第2.1节 导数

DIFFERENTIATION

2.1 DERIVATIVES

We are now ready to explain what is meant by the slope of a curve or the velocity of a moving point. Consider a real functionf and a real numbera in the domain of f. When x has valuea, f(x) has value f(a). Now suppose the value of x is changed froma to a hyperreal numbera+Δxwhich is infinitely close to but not equal toa. Then the new value off(x) will be f(a+ Δx).In this process the value ofx will be changed by a nonzero infinitesimal amountΔx,while the value off(x) will be changed by the amount

f(a+Δx) - f(a).

The ratio of the change in the value off(x)to the change in thevalue of x is

f(a+Δx) - f(a)

Δx

The ratio is used in the definition of the slope offwhich we nowgive.

DEFINITION

Sis said to be the slope of f at a if

for every nonzero infinitesimalΔx

Theslope, when it exists, is infinitely close to the ratio of the changeinf(x) to an infinitely small change inx. Given acurve y=f(x), the slope off at a is also calledthe slope of the curvey=f(x) at x=a. Figure 2.1.1shows a nonzero infinitesimal Δx and a hyperreal straightline through the two points on the curve ata anda+Δx.The quantity

isthe slope of this line, and its standard part is the slope of thecurve.

Figure2.1.1

Theslope offat a does not always exist. Here is a listof all the possibilities.

(1)theslope offat a exists if the ratio

isfiniteand has the same standard part for all infinitesimalΔx≠ 0. It has the value

(2)the slope offat a can fail to exist in any of four ways:

(a)f(a) is undefined.

(b)f(a+Δ x) is undefined for some infinitesimal Δx≠ 0.

(c)theterm _____________ is infinitefor some infinitesimalΔx≠ 0.

(d)theterm____________ has differentstandard parts for different

infinitesimalsΔx≠ 0.

Wecan consider the slope off at any pointx, which givesus a new function of x.

DEFINITION

Letf be a real function of one variable. The derivative of f is the newfunction f ' whose value at x is the slope of f at x. Insymbols,

Wheneverthe slope exists.

Thederivativef ' (x) is undefined if the slope offdoes not exist at x.

Fora given pointa, the slope of f ata and the derivativeof f at a are the same thing. We usually use the word“slope” to emphasize the geometric picture and “derivative”to emphasize the fact thatf ' is a function.

Theprocess of finding the derivative off is calleddifferentiation. We say thatf is differentiableata if f ' (a) is defined; i.e., the slope offat a exists.

Independentand dependent variables are useful in the study of derivatives. Letus briefly review what they are. A system of formulas is a finite setof equations and inequalities. If we are given a system of formulaswhich has the same graph as a simple equationy=f(x), we saythaty is a function of x, or that y depends on x, and we call xthe independent variable and y the dependent variable.

Wheny=f(x), we introduce a new independent variable Δx anda new dependent variable Δy, with the equation

(1)Δy= f(x+Δx)- f(x).

Thisequation determines Δyas a real function of the twovariablesx and Δx, when x and Δx varyover the real numbers. We shall usually want to use the Equation 1for Δy whenx is a real number and Δx isnonzero infinitesimal. The Transfer Principle implies that Equation 1also determines Δyas a hyperreal function of two variableswhen x and Δx are allowed to vary over the hyperrealnumbers.

Δyis called the increment ofy. Geometrically, the increment Δyis the change iny along the curve corresponding to the changeΔx inx. The symboly' is sometimes used forthe derivative, y' = f ' (x).Thus the hyperrealequation

nowtakes the short form

Theinfinitesimal Δx may be either positive or negative, but notzero. The various possibilities are illustrated in Figure 2.1.2 usingan infinitesimal microscope. The signs of Δx and Δyare indicated in the captions.

Ourrules for standard parts can be used in many cases to find thederivative of a function. There are two parts to the problem offinding the derivative f ' of a function f:

(1)findthe domainf '.

(2)Findthe value off '(x) when it is defined.

Figure2.1.2

EXAMPLE1 Find the derivative of the function

f(x)=x3.

Inthisand the following examples we letx vary over the real numbersand Δx vary overthenonzero infinitesimals. Let us introduce the new variableywith the equation y= x3. We first find Δy/Δx.

Nextwe simplify the expression for Δy/Δx.

