Definition of Average Rate of Change
The expression
is called the difference quotient for f at a and represents the average rate of change of y = f(x) from a to a + h .
Geometrically, the rate of change of f from a to a+h the slope of the secant line through the point P(a, f(a)) and Q(a+h, f(a+h)).
If f(t) is the position function of a particle that is moving on a straight line, then in the time interval from t = a to t = a+h, the change in position is f(a+h) − f(a), and the average velocity of the particle over the time interval is
Example 1. The displacement of a particle moving in a straight line is given by the equation of motion f(t) = t3 − 4t + 3. Find the average velocity of the particle over the interval 0 ≤ t ≤ 4.
Solution :
Exercises – Rate of Change
Multiple Choice Questions
1. The traffic flow at a particular intersection is modeled by the function f defined by f(t) = 25+6cos(x/3) for 0 ≤ t ≤ 120. What is the average rate of change of the traffic flow over the time interval 30 ≤ t ≤ 40.
(A) 0.743
(B) 0.851
(c) 0.935
(d) 1.176
2. The rate of change of the altitude of a hot air balloon rising from the ground is given by y(t) = t3−3t2 + 3t for 0 ≤ t ≤ 10. What is the average rate of change in altitude of the balloon over the time interval 0 ≤ t ≤ 10.
(A) 56
(B) 73
(c) 85
(d) 94
Free Response Questions
| t (sec) |
0 | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 |
| f(t) (ft/sec) |
0 | 28 | 43 | 67 | 82 | 85 | 74 | 58 | 42 | 35 |
3. The table above shows the velocity of a car moving on a straight road. The car’s velocity v is measured in feet per second.
(a) Find the average velocity of the car from t = 60 to t = 90.
(b) The instantaneous rate of change of f with respect to x at x = a can be approximated by finding the average rate of change of f near x = a. Approximate the instantaneous rate of change of f at x = 40 using two points, x = 30 and x = 50.
