Docsity
Docsity

Prepare for your exams
Prepare for your exams

Study with the several resources on Docsity


Earn points to download
Earn points to download

Earn points by helping other students or get them with a premium plan


Guidelines and tips
Guidelines and tips

Instantaneous Rate of Change: Velocity of a Moving Object, Study notes of Physics

The concept of instantaneous rate of change, specifically as it relates to the velocity of a moving object. It covers the calculation of average velocity over smaller time intervals and the approach to the instantaneous velocity using the limit notation. The document also defines the concept of speed and the relationship between velocity and the derivative of a function.

Typology: Study notes

2021/2022

Uploaded on 09/12/2022

vernon
vernon ๐Ÿ‡บ๐Ÿ‡ธ

4.8

(4)

216 documents

1 / 5

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
2.1 INSTANTANEOUS RATE OF CHANGE 1
2.1 Instantaneous Rate of Change
In this section, we discuss the concept of the instantaneous rate of change of
a given function. As an application, we use the velocity of a moving object.
The motion of an object along a line at a particular instant is very difficult
to define precisely. The modern approach consists of computing the average
velocity over smaller and smaller time intervals. To be more precise, let s(t)
be the position function or displacement of a moving object at time t. We
would like to compute the velocity of the object at the instant t=t0.
Average Velocity
We start by finding the average velocity of the object over the time interval
t0โ‰คtโ‰คt0+ โˆ†tgiven by the expression
v=Distance Tr aveled
Elapsed T ime =s(t0+ โˆ†t)โˆ’s(t0)
โˆ†t.(2.1.1)
Geometrically, the average velocity over the time interval [t0, t0+ โˆ†t] is just
the slope of the line joining the points (t0, s(t0)) and (t0+ โˆ†t, s(t0+ โˆ†t)) on
the graph of s(t).(See Figure 2.1.1)
Figure 2.1.1
Example 2.1.1
A freely falling body experiencing no air resistance falls s(t) = 16t2feet in t
seconds. Complete the following table
time interval [1.8,2] [1.9,2] [1.99,2] [1.999,2] [2,2.0001] [2,2.001] [2,2.01]
Average velocity
pf3
pf4
pf5

Partial preview of the text

Download Instantaneous Rate of Change: Velocity of a Moving Object and more Study notes Physics in PDF only on Docsity!

2.1 INSTANTANEOUS RATE OF CHANGE 1

2.1 Instantaneous Rate of Change

In this section, we discuss the concept of the instantaneous rate of change of a given function. As an application, we use the velocity of a moving object. The motion of an object along a line at a particular instant is very difficult to define precisely. The modern approach consists of computing the average velocity over smaller and smaller time intervals. To be more precise, let s(t) be the position function or displacement of a moving object at time t. We would like to compute the velocity of the object at the instant t = t 0.

Average Velocity We start by finding the average velocity of the object over the time interval t 0 โ‰ค t โ‰ค t 0 + โˆ†t given by the expression

v =

Distance T raveled Elapsed T ime

s(t 0 + โˆ†t) โˆ’ s(t 0 ) โˆ†t

Geometrically, the average velocity over the time interval [t 0 , t 0 + โˆ†t] is just the slope of the line joining the points (t 0 , s(t 0 )) and (t 0 + โˆ†t, s(t 0 + โˆ†t)) on the graph of s(t).(See Figure 2.1.1)

Figure 2.1.

Example 2.1. A freely falling body experiencing no air resistance falls s(t) = 16t^2 feet in t seconds. Complete the following table

time interval [1.8,2] [1.9,2] [1.99,2] [1.999,2] [2,2.0001] [2,2.001] [2,2.01] Average velocity

Solution. From time t = 1.8 to time t = 2, formula (2.1.1) gives

s(2) โˆ’ s(1.8) 2 โˆ’ 1. 8

โˆ’16(2)^2 โˆ’ [โˆ’16(1.8)^2 ]

= โˆ’ 60 .8 ft/sec.

