第6.5节 平均值

6.5AVERAGES

Given n numbersy1…,yn,their average value is defined as

_____________________

Ifall they1arereplaced by the average valueyave,the sum will be unchanged,

y1+… + yn=yave+… +yave=n yave.

Iffiscontinuous function on a closed interval [a, b], what is meant by the average value off betweenaandb(Figure6.5.1)? Let us try to imitate the procedure for finding the averageofn numbers.Take an infinite hyperreal numberHanddivide the interval [a ,b]into infinitesimal subintervals of length dx=(b-a)/H.Let

Figure6.5.1

Us“sample”thevalue of f attheH pointsa, a+dx, a+2dx,…，a+ (H-1)dx.

Thenthe average value offshouldbe infinitely close to the sum of the values offata,a+ dx, ……,a+ (H-1)dx,divided byH.Thus

Sincedx=_________, ______________andwe have

fave≈__________________.

________________.

Takingstandard parts, we are led to

DEFINITION

Letf be continuous on [a, b]. The average value of f between a and b is

fave= ______________

Geometrically,the area under the curvey=f(x)isequal to the area under the constant curvey=favebetweenaanb,

fave·(b-a)= _____ f(x)dx.

EXAMPLE1Findthe average value ofy=___fromx=1tox=4( Figure 6.5.2).

Figure6.5.2

Recallthat in Section 3.8, we defined the average slope of a functionFbetweenaandbasthe quotient

averageslope=________________

Usingthe Fundamental Theorem of Calculus we can find the connectionbetween the average value of F'andthe average slope ofF.

THEOREM1

LetF be an antiderivative of a continuous function f on an open intervalI.

Thenfor any a < b in I , the average slope of F between a and b isequal to the average value of f between a and b,

___________________________.

PROOFBy the Fundamental Theorem,

F(b)-F(a) =_____ f (x) dx.

THEOREM2 (Mean Value Theorem for Integrals)

Letf be continuous on [ a, b]. Then there is a point c strictly betweena and b where the value of f is equal to its average value,

PROOFTheorem 2 is illustrated in Figure 6.5.3. We can make f continuous on

thewholereal line by defining f(x) = f(a) forx <aandf(x)=f(b)forx > b.

BytheSecond Fundamental Theorem of Calculus , fhasan antiderivativeF.

Bythe Mean Value Theorem there is a pointcstrictlybetweena andbat

whichF'(c)isequal to the average slope ofF,

_____________________

ButF(c) = f(c)andF(b) - F(a)=______dx,so

f(c)=____________

Figure6.5.3

EXAMPLE2

Acarstats at rest and moves with velocity v= 3t².Find its average

velocitybetweentimest=0andt=5.At what point of time is its velocity

equaltothe average velocity?

__________________________=25

Tofind the value oftwherev=vave, we put

3t²=25,t=_________=5/_________.

Supposea car drives from cityAtocityB andback, a distance of 120 miles each way. FromAtoBittravels at a speed of 30 mph, and on the return trip it travels at 60mph. What is the average speed?

Ifwe choose distance as the independent variable we get one answer, andif we choose time we get another.

Averagespeed with respect to time : the car takes 120/ 30 = 4 hours to gofrom A toBand

120/60=2 hours to return toA.The total trip takes 6 hours.

_________________________

Averagespeed with respect to distance : The car goes 120 miles at 30 mph and120 miles at 60 mph, with a total distance of 240 miles. Therefore

___________________

In general, ify is given both as a function of s and of t, y=f(s) = g(t), then there is one average ofy with respectto s, and another with respect to t.

EXAMPLE3 A car travels with velocityv= 4t + 10, where t istime. Between times t = 0 andt=4 find the averagevelocity with respect to (a) time, and (b) distance.

________________________________(Figure 6.5.5(a)).

Figure6.5.5

(b)lets be the distance, and puts = 0 when t =0.Since ds =vdt = (4t+10)dt,

attimet = 4 we have

_______________________

Then_____________________________

PROBLEMSFOR SECTION 6.5

InProblems 1-8, sketch the curve, find the average value of thefunction, and sketch the rectangle which has the same area as theregion under the curve.

1f(x) = 1 +x, -1≤ x ≤12 f(x) = 2 - _____x, 0 ≤ x ≤ 4

3f(x) = 4 -x², -2 ≤ x ≤24 f(x) = 1 + x²,-2 ≤x ≤ 2

5f(x) = _____, 1 ≤x≤ 56 f(x) = x3, 0 ≤ x ≤2

7f(x) = ______, 0 ≤x≤ 88 f(x) = 1 - x4, -1 ≤ x≤ 1

InProblems 9-22, find the average value off(x).

9f(x) =x² - ______,0 ≤ x ≤ 310 f(x) = _____ +______, 1 ≤x ≤ 9

11f(x) = 6x, - 4 ≤x ≤212 f(x) = ________,___ ≤ x ≤ ____

13f(x)=_______,- 3 ≤ x ≤314 f(x) = 5x4 - 8x3 + 10, 0 ≤x ≤10

15f(x) =sinx,0 ≤ x ≤ π16 f(x)=sinx,0 ≤x ≤ 2π

17f(x) =sinxcosx,0 ≤ x ≤ π/218 f(x)=x +sinx,0 ≤x ≤ 2π

19f(x) =ex, -1≤ x ≤ 120 f(x)=e x- 2x,0 ≤ x ≤ 2

21f(x)=___,1 ≤ x ≤ 422 f(x) =_____,0≤x≤4

InProblems 23-28, find a pointc in the given interval such thatf(c) is equal to the average value off(x).

23f(x)=2x,-4 ≤ x ≤ 6 24f(x)=3x²,0 ≤ x ≤ 3

25f(x)=____,0 ≤ x ≤ 2 26f(x)=x²- x²,-1 ≤ x ≤ 1

27f(x) =x2/3,0 ≤x ≤ 228 f(x)=|x-3|1≤x ≤ 4

29What is the average distance between a pointx in the interval[5,8] and the origin ?

30What is the average distance between a point in the interval [ -4,3 ]and the origin ?

31Find the average distance from the origin to a point on the curvey=x3/2, 0 ≤ x≤ 3,

withrespecttox.

32A particle moves with velocityv= 6tfrom timet = 0 to t=10. Find its averagevelocity

withrespectto (a) time , (b) distance.

33An object moves with velocityv=f(t) from t= a to t=b. Thus its average velocity with

respecttotime is

_________________

Showthatits average velocity with respect to distance is