Thenwe take the standard part,

Therefore,

Wehave shown that the derivative of the function

f(x)=x3

isthe functionf '(x) =3x2

withthe whole real line as domain. f (x)and f '(x) areshown in Figure 2.1.3.

Figure2.1.3

EXAMPLE2 Findf '(x) givenf(x) =_____

Case1x<0. Since____ is notdefined, f '(x)does not exist.

Case2x=0. Whenx is a negative infinitesimal, the term

isnot defined because____is undefined.When Δx is a positive infinitesimal, the term

isdefined but its value is infinite. Thus for two reasons,f '(x)does not exist.

Case3x >0. Lety=____.Then

Wethenmake the computation

Takingstandard parts,

Therefore,whenx >0,f '(x) = __________.

Sothe derivative off(x)= _________.

isthe functionf '(x)= __________,

andthe set of allx >0 is its domain (see Figure 2.1.4)

Figure2.1.4

EXAMPLE3 Find the derivative off(x) = 1/x.

Case1x=0. Then 1/x is undefined sof '(x) isundefined.

Case2x≠ 0.

Simplifying,

Takingthe standard part,

Thusf '(x) =-1/x²

Thederivative of the functionf(x) = 1/x is the functionf '(x) = -1/x²

Whosedomain is the set of allx≠ 0. Both functions are graphedin Figure 2.1.5.

Figure2.1.5

EXAMPLE4 Find the derivative off(x) = |x|.

Case1x>0. In this case |x|=x, and we have

y=x,
y+ Δy= x+Δx,

Δy=Δx,

______= 1,f '(x) =1

Case2x<0. Now |x| = -x, and

y=-x,
y+ Δy=-(x+Δx,)

Δy=-(x+Δx)- (-x) = - Δx,

______= -1,f '(x)= -1.

Case3x=0. Then

and

Thestandard part ofy/x is then 1 for some values ofx and-1 for others. Therefore f '(x) does not exist when x=0.

insummary,

Figure2.1.6 showsf(x) andf '(x).

Figure2.1.6

Thederivative has a variety of applications to the physical, life, andsocial sciences. It may come up in one of the following contexts.

Velocity:If an object moves according to the equations= f(t) wheretis time and s is distance, the derivativev= f '(t) iscalled the velocity of the object is timet.

Growthrates: A populationy(of people, bacteria, molecules,etc.) grows according to the equation y= f(t) wheretis time. Then the derivative y' = f '(t) is the rateofgrowth of the populationy at time t.

Marginalvalues(economics ): Suppose the total cost (or profit, etc.) ofproducingx items isy=f(x) dollars. Then the cost ofmaking one additional item is approximately the derivativey'= f '(x)because y' is the change in y per unit change inx. This derivative is called themarginal cost.

EXAMPLE5 A ball thrown upward with initial velocityb ft persec will be at a height

y= bt -16t²

feetaftertseconds.Find the velocity at timet.Let t be real and Δt≠0,

Infinitesimal.

Attimetsec,v=y'=b-32tft/sec.

Bothfunctions are graphed in Figure 2.1.7.

Figue2.1.7

EXAMPLE6 Suppose a bacterial culture grows in such a way that at timet there aret3

bacteria.Findthe rate of growth at timet= 100 sec.

y=t3 y'=t3 by Example 1.

Att= 1000,y'=3,000,000 bacteria/sec.

EXAMPLE7 Suppose the cost of making x needles is_____dollars. What is the marginal cost after 10,000 needles have beenmade?

y=______,y' =_______ by Example 2.

Atx= 10,000,y' =_______ =______dollars per needle.

Thusthe marginal cost is one half of a cent per needle.

PROBLEMSFOR SECTION 2.1

Findthe derivative of the given function in Problems 1-21.

1f(x) =x²2f(t) = t² + 3

3f(x) =1-2x²4f(x) =3x² + 2

5f(t) =4t6f(x)=2-5x

7f(t)=4t3 8f(t) = -t3

9f(u) =_____10 f(u) =_____

11g(x)=______12g(x)=______

13g(t)=______14g(x)=t -3

15f(y)=3y-1 + 4y16f(y)=2y3 + 4y²

17f(x)=ax +b18f(x)=ax²

19f(x)=________20f(x)=1/(x+2)

21f(x)=1/(3-2x)

Theslope, when it exists, is infinitely close to the ratio of the changeinf(x) to an infinitely small change inx. Given acurve y=f(x), the slope off at a is also calledthe slope of the curvey=f(x) at x=a. Figure 2.1.1shows a nonzero infinitesimal Δx and a hyperreal straightline through the two points on the curve ata anda+Δx.The quantity

isthe slope of this line, and its standard part is the slope of thecurve.