Using formula (2.1.1) on each of the remaining intervals, we find

time interval [1.8,2] [1.9,2] [1.99,2] [1.999,2] [2,2.0001] [2,2.001] [2,2.01] Average velocity 60.8 62.4 63.84 63.98 64.0016 64.016 64.

Instantaneous Velocity and Speed The next step is to calculate the average velocity on smaller and smaller time intervals ( that is, make โˆ†t close to zero). The average velocity in this case approaches what we would intuitively call the instantaneous velocity at time t = t 0 which is defined using the limit notation by

v(t 0 ) = lim โˆ†tโ†’ 0

s(t 0 + โˆ†t) โˆ’ s(t 0 ) โˆ†t

Geometrically, the instantaneous velocity at t 0 is the slope of the tangent line to the graph of s(t) at the point (t 0 , s(t 0 )).(See Figure 2.1.2)

Figure 2.1.

Example 2.1. For the distance function in Example 2.1.1, find the instantaneous velocity at t = 2.

Example 2.1. (a) Find f โ€ฒ(1) for f (x) = x^2. (b) Find the equation of the tangent line to the graph of f (x) at the point (1, f (1)).

Solution. Completing the following chart

x [0.9,1] [0.99,1] [0.999,1] [1,1.0001] [1,1.001] [1,1.01] [1,1.1] f (b)โˆ’f (a) bโˆ’a 1.9^ 1.99^ 1.999^ 2.0001^ 2.001^ 2.01^ 2.

we see that f โ€ฒ(1) = 2. (b) The equation of the tangent line is

y โˆ’ f (1) = f โ€ฒ(1)(x โˆ’ 1)

or y โˆ’ 1 = 2(x โˆ’ 1).

In point-intercept form, we have y = 2x โˆ’ 1

Example 2.1.4 (Numerical Estimation of the Derivative) Find approximate values for f โ€ฒ(x) at each of the xโˆ’values given in the fol- lowing table

x 0 5 10 15 20 f (x) 100 70 55 46 40

Solution. The derivative can be estimated by using the average rate of change or the difference quotient

f โ€ฒ(a) โ‰ˆ

f (a + h) โˆ’ f (a) h

If a is a left-endpoint then f โ€ฒ(a) is estimated by

f โ€ฒ(a) โ‰ˆ

f (b) โˆ’ f (a) b โˆ’ a

where b > a. If a is a right-endpoint then f โ€ฒ(a) is estimated by

f โ€ฒ(a) โ‰ˆ

f (a) โˆ’ f (b) a โˆ’ b

2.1 INSTANTANEOUS RATE OF CHANGE 5

where b < a. If a is an interior point then f โ€ฒ(a) is estimated by

f โ€ฒ(a) โ‰ˆ

f (a) โˆ’ f (b) a โˆ’ b

f (c) โˆ’ f (a) c โˆ’ a

where b < a < c. For example,

f โ€ฒ(0) โ‰ˆ

f (5) โˆ’ f (0) 5

f โ€ฒ(5) โ‰ˆ

f (10) โˆ’ f (5) 5

f (5) โˆ’ f (0) 5

f โ€ฒ(10) โ‰ˆ

f (15) โˆ’ f (10) 5

f (10) โˆ’ f (5) 5

f โ€ฒ(15) โ‰ˆ

f (20) โˆ’ f (15) 5

f (15) โˆ’ f (10) 5

f โ€ฒ(20) โ‰ˆ

f (20) โˆ’ f (15) 5

The quantity f^ (10) 10 โˆ’โˆ’f 5 (5) is known as the right slope estimation of f โ€ฒ(5). Similarly, we can estimate f โ€ฒ(5) by using a left slope estimation,i.e.,

f โ€ฒ(5) โ‰ˆ

f (5) โˆ’ f (0) 5 โˆ’ 0

An improved estimation consists of taking the average of the left slope and the right slope, that is,

f โ€ฒ(5) โ‰ˆ