Calc1231W13T1 - Average of a function

Sunday, 21 October 2018

5:21 PM

Consider the discrete function f@) defined on [0, 8]; 
9 
6 
1 
3 
4 
7 
6 
2 
The sum of the values of f(x) from O to 8 is 
The average value of f@) on the interval [0, 8] is: 
and 
and 
and 
and 
and 
and 
and 
and 
8 
6 
4 
2 
8 
n=l 
f(n) — 
38 
4 75

 

Consider the continuous function f@) defined as follows 
The area under the function f@) from 0 to 8 is 
The average value of f@) is 
—21 x +265 
50 
8 
936 
8 
1 
8-0 
2 
117 
3 
4 
5 
6 
7 
8

 

The average value f of a function f on [a, b] is defined by the formula 
1 
f@) dc. 
• the average value of f@) = 5 on [2, 5] is f 
• the average value of g(x) — 2m + 1 on [1, 5] is 
• the average value of h@) 
x +1 on [5, 10] is h 
7 
178/3

 

The Mean Value Theorem for Integrals states that if f@) is continuous on the interval [a, b] then for some c e (a, b): 
f(c) = j 
dc. 
For a fixed interval [a, b] 
[2, 8] and the continuous functions f, g and h specified below; the Mean Value Theorem for Integrals guarantees that for some c e (2, 8) the function value at c 
equals the function average. Using the slider below, (or otherwise) find c: 
. f@) = 2m + 1 then f(c) 
• Hence, c 
5 
= 2m + I 
= 5.7 
(5.7, 12.4) 
Similarly if we change the interval to [a, b] = [4, 7] then the functions g, h defined below are continuous on this interval. The Mean Value Theorem for integrals then states that for some c, 
the function value at c equals the function average, find c: 
• g@) = 5 then c 
• h(x) = 6m — 3 then c 
33/6

 

The following example has been adapted from sample final exams, which are available from the UNSW library 
The following MAPLE session may assist you with this question 
t— > t— sin(t) : 
t— > 1— cos(t) : 
:= diff(x(t) , t) : 
:= diff(y(t) , t) : 
> Al := simplify(sqrt(xp +yp )); 
> yp 
> 12 
:= int(A1 * (1 — .2*Pi); 
2 — 2 cos(t) 
32 
3 
The area of the surface of revolution formed when the curve with parametric equation x = t — sin(t),y = 1 — cos(t) between t 
27T 
27ry(t) 
Using this formula and the Maple session, the surface area is 
Area 
Note: the Maple syntax for T is Pi. 
0 and t 
2K is rotated about the x-axis is

 

 

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