Theslope

Figure2.1.1

Theslope offat a does not always exist. Here is a listof all the possibilities.

(1)theslope offat a exists if the ratio

isfiniteand has the same standard part for all infinitesimalΔx≠ 0. It has the value

(2)the slope offat a can fail to exist in any of four ways:

(a)f(a) is undefined.

(b)f(a+Δ x) is undefined for some infinitesimal Δx≠ 0.

(c)theterm _____________ is infinitefor some infinitesimalΔx≠ 0.

(d)theterm____________ has differentstandard parts for different

infinitesimalsΔx≠ 0.

Wecan consider the slope off at any pointx, which givesus a new function of x.

DEFINITION

Letf be a real function of one variable. The derivative of f is the newfunction f ' whose value at x is the slope of f at x. Insymbols,

Wheneverthe slope exists.

Thederivativef ' (x) is undefined if the slope offdoes not exist at x.

Fora given pointa, the slope of f ata and the derivativeof f at a are the same thing. We usually use the word“slope” to emphasize the geometric picture and “derivative”to emphasize the fact thatf ' is a function.

Theprocess of finding the derivative off is calleddifferentiation. We say thatf is differentiableata if f ' (a) is defined; i.e., the slope offat a exists.

Independentand dependent variables are useful in the study of derivatives. Letus briefly review what they are. A system of formulas is a finite setof equations and inequalities. If we are given a system of formulaswhich has the same graph as a simple equationy=f(x), we saythaty is a function of x, or that y depends on x, and we call xthe independent variable and y the dependent variable.

Wheny=f(x), we introduce a new independent variable Δx anda new dependent variable Δy, with the equation

(1)Δy= f(x+Δx)- f(x).

Thisequation determines Δyas a real function of the twovariablesx and Δx, when x and Δx varyover the real numbers. We shall usually want to use the Equation 1for Δy whenx is a real number and Δx isnonzero infinitesimal. The Transfer Principle implies that Equation 1also determines Δyas a hyperreal function of two variableswhen x and Δx are allowed to vary over the hyperrealnumbers.

Δyis called the increment ofy. Geometrically, the increment Δyis the change iny along the curve corresponding to the changeΔx inx. The symboly' is sometimes used forthe derivative, y' = f ' (x).Thus the hyperrealequation

nowtakes the short form

Theinfinitesimal Δx may be either positive or negative, but notzero. The various possibilities are illustrated in Figure 2.1.2 usingan infinitesimal microscope. The signs of Δx and Δyare indicated in the captions.

Ourrules for standard parts can be used in many cases to find thederivative of a function. There are two parts to the problem offinding the derivative f ' of a function f:

(1)findthe domainf '.

(2)Findthe value off '(x) when it is defined.

Figure2.1.2

EXAMPLE1 Find the derivative of the function

f(x)=x3.

Inthisand the following examples we letx vary over the real numbersand Δx vary overthenonzero infinitesimals. Let us introduce the new variableywith the equation y= x3. We first find Δy/Δx.

Nextwe simplify the expression for Δy/Δx.

Thenwe take the standard part,

Therefore,

Wehave shown that the derivative of the function

f(x)=x3

isthe functionf '(x) =3x2

withthe whole real line as domain. f (x)and f '(x) areshown in Figure 2.1.3.

Figure2.1.3

EXAMPLE2 Findf '(x) givenf(x) =_____

Case1x<0. Since____ is notdefined, f '(x)does not exist.

Case2x=0. Whenx is a negative infinitesimal, the term

isnot defined because____is undefined.When Δx is a positive infinitesimal, the term

isdefined but its value is infinite. Thus for two reasons,f '(x)does not exist.

Case3x >0. Lety=____.Then

Wethenmake the computation

Takingstandard parts,

Therefore,whenx >0,f '(x) = __________.

