math calc2

Average Value of a Function

Definition 1 Let f be continuous on the closed interval [a,b]. The average value of f on [a,b] is given by

fave = 1

b−a

∫ b a

f(x) dx

� 1, 2 Find the average value of the function on the given interval.

1. f(x) = cos x, [0,π/2]

2. g(t) = t2 √

1 + t3, [0, 2]

� 3, 4 A function f and interval I are given.

a) Find fave on I.

b) Find all numbers c in I for which f(c) = fave.

c) Illustrate your solution(s) to Part (b) by sketching the graph of f and a rectangle whose area is the same as the area under the graph of f.

3. f(x) = ln x, [1,e]

4. f(x) = 2 sin x− sin 2x, [0,π] (Hint: You may use TRACE to approximate the c values.)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5. Find all numbers b such that the average value of f(x) = 2 + 6x−3x2 on the interval [0,b] is equal to 3.

6. Suppose that the area bound by the graph of f over [1, 4] is 9. Find fave.

7. Suppose that fave = 5 on [−3, 3]. Find ∫ 3 −3 f(x) dx.

Applications

8. A cup of coffee has temperature 95◦C in a room where the temperature is 20◦C. According to Newton’s Law of Cooling, the temperature of the coffee after t minutes is given by

T(t) = 20 + 75e−t/50

a) Find Tave over the first half hour.

b) Solve by hand to find the time at which Tave is realized.

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9. Calculus shows that the height of an object that is shot straight up in the air is given by

h(t) = a

2 t2 + v0t + h0

where v0 is the initial velocity, h0 is the initial height, and a = −32 ft/sec2 is the gravitational constant. Find the average height of a dart that is shot in the air vertically from ground level with initial velocity 64 ft/sec.

10. Find the average area of the circles whose radii r vary from 0 to 1.

Hint: The differential is dr.

11. Consider an automobile that travels along a straight path with position s(t) and velocity v(t) over a time interval [t1, t2]. By definition, its average velocity is given by

∆s

∆t = s(t2) −s(t1) t2 − t1

Show that the integral definition of vave is consistent with this definition. This shows that the average velocity is equivalent to the average of the velocities.

Hint: Apply the average value integral to v(t) and invoke the Fundamental Theorem of Calculus to evaluate.

Solutions to Selected Problems

1. 2 π

2. 26 9

3. a) 1

e− 1

b) c = e 1

e−1

c) Graph

4. a) 4 π

b) c ≈ 1.24 and 2.81

c) Graph

5. b = 3 ± √

5

2

6. 3

7. 30

8. a) Tave ≈ 76.4◦

b) t = 14.25 min

9. have = 128 3

10. π 3

11. Proof

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