxx
x
)(lim0 xxxx
xxxx
0lim
1
xxxx
1lim0 x2
1
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xmxf
21)('
Therefore, at the point (1,1), the slope is ½, and at the point (4,2),the slope is ¼.
What happens at the point (0,0)? The slope is undefined, since it produces divisionby zero.
21
m
41
m
1 2 3 4
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Find the derivative with respect to t for the
function .2t
y
ttfttf
dxdy
t
)()(lim0
tttt
t
22
lim0
1
)()(22
lim0 t
tttttt
t
ttttttt
t
1)(222lim
0
2
2t
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Theorem 3.1 Alternate Form of the Derivative
The derivative of f at x = c is given by
cxcfxfcf
cx
)()(lim)('
(c, f(c)))()( cfxf
cxx c x
(x, f(x))
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Derivative from the left and from the right.
cxcfxf
cx
)()(lim cxcfxf
cx
)()(lim
Example of a point that is not differentiable.
2)( xxf is continuous at x = 2 but let’s look at it’s one sided limits.
2)2()(lim
2 xfxf
x
202
lim2 xx
x-1
2)2()(lim
2 xfxf
x
202
lim2 xx
x1
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The 1-sided limits are not equal.
, x is not differentiable at x = 2. Also, thegraph of f does not have a tangent line at the point (2, 0).
A function is not differentiable at a point atwhich its graph has a sharp turn or a verticaltangent line(y = x1/3 or y = absolute value of x). Differentiability can also be destroyed by a discontinuity ( y = the greatest integer of x).