6.5  AVERAGES

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

                        _____________________

 

If all the y1are 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 Hand 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 fshould be infinitely close to the sum of the values of fat 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=favebetween 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 Fbetween 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)

 

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

 

                         ________________________

 

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 and f(x)=f(b) for x > b.

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

By the Mean Value Theorem there is a point cstrictly 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 twhere v=vave , we put

 

                        3t ²= 25,  t= _________= 5/_________.

 

Suppose a car drives from city Ato city B and back, a distance of 120 miles each way. From Ato 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 xcosx,  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