Download secant line slope

2/3/2011
Math 103 – Rimmer
3.1/3.2 The Derivative
The limit of the slopes of the secant lines is the slope of the tangent line.
secant line slope
m = lim
x →a
f ( x) − f (a)
x−a
f ′(a)
the slope of the tangent line
to f ( x ) at x = a.
Math 103 – Rimmer
3.1/3.2 The Derivative
Another expression for the slope of the tangent line.
secant line slope
m = lim
h →0
f ( a + h) − f ( a )
h
f ′(a)
the slope of the tangent line
to f ( x ) at x = a.
1
2/3/2011
Math 103 – Rimmer
3.1/3.2 The Derivative
If you zoom in on the point of tangency, the function is
"locally linear" there.
http://www.stewartcalculus.com/tec/
Module 3.1
Math 103 – Rimmer
3.1/3.2 The Derivative
List the following numbers from smallest to largest.
least steep
+
most steep
+
−
+
g ′ ( 0 ) < 0 < g ′ ( 4 ) < g ′ ( 2 ) < g ′ ( −2 )
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2/3/2011
a = −1
Find the equation of the tangent line to
the graph of the function y = 2 x3 − 5 x at ( −1,3) .
f ( −1)
f ( x) − f (a)
f ( x ) − f ( −1)
2 x3 − 5 x − 3
= lim
= lim
x → −1
x → −1
x−a
x +1
x − ( −1)
m = lim
x →a
−1
Math 103 – Rimmer
3.1/3.2 The Derivative
( x + 1) ( 2 x 2 − 2 x − 3)
x → −1
( x + 1)
= lim
2 0 −5 −3
−2
2
3
2 −2 −3
0
x
x2
= lim ( 2 x 2 − 2 x − 3)
x → −1
const.
2
= 2 ( −1) − 2 ( −1) − 3 = 1
Equation of the tangent line:
m =1
y = mx + b ⇒ 3 = 1( −1) + b
( −1, 3)
⇒b=4
y
x
y = x+4
Math 103 – Rimmer
3.1/3.2 The Derivative
We could have used the other formula to get the slope of the
tangent line.
m = lim
f ( a + h) − f ( a )
h
h →0
= lim
f ( −1 + h ) − f ( −1)
h→0
3
f ( −1 + h ) = 2 ( −1 + h ) − 5 ( −1 + h )
f ( −1 + h ) = 2 ( h3 − 3h 2 + 3h − 1) + 5 − 5h
f ( −1 + h ) = 2h3 − 6h 2 + 6h − 2 + 5 − 5h
f ( −1 + h ) = 2h − 6h + h + 3
3
2
y = 2 x3 − 5 x at ( −1,3)
h
3
( −1 + h ) = ( −1 + h )( −1 + h )( −1 + h )
3
( −1 + h ) = (1 − 2h + h2 ) ( −1 + h )
×
1
−2 h
−1
−1
2h
h
h
−2 h 2
h2
−h 2
h3
( −1 + h )
3
= h3 − 3h 2 + 3h − 1
h ( 2h 2 − 6h + 1)
2h 3 − 6 h 2 + h + 3 − ( 3 )
2h3 − 6h 2 + h
= lim
= lim
m = lim
h→0
h→0
h →0
h
h
h
= lim ( 2h 2 − 6h + 1) = 1
h→0
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2/3/2011
Math 103 – Rimmer
3.1/3.2 The Derivative
∆y f ( x2 ) − f ( x1 )
=
∆x
x2 − x1
This is called a
difference quotient
This is the average rate of change of y = f ( x )
with respect to x over the interval [ x1 , x2 ] .
f ( x2 ) − f ( x1 )
∆y
= lim
∆x → 0 ∆x
x2 → x1
x2 − x1
lim
The derivative f ′ ( a ) is the instantaneous rate of
change of y = f ( x ) with respect to x when x = a.
Interpreting the derivative as a rate of change.
Math 103 – Rimmer
3.1/3.2 The Derivative
The cost of producing x ounces of gold from a new gold mine is C ( x ) dollars.
