上海财经大学英语高数课件02

d y d d f ( x) d f ( x) n n f ( x) y n ( n1 ) D f ( x) Dx f ( x) n dx dx dx dx
( n) ( n)
n
n1
n
Example If y=x4-3x2+6x+9, find y ', y ' ', y ' ' ', y(4). Example If
(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 dx=x
Q
dy
S
t
f(x)=RS
o
x
x
As x=dx is very small, y=dy ,i.e., f(t)-f(x) f '(x) t.
Example The equations xy=c (c0) represents a faFra Baidu bibliotekily
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.
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
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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.
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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
can find the derivative of f(x) even though we have not gotten the expression of f(x). Fortunately it is not necessary to solve the equation for y in terms of x to find the derivative. We will use the method called implicit differentiation to find the derivative. Differentiating both sides of the equation, we obtain that [f(x)+xf ' (x)]cos[xf(x)]=2x+3[f(x)]2f ' (x). Then
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Orthogonal: Two curves are called Orthogonal, if
at each point of intersection their tangent lines are perpendicular. If two families of curves satisfy that every curve in one family is orthogonal to every curve in the another family, then we say the two families of curves are orthogonal trajectores of each other.
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.
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.
(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).
1 f(x)= x
, find f(n)(x).
Example If f(x)=sinx, g(x)=cosx, find f(n)(x) and g(n)(x) .
Example Find y ' ' , if x4+y3 =x-y .
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2.8 Related rates (omitted)
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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
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),
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).
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) 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 .
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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:
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