微積分一:講義2-1

第二章導數

2.1 導數的定義

Case 1: 0    ₂  m  tan   0, then 0  m  . Case 2:  ₂  m  tan ₂   (No slope). Case 3: ₂      −  m  0.

Case 4:  0 or   m  tan 0  tan   0.

1. Line Tangent to a Curve

10 5 0 -5 -10 100 75 50 25 x y x y

2. Finding Slopes of Tangents Example gx  5x − x2 5 3.75 2.5 1.25 0 6.25 5 3.75 2.5 1.25 0 x y x y Slope of line S  g1  h − g1 1  h − 1  g1  h − g1 h As h approaches zero. The behave of right hand side :

The behave of left hand side: ; x

Since gx  5x − x2_{, we have that}

g1  h − g1 h  51  h − 1  h2 − 5  1 − 12 h  3 − h. Hence lim h→0 g1  h − g1 h  3.

Theorem Suppose c is the abccissa of a point on the graph of a function fx where

the tangent is nonvertical. Then lim

h→0

fc  h − fc h exixt and its value is the slope of the tangent.

Example Find the slope of the tangent to the graph of

fx  4x at the point whose x coordinate is 3.

10 7.5 5 2.5 0 1000 750 500 250 0 x y x y

The pointx, Px is an arbitrary fixed point on the graph of the function. Let

mx represent the slope of line T. Then mx  lim

h→0

Px  h − Px

h .

Since Px  256x − 3x3_{, it follows that}

Px  h − Px h  256 − 9x 2 _{− 9xh − 3h}2_. We have lim h→0 Px  h − Px h  256 − 9x 2_.

Thus, a formula for the slope of line T is

mx  256 − 9x2_.

Definition The derivative of a function fx is a function f′x whose formula can be

derived by computing

lim

h→0

fx  h − fx

h .

The Connection between Derivative and Slope of Tangents

Theorem Suppose c is the abccissa of a point on the graph of a function fx where

the tangent is nonvertical. Then fx is differential at x  c and f′c is the

slope of the tangent.

Theorem Suppose a function fx is differentiable at x  c. Then the graph has a

nonvertical tangent at the pointc, fc and the slope of the tangent is f′c.

Notation for derivatives

f′x, dy

dx , y

′_. d

dxfx, Dxy

The notation is real, you can write

dy

dx lim_h→0

fx  h − fx h

 f′_x.

and said to the derivative of f with respect to x.

Example Find a formula for dy/dx, if

y  2 x

9, 6.

Example Find f′x for fx  x3  2x

f′x 

h→0

lim fx  h − fx h

 3x2  2

The connection between Differantiable and Continuous

Alternative form of derivative:

x→c

lim fx − fc_{x− c}  f′c.

A: If f is discintinuous at c, then f is not differentiable at c. Example fx  x, then x→0lim− fx − f0 x− 0   , _x→0lim fx − f0 x− 0  0.

B: Is f differentiable at c, if function f is continuous at c?

Ans: No!

Example fx  |x − 2|, clearly f is continuous at x  2, but

x→2lim−

fx − f2

x− 2  −1 ≠ _x→2lim

fx − f2

x− 2  1.

Thus f is not differentiable at x  2.

Example fx  x13, clearly f is continuous at x  0, but

x→0

lim fx − f0

x− 0  

Thus f is not differentiable at x  0.