The Derivative as a Rate of Change

In Alg I and Alg II you used the slope of a line to estimate the rate of change of a function with respect to its independent variable. We know this would be the average rate of change over an interval.

We now know that if we take the derivative of any function at a point, it would give us the rate of change of that function for that value, that particular item, or a particular moment in time.

Area of a Circle: r The rate of change in the area with respect to the radius. Units would be

r Ex: As the radius of a circle changes, so does the area. Find the rate of change in the area of a circle when its radius is 8cm.

We will focus on some familiar rates having to do with motion.

There are a lot of terms that we need to define.

Suppose the position of a moving particle is given in the form of a function of time, s(t). (Note: s is not my choice, it is the typical letter used for these problems.)

For our purposes (and abilities at this point) we say that the particle is moving along a number line.

If s(2)=5 then the particle is at 5 when t=2. If s(4)=-6 then the particle is at -6 when t=4. 0

Whenever the term initial is used, it means when t=0. Ex. The initial position of the particle would be found by s(0).

Displacement: the change in the position over an interval of time

What would a displacement of -5 mean? The moving particle ended up 5 units to the left of where it started.

What would a displacement of 0 mean? It does not necessarily mean that the particle didn’t move. It just means that it ended up where it started.

IB Only: “Displacement function”: is just the position function.

Velocity: The rate of change in the position with respect to time

Average Velocity: The rate of change in the position over an interval of time

Finding Average Velocity: is just like finding the slope of the secant line. Average velocity doesn’t tell you much about the particle’s movements between

Instantaneous Velocity: The rate of change in the position at a given moment in time:

A positive velocity means the particle is moving forward. A negative velocity means that the particle is moving backwards.

What would a velocity of 0 mean? That the particle is not moving, or is“at rest”.

What must happen before a particle can change directions? The particle must stop, that is, its velocity must reach zero.

The Units of Velocity: The notation is a great reminder of the units. Whether average velocity or instantaneous:

Speed: (The rate of change in position with respect to time, but without direction) Note: When a question asks “how fast”, it is asking for the speed. This can be average or instantaneous velocity

Acceleration: The rate of change in velocity with respect to time

Position function: Velocity function: Acceleration function:

The Units of Acceleration: The notation is a great reminder of the units: If the units of both times are the same

Just like velocity, acceleration can be positive, negative or zero.

What would a positive acceleration mean? The velocity is increasing. Example: Think of pressing on the accelerator.

What would a negative acceleration mean? The velocity is decreasing. Example: Think of letting up on the accelerator.

What would a zero acceleration mean? The velocity is constant. Example: Think of “cruise control”.

Be careful about using terms like “speeding up” and “slowing down” These are specifically defined.

t s(t) velocity is + Acceleration is – = Slowing down

t s(t) velocity is + Acceleration is + = Speeding Up

t s(t) velocity is – Acceleration is – = Speeding Up

t s(t) velocity is - Acceleration is + = Slowing down

If the velocity and the acceleration of a particle at a given time, t: A. Are the same sign, then the particle is speeding up. B. are opposite signs, then the particle is s ss slowing down.

Example: Given a particle moving along a line. Its position in units on the line at t sec (t>0) is given by:

What is the particle’s initial position?

Find a function for the velocity of the particle at any time t?

Find the particle’s displacement from t=0 to t=3?

Find the particle’s average velocity from t=0 to t=3?

What is the particle’s initial velocity?

Find a function for the acceleration of the particle at any time t.

Find the speed of the function at t=3?

When is the particle “at rest”? Only one of those times is positive. At t = 3.2 sec approx. the particle stops

Is the particle speeding up or slowing down at t=4. Both are same sign, therefore speeding up

t=0 S=0 t=1 S= -14 t=3 S= -36 t=3.2 S= t=6, S= 36

Concept Questions!

In the next slide we are given the graph of a position function, s(t), for a particle in motion on a number line at time t in seconds.

ca f e d b S(t) t Describe the initial position of the particle relative to the origin? To the left of the origin

ca f e d b S(t) t When is the particle at the origin? t=a, t=c, t=e

ca f e d b S(t) t When is the particle moving forward? (0,b) (d,f)

ca f e d b S(t) t When is the particle moving backward? (,b,d,)

ca f e d b S(t) t When is the particle at rest? t=b, t=d

ca f e d b S(t) t When is the particle to the right of the origin? (a,c) (e,f)

ca f e d b S(t) t At t=c is the acceleration positive or negative ? positive

ca f e d b S(t) t At t=c is the particle speeding up or slowing down? Slowing down

ca f e d b S(t) t At t=e is the acceleration positive or negative ? positive

ca f e d b S(t) t At t=e is the particle speeding up or slowing down? Speeding Up

In the next slide we are given the graph of the velocity function, v(t), for a particle in motion at time t in seconds.

ca f e d b v(t) t When is the particle moving forward? (a,c) (e,f)

ca f e d b v(t) t When is the particle moving backward? (0,a) (c,e)

ca f e d b v(t) t When is the particle at rest? t=a, t=c, t=e

ca f e d b v(t) t When is the acceleration of the particle positive? (0,b) (d,f)

ca f e d b v(t) t When is the acceleration of the particle negative? (b,d)

ca f e d b v(t) t At t=g is the particle speeding up or slowing down ? Slowing down g

ca f e d b v(t) t At t=g is the particle speeding up or slowing down? Speeding up g

ca f e d b v(t) t At t=g is the particle speeding up or slowing down? Speeding up g

ca f e d b v(t) t When is the acceleration zero ? t=b, t=d