3.7 Rates of Change in the Natural and Social Sciences Remember... Average rate of change slope of a secant (between two points) Instantaneous rate of change slope of a tangent derivative We will assume that in calculus the phrase "rate of change" refers to the instantaneous rate of change. Particle Motion along the x-axis Three closely related concepts that you need to keep straight: 1. Position: x(t) or s(t) - determines where the particle is located on the x-axis at a given time t 2. Velocity: v(t) = s'(t) - determines how fast the position is changing at a time t as well as the direction of movement 3. Acceleration: a(t) = v'(t) = s''(t) - determines how fast the velocity is changing at time t; the sign indicates if the velocity is increasing or decreasing
If x(t) represents the position of a particle along the x-axis at any time t, then the following statements are true: 1. "Initially" means when = 0. 2. "At the origin" means when = 0. 3. "At rest" means = 0. 4. If the velocity of the particle is positive, then the particle is moving to the. 5. If the velocity of the particle is, then the particle is moving to the left. 6. To find the average velocity over a time interval, divide the change in by the change in time. 7. Instantaneous velocity is the velocity at a single moment (instant!) in time. 8. If the acceleration of the particle is positive, then the is increasing. 9. If the acceleration of the particle is, then the velocity is decreasing. 10. In order for a particle to change direction, the must change signs. 11. For a particle to "speed up", v(t) and a(t) must have the signs. 12. For a particle to "slow down", v(t) and a(t) must have signs.
Example 1 Find the total distance traveled on the interval 0 t 6 for the position of 1 a particle using the function s(t) = t 3 - t 2-3t + 4. 3 Example 2 Bugs Bunny has been captured by Yosemite Sam and forced to "walk the plank". Instead of waiting for Yosemite Sam to finish cutting the board from underneath him, Bugs finally decides just to jump. Bugs' position, s, is given by s(t) = -16t 2 + 16t + 320, where s is measured in feet and t is measured in seconds. a) What is Bugs' displacement from t = 1 to t = 2 seconds? b) When will Bugs hit the ground? c) What is Bugs' velocity at impact? d) What is Bugs' speed at impact? Include units. e) Find Bugs' acceleration as a function of time. Include units.
Example 3 Suppose the graph below shows the velocity of a particle moving along the x-axis. Justify each response. a) Which way does the particle move first? b) When does the particle stop? c) When does the particle change direction? d) When is the particle moving left? e) When is the particle moving right? f) When is the particle speeding up? g) When is the particle slowing down? h) When is the particle moving the fastest? i) When is the particle moving at a constant speed?
j) Graph the particle's acceleration for 0 < t < 10. k) Graph the particle's speed for 0 < t < 10.
Example 4 Table 1 contains data on temperature T on the surface of Mars at Martian time t, collected by the NASA Pathfinder space probe. a) Calculate the average rate of change from temperature T from 6:11 am to 9:05 am. b) Use the figure at the right to estimate the rate of change at t = 12:28 pm. Example 5 Let A = πr 2 be the area of a circle with radius r. a) Compute da at r = 2 and r = 5. dr b) Why is larger at r = 5? da dr
Example 6 A truck enters the off-ramp of a highway at t = 0. Its position after t seconds is s(t) = 25t - 0.3t 3 meters for 0 t 5. a) How fast is the truck going at the moment it enters the off-ramp? b) Is the truck speeding up or slowing down? Example 7 The position of a particle is given by the equation s = f(t) = t 3-6t 2 + 9t where t is measured in seconds and s is in meters. a) Find the velocity at time t. b) What is the velocity after 2 s? After 4 s? c) When is the particle at rest?
d) When is the particle moving forward (that is, in the positive direction)? e) Draw a diagram to represent the motion of the particle. f) Find the total distance traveled by the particle during the first five seconds. g) Find the acceleration at time t and after 4 seconds.
h) Graph the position, velocity, and acceleration for 0 t 5. i) When is the particle speeding up? When is it slowing down?