上海财经大学英语高数课件02

d 3 y d d 2 f ( x) d 3 f ( x) 3 3 f ' ' ' ( x) y ' ' ' 3 ( ) D f ( x ) D x f ( x) 2 3 dx dx dx dx
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And we can define f' ' ' ' (x)=[f ' ' ' (x)] '. From now on instead of using f' ' ' ' (x) we use f(4)(x) to represent f ' ' ' ' (x). In general, we define f(n)(x)=[f(n-1)(x)] ', which is called the nth derivative of f(x). We also like to use the following notations, if y=f(x),
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Corollary: If the differential of f(x) is df(x)= A(x) x,
then f(x) is differentiable and A(x)=f '(x).
Proof: From the definition,
f (t ) f ( x) A( x)t B(t , x) f ' ( x) lim lim A( x). tx t 0 tx t
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Example Use differentials to find an approximate (65)1/3 .
From definition of the differential, we can easily get
If f(x) is differentiable at x=a, and x is very closed to a, then f(x) f(a)+f '(a)(x-a). The approximation is called Linear approximation or tangent line approximation of f(x) at a. And function L(x)= f(a)+f '(a)(x-a) is called the linearization of f(x) at a.
(b) Find the equation of the tangent to the curve x3+y3 =28 at point (1,3).
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Example (a) If x3+y3 =6xy, find y'. (b) Find the equation of the tangent to the folium of Descartes x3+y3 =6xy at point (3,3).
d 2 y d df ( x) d 2 f ( x) 2 2 f ' ' ( x) y ' ' 2 ( ) D f ( x ) D x f ( x) 2 dx dx dx dx
Similarly f ' ' ' (x)=[f '' (x)] '
of f(x), and
is called the third derivative
2.9 Differentials, Linear and Quadratic Approximations
Definition: Let x=x-x0, f(x) =f(x)-f(x0). If there
exists a constant A(x0) which is independent of x and x such that f(x)=A(x0) x+B(x, x0) where
f(x)cos[ xf ( x)]-2 x y cos(xy) 2 x f ' ( x) 2 2 3[ f ( x)] xcos[xf ( x)] 3 y x cos(xy)
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Example (a) If
x3+y3
=27, find
dy . dx
Corollary: (a) If f(x)=x, then dx=df(x)=x.
(b) If f(x) is differentiable, then differential
of f(x) exists and df(x)=f '(x)dx.
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Example (a) Find dy, if y=x3+5x4.
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3) Derivatives of implicit function Suppose y=f(x) is an implicit function defined by sin(xy)= x2+y3. Then sin[xf(x)]=x2+ [f(x)]3. From the equation, we
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2.7 Higher derivatives
Derivative f' (x) of differentiable function f(x) is also a
function. If f' (x) is differentiable, then we have [f ' (x)] '. We will denote it by f ' ' (x), i.e., f' ' (x)=[f ' (x)] '. The new function f ' ' (x) is called the second derivative of f(x). If y=f(x), we also can use other notations:
Chapter 2
Derivatives
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2.6 Implicit differentiation
1) Explicit function: The function which can be
described by expressing one variable explicitly in terms of another variable (other variables) are generally called explicit function---for example, y=xtanx, or y=[1+x2+x3]1/2 , or in general y=f(x).
B(x, x 0 ) lim 0 . Then A x is called B(x, x0) satisfies x 0 x
differential of f(x) at x0. Generally A x is denoted by df(x)|x=x = A(x0) x. Replacing x0 by x, the differential 0 is denoted by df(x) and df(x)= A(x) x.
Example The equations xy=c (c0) represents a family
of hyperbolas. And the The equations x2-y2=k (k0) represents another family of hyperbolas with asymptotes y=x. Then the two families of curves are trajectores of each other.
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Example Find the linearization of the function
f(x)=(x+3)1/2 and approximations the numbers (3.98) 1/2 and (4.05)1/2.
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Quadratic approximation to f(x) near x=a: Suppose f(x) is a function which the second derivative
(b) Find the value of dy when x=2 and dx=0.1.
Solution:
Geometric meaning of differential of f(x), df(x)=QS
y
y=f(x) R P d=RS
o
x
x
As x=dx is very small, y=dy ,i.e., f(t)-f(x) f '(x) t.
2) Implicit function: The functions which are defined
implicitly by a relation between variables--x and y--are generally called implicit functions--- such as x2+y2 =4, or 7sin(xy)=x2+y3 or, in general F(x,y)=0.If y=f(x) satisfies F(x, f(x))=0 on an interval I, we say f(x) is a function defined on I implicitly by F(x,y)=0, or implicit function defined by F(x,y)=0 .