What is the meaning of C ′ ( x ) ? What are its units?
change in C ∆C
C ′ ( x ) measures the ratio:
=
change in x ∆x
C ′ ( x ) is the rate of change of production cost with respect to
the number of ounces produced, this is called marginal cost.
The units for C ′ ( x ) are dollars per ounce.
What does C′ ( 800 ) = 17 mean ?
C′ ( 800 ) is a ratio so let's turn 17 into a fraction.
C′ ( 800 ) =
17
1
When you are producing 800 ounces of gold and you
increase production by 1 to 801 ounces, cost will increase by $17.
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2/3/2011
Math 103 – Rimmer
3.1/3.2 The Derivative
Let the number a vary.
f ′ ( x ) = lim
f ( x + h) − f ( x)
h
h →0
f ′ ( x ) can be thought of as a new function, it is called the derivative of f .
If f ′ ( a ) exists, then f is called differentiable at a.
f is called differentiable on ( a, b ) if it is differentiable
for all numbers in ( a, b ) .
Math 103 – Rimmer
3.1/3.2 The Derivative
Find the derivative of the function using the definition of the derivative.
f ( x + h) − f ( x)
f ′ ( x ) = lim
f ( x ) = 4 x − 7 x2
h →0
h
2
2
f ( x + h ) = 4 ( x + h ) − 7 ( x + h ) = 4 x + 4h − 7 ( x + 2 xh + h )
2
f ( x + h ) = 4 x + 4h − 7 x 2 − 14 xh − 7h 2
− f ( x ) = −4 x
+ 7 x2
f ( x + h ) − f ( x ) = 4h − 14 xh − 7 h 2 = h ( 4 − 14 x − 7 h )
f ′ ( x ) = lim
f ( x + h) − f ( x)
h →0
h
= lim
h→0
h ( 4 − 14 x − 7 h )
h
= lim ( 4 − 14 x − 7h )
h→0
f ′ ( x ) = 4 − 14 x
5
2/3/2011
Math 103 – Rimmer
3.1/3.2 The Derivative
Find the derivative of the function using the definition of the derivative.
1
f ( x + h) − f ( x)
f ( x) =
f ′ ( x ) = lim
h →0
x
h
1
1
1
f ( x + h) =
f ( x + h) − f ( x) =
−
x+h
x+h
x
f ′ ( x ) = lim
h →0
= lim
h→0
= lim
h →0
1
1
−
x x + h = lim x − x + h
x+h
x
⋅
⋅
h
x x + h h→0 h x ( x + h )
x − ( x + h)
h x ( x + h)
(
x + x+h
−1
x ( x + h)
(
x + x+h
)
)
= lim
h→0
x
2
(
)
x+h)
x + x+h
x+
−h
h x ( x + h)
−1
=
(
(
x+ x
)
=
(
−1
2x x
x + x+h
f ′( x) =
)
−1
2 x 3/2
Math 103 – Rimmer
3.1/3.2 The Derivative
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2/3/2011
Math 103 – Rimmer
3.1/3.2 The Derivative
Match the graph of each function in (a)-(d) with the graph of its derivative in I-IV.
The main connection:
function: sign of the slope of the tangent line
derivative: + ⇒ above x − axis, − ⇒ below x − axis
0 ⇒ "touches" x − axis
(a): sign of the slope of the tangent line
−→0→+→0→−
deriv.: below,then 0, then above, then 0, then below
(a) ⇔ II
(b): sign of the slope of the tangent line
+ → dne → − → dne → +
deriv.: above, then jump to below, then jump to above
(b) ⇔ IV
(c): sign of the slope of the tangent line
−→0→+
deriv.: below,then 0, then above
(c) ⇔ I
(d): sign of the slope of the tangent line
+→0→−→0→+→0→−
deriv.: above, then 0, then below, then 0,
then above, then 0, then below
(d) ⇔ III
Math 103 – Rimmer
3.1/3.2 The Derivative
Animation of the graph of the derivative function
http://www.stewartcalculus.com/tec/
Module 3.2
7