Sothe derivative off(x)= _________.

isthe functionf '(x)= __________,

andthe set of allx >0 is its domain (see Figure 2.1.4)

Figure2.1.4

EXAMPLE3 Find the derivative off(x) = 1/x.

Case1x=0. Then 1/x is undefined sof '(x) isundefined.

Case2x≠ 0.

Simplifying,

Takingthe standard part,

Thusf '(x) =-1/x²

Thederivative of the functionf(x) = 1/x is the functionf '(x) = -1/x²

Whosedomain is the set of allx≠ 0. Both functions are graphedin Figure 2.1.5.

Figure2.1.5

EXAMPLE4 Find the derivative off(x) = |x|.

Case1x>0. In this case |x|=x, and we have

y=x,
y+ Δy= x+Δx,

Δy=Δx,

______= 1,f '(x) =1

Case2x<0. Now |x| = -x, and

y=-x,
y+ Δy=-(x+Δx,)

Δy=-(x+Δx)- (-x) = - Δx,

______= -1,f '(x)= -1.

Case3x=0. Then

and

Thestandard part ofy/x is then 1 for some values ofx and-1 for others. Therefore f '(x) does not exist when x=0.

insummary,

Figure2.1.6 showsf(x) andf '(x).

Figure2.1.6

Thederivative has a variety of applications to the physical, life, andsocial sciences. It may come up in one of the following contexts.

Velocity:If an object moves according to the equations= f(t) wheretis time and s is distance, the derivativev= f '(t) iscalled the velocity of the object is timet.

Growthrates: A populationy(of people, bacteria, molecules,etc.) grows according to the equation y= f(t) wheretis time. Then the derivative y' = f '(t) is the rateofgrowth of the populationy at time t.

Marginalvalues(economics ): Suppose the total cost (or profit, etc.) ofproducingx items isy=f(x) dollars. Then the cost ofmaking one additional item is approximately the derivativey'= f '(x)because y' is the change in y per unit change inx. This derivative is called themarginal cost.

EXAMPLE5 A ball thrown upward with initial velocityb ft persec will be at a height

y= bt -16t²

feetaftertseconds.Find the velocity at timet.Let t be real and Δt≠0,

Infinitesimal.

Attimetsec,v=y'=b-32tft/sec.

Bothfunctions are graphed in Figure 2.1.7.

Figue2.1.7

EXAMPLE6 Suppose a bacterial culture grows in such a way that at timet there aret3

bacteria.Findthe rate of growth at timet= 100 sec.

y=t3 y'=t3 by Example 1.

Att= 1000,y'=3,000,000 bacteria/sec.

EXAMPLE7 Suppose the cost of making x needles is_____dollars. What is the marginal cost after 10,000 needles have beenmade?

y=______,y' =_______ by Example 2.

Atx= 10,000,y' =_______ =______dollars per needle.

Thusthe marginal cost is one half of a cent per needle.

PROBLEMSFOR SECTION 2.1

Findthe derivative of the given function in Problems 1-21.

1f(x) =x²2f(t) = t² + 3

3f(x) =1-2x²4f(x) =3x² + 2

5f(t) =4t6f(x)=2-5x

7f(t)=4t3 8f(t) = -t3

9f(u) =_____10 f(u) =_____

11g(x)=______12g(x)=______

13g(t)=______14g(x)=t -3

15f(y)=3y-1 + 4y16f(y)=2y3 + 4y²

17f(x)=ax +b18f(x)=ax²

19f(x)=________20f(x)=1/(x+2)

21f(x)=1/(3-2x)

22Find the derivative of f(x) = 2x² at the point x=3.

23Find the slope of the curve f(x) = ________at the point x=5.

24An object moves according to the equation y= 1/(t+2), t≥ 0.

Findthevelocity as a function of t.

25A particle moves according to the equation y=t4. Findthe velocity as a function of t.

26Suppose the population of a town grows according to the equation y=100t + t2. Find the

rateofgrowth at time t= 100 years.

27Suppose a company makes a total profit of 1000x- x²dollars on x items. Find the

Marginalprofitin dollars per item when x = 200, x=500, and x=1000.

28Find the derivative of the function f(x) = |x + 1|.

29Find the derivative of the function f(x) = |x 3|.

30Find the slope of the parabola y= ax² + bx + cwhere a, b, c are constants.