第6.5节 平均数

6.5 AVERAGES

Given n numbers y1 …, yn, their average value is defined as

_____________________

If all the y1 are replaced by the average value yave, the sum will be unchanged,

y1 + … + yn = yave + … + yave = n yave .

If f is continuous function on a closed interval [a, b] , what is meant by the average value of f between a and b (Figure 6.5.1)? Let us try to imitate the procedure for finding the average of n numbers. Take an infinite hyperreal number H and divide the interval [a ,b] into infinitesimal subintervals of length dx=(b-a)/ H. Let

Figure 6.5.1

Us “sample” the value of f at the H points a, a+dx, a+2dx, …，a + (H-1)dx.

Then the average value of f should be infinitely close to the sum of the values of f at a, a+ dx, ……, a+ (H-1)dx, divided by H. Thus

Since dx = _________ , ______________ and we have

fave ≈__________________.

________________.

Taking standard parts, we are led to

DEFINITION

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

fave = ______________

Geometrically, the area under the curve y=f(x) is equal to the area under the constant curve y=fave between a an b,

fave·(b-a) = _____ f(x)dx.

EXAMPLE 1 Find the average value of y=___ from x=1 to x=4 ( Figure 6.5.2).

Figure 6.5.2

Recall that in Section 3.8, we defined the average slope of a function F between a and b as the quotient

Average slope = ________________

Using the Fundamental Theorem of Calculus we can find the connection between the average value of F 'and the average slope of F.

THEOREM 1

Let F be an antiderivative of a continuous function f on an open interval I.

Then for any a < b in I , the average slope of F between a and b is equal to the average value of f between a and b,

___________________________.

PROOF By the Fundamental Theorem,

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

THEOREM 2 (Mean Value Theorem for Integrals)

PROOF Theorem 2 is illustrated in Figure 6.5.3. We can make f continuous on the whole real line by defining f(x) = f(a) for x <a and f(x)=f(b) for x > b.

By the Second Fundamental Theorem of Calculus , f has an antiderivative F.

By the Mean Value Theorem there is a point c strictly between a and b at which F'(c) is equal to the average slope of F,

_____________________

But F(c) = f(c) and F(b) - F(a) = ______ dx, so

f(c)=____________

Figure 6.5.3

EXAMPLE 2

A car stats at rest and moves with velocity v= 3t². Find its average velocity between times t=0 and t=5. At what point of time is its velocity equal to the average velocity?

__________________________=25

To find the value of t where v=vave , we put

3t² = 25, t= _________ = 5/ _________.

Suppose a car drives from city A to city B and back, a distance of 120 miles each way. From A to B it travels at a speed of 30 mph, and on the return trip it travels at 60 mph. What is the average speed?

If we choose distance as the independent variable we get one answer, and if we choose time we get another.

Average speed with respect to time : the car takes 120/ 30 = 4 hours to go from A to B AND 120/60 =2 hours to return to A. The total trip takes 6 hours.

_________________________

Average speed with respect to distance : the car goes 120 miles at 30 mph and 120 miles at 60 mph, with a total distance of 240 miles. Therefore

___________________

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

EXAMPLE 3 A car travels with velocity v= 4t + 10, where t is time. Between times t = 0 and t=4 find the average velocity with respect to (a) time, and (b) distance.

________________________________ (Figure 6.5.5(a)).

Figure 6.5.5

(b) let s be the distance, and put s= 0 when t=0. Since ds=vdt = (4t + 10 ) dt, at time t = 4 we have

_______________________

Then _____________________________

PROBLEMS FOR SECTION 6.5

In Problems 1-8, sketch the curve, find the average value of the function, and sketch the rectangle which has the same area as the region under the curve.

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

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

5 f(x) = _____, 1 ≤ x ≤ 5 6 f(x) = x3, 0 ≤ x ≤ 2

7 f(x) = ______, 0 ≤ x ≤ 8 8 f(x) = 1 - x4, -1 ≤ x ≤ 1

In Problems 9-22, find the average value of f(x).

9 f(x) = x² - ______, 0 ≤ x ≤ 3 10 f(x) = _____ + ______, 1 ≤ x ≤ 9

11 f(x) = 6x, - 4 ≤ x ≤ 2 12 f(x) = ________, ___ ≤ x ≤ ____

13 f(x) =_______, - 3 ≤ x ≤ 3 14 f(x) = 5x4 - 8x3 + 10, 0 ≤ x ≤ 10

15 f(x) =sin x, 0 ≤ x ≤ π 16 f(x) =sin x, 0 ≤ x ≤ 2π

17 f(x) =sin x cos x, 0 ≤ x ≤ π/2 18 f(x) =x + sin x, 0 ≤ x ≤ 2π

19 f(x) =e x, -1 ≤ x ≤ 1 20 f(x) =e x - 2x, 0 ≤ x ≤ 2

21 f(x) =___, 1 ≤ x ≤ 4 22 f(x) =_____, 0 ≤ x ≤ 4

In Problems 23-28, find a point c in the given interval such that f(c) is equal to the average value of f(x).

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

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

27 f(x) =x2/3, 0 ≤ x ≤ 2 28 f(x) =|x - 3| 1 ≤ x ≤ 4

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

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

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

with respect to x.

32 A particle moves with velocity v= 6t from time t = 0 to t=10. Find its average velocity

with respect to (a) time , (b) distance.

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

respect to time is

_________________

Show that its average velocity with respect to distance is

____________________.