SHW6B Answers:  Real World Problems pp. 74-76 continued.

Definition. The Average Val ue of a function f(x)   on an interval [a, b] is defined :  

`Average Value ` = 1/(b-a)*Int(f(x),x = a .. b)

Example 1.

[Maple Plot]

`Function: `*f(x) = -x^2+1, `on the interval `*[-1, 1]

`Average Value ` = 1/2*Int(-x^2+1,x = -1 .. 1)

`Average Value ` = .6666666667

`Ave Value = .6667 = 66.67 % of Max Value`

``

`Find x-intercepts:`

f(x) = 0

-x^2+1 = 0

` x-intercept(s): `*x = -1.000000000, 1.

``

`Derivative of Function: `*`f '`(x) = -2*x

`Find bumps:`

`f '`(x) = 0

-2*x = 0

` x-bump(s): `*x = 0.

` bump point(s) ` = [0., 1.]

SHW6B: For each of the following compute the average value of the function on the given interval and sketch the graph.

Exercise 3. p. 74.

[Maple Plot]

`Function: `*f(x) = 12*x-2*x^2, `on the interval `*[0, 6]

`Average Value ` = 1/6*Int(12*x-2*x^2,x = 0 .. 6)

`Average Value ` = 12.

`Ave Value = 12.00 = 66.67 % of Max Value`

``

`Find x-intercepts:`

f(x) = 0

12*x-2*x^2 = 0

` x-intercept(s): `*x = 0., 6.

``

`Derivative of Function: `*`f '`(x) = 12-4*x

`Find bumps:`

`f '`(x) = 0

12-4*x = 0

` x-bump(s): `*x = 3.

` bump point(s) ` = [3., 18.]

Exercise 5. p. 74.

[Maple Plot]

`Function: `*f(x) = x*(4-x^2)^(1/2), `on the interval `*[0, 2]

`Average Value ` = 1/2*Int(x*(4-x^2)^(1/2),x = 0 .. 2)

`Average Value ` = 1.333333333

`Ave Value = 1.333 = 66.67 % of Max Value`

``

`Find x-intercepts:`

f(x) = 0

x*(4-x^2)^(1/2) = 0

` x-intercept(s): `*x = -2.000000000, 0., 0., 2.000000000

``

`Derivative of Function: `*`f '`(x) = (4-x^2)^(1/2)-x^2/(4-x^2)^(1/2)

`Find bumps:`

`f '`(x) = 0

4-2*x^2 = 0

` x-bump(s): `*x = -1.414213562, 1.414213562

` bump point(s) ` = [-1.414213562, -2.000000000], [1.414213562, 2.000000000]

Exercise 7. (a) p. 74.

[Maple Plot]

`Function: `*f(x) = 500*x-x^2, `on the interval `*[0, 500]

`Average Value ` = 1/500*Int(500*x-x^2,x = 0 .. 500)

`Average Value ` = 41666.66667

`Ave Value = .4167e5 = 66.67 % of Max Value`

``

`Find x-intercepts:`

f(x) = 0

500*x-x^2 = 0

` x-intercept(s): `*x = 0., 500.

``

`Derivative of Function: `*`f '`(x) = 500-2*x

`Find bumps:`

`f '`(x) = 0

500-2*x = 0

` x-bump(s): `*x = 250.

` bump point(s) ` = [250., 62500.]

Exercise 9. p. 74.

[Maple Plot]

`Function: `*f(x) = x*(16-x^2)^(1/2), `on the interval `*[0, 4]

`Average Value ` = 1/4*Int(x*(16-x^2)^(1/2),x = 0 .. 4)

`Average Value ` = 5.333333333

`Ave Value = 5.333 = 66.67 % of Max Value`

``

`Find x-intercepts:`

f(x) = 0

x*(16-x^2)^(1/2) = 0

` x-intercept(s): `*x = -4.000000000, 0., 0., 4.000000000

``

`Derivative of Function: `*`f '`(x) = (16-x^2)^(1/2)-x^2/(16-x^2)^(1/2)

`Find bumps:`

`f '`(x) = 0

16-2*x^2 = 0

` x-bump(s): `*x = -2.828427125, 2.828427125

` bump point(s) ` = [-2.828427125, -8.000000001], [2.828427125, 8.000000001]

Exercise 13. p. 75.

[Maple Plot]

`Function: `*f(x) = 200*x-5*x^2, `on the interval `*[0, 40]

`Average Value ` = 1/40*Int(200*x-5*x^2,x = 0 .. 40)

`Average Value ` = 1333.333333

`Ave Value = 1333. = 66.67 % of Max Value`

``

`Find x-intercepts:`

f(x) = 0

200*x-5*x^2 = 0

` x-intercept(s): `*x = 0., 40.

``

`Derivative of Function: `*`f '`(x) = 200-10*x

`Find bumps:`

`f '`(x) = 0

200-10*x = 0

` x-bump(s): `*x = 20.

` bump point(s) ` = [20., 2000.]

Exercise 15. p. 76.

[Maple Plot]

`Function: `*f(x) = x-3/4*x^2, `on the interval `*[0, 4/3]

`Average Value ` = 3/4*Int(x-3/4*x^2,x = 0 .. 4/3)

`Average Value ` = .2222222222

`Ave Value = .2222 = 66.67 % of Max Value`

``

`Find x-intercepts:`

f(x) = 0

x-3/4*x^2 = 0

` x-intercept(s): `*x = 0., 1.333333333

``

`Derivative of Function: `*`f '`(x) = 1-3/2*x

`Find bumps:`

`f '`(x) = 0

2-3*x = 0

` x-bump(s): `*x = .6666666667

` bump point(s) ` = [.6666666667, .3333333333]