DERIVATIVES: RATES OF CHANGEMR. VELAZQUEZ

AP CALCULUS

VELOCITY AS A RATE OF CHANGE

Let’s suppose an object moves along a straight line where its displacement 𝑠 (distance from its point of origin) is represented by some function of time 𝑡, or 𝑠 = 𝑓(𝑡). We will call 𝑓(𝑡) the position function of the object.

Now we ask, what is the average velocity of the object over a certain time interval?

Drawing a graph of the function over time can help us better understand this velocity. But recall the difference quotient(slope of the secant) of a function over the interval 𝑎, 𝑏 shown below:

𝑓 𝑏 − 𝑓 𝑎

𝑏 − 𝑎

𝑠

𝑡

The average velocity is the slope of the bold dashed line.

𝑎 𝑏

Note that as 𝑏 and 𝑎 get closer together, we get a velocity

that better represents the instantaneous speed at 𝑎.

𝑓(𝑡)

EXAMPLES:

If 𝑠 = 𝑓 𝑡 = 𝑡2 + 2𝑡, find the average velocity on

the intervals 0, 1 , 0, 2 and 1, 2 .

If 𝑠 = 𝑓 𝑡 = 𝑡3 − 9𝑡, find the average velocity on

the intervals 0, 1 , 0, 3 , 0, 4 and 1, 4 .

INSTANTANEOUS RATE OF CHANGE

Now, suppose that we define ℎ as the difference between 𝑎and 𝑏. This implies that 𝑏 = 𝑎 + ℎ

Here, note that as ℎ → 0, the value of the average velocity

becomes closer and closer to the instantaneous velocity.

Graphically, the secant line becomes closer and closer to the

tangent line at 𝑡 = 𝑎.

Knowing this, we can define the instantaneous velocity at

𝑡 = 𝑎 using a limit:

𝑣 𝑎 = limℎ→0

𝑓 𝑎 + ℎ − 𝑓(𝑎)

ℎ

𝑡

𝑠

Note that as ℎ → 0, the secant line becomes more like a

tangent line, which only meets the function at a single point.

EXAMPLES:

If 𝑠 = 𝑓 𝑡 = 𝑡2 + 2𝑡, find the instantaneous velocity at

𝑡 = 0, 𝑡 = 1, and 𝑡 = 2.

If 𝑠 = 𝑓 𝑡 = 𝑡3 − 9𝑡, find the instantaneous velocity at

𝑡 = 1 and 𝑡 = 3.

DERIVATIVE DEFINED AS A RATE OF CHANGE

We can extend this same tangent line logic to any function of one

variable, commonly 𝑦 = 𝑓(𝑥). We simply replace the 𝑠 with 𝑦 and

t = 𝑎 with 𝑥.

The slope of the line tangent to 𝑓(𝑥) for any 𝑥 can then be defined

as the limit as ℎ → 0 of the difference quotient. We call this slope

the derivative of the function at 𝒙.

𝑑

𝑑𝑥𝑓(𝑥) = 𝑓′ 𝑥 = lim

ℎ→0

𝑓 𝑥 + ℎ − 𝑓(𝑥)

ℎ

Leibnizian Notation Newtonian Notation

NOTATION OF DERIVATIVES

Note that all the above refer to the same thing—the derivative

of a function of 𝑥, or essentially the slope of the line tangent to

the function at a given 𝑥.

We can also plug in specific values for 𝑥 into the limit and find

the slope of the tangent line at that 𝑥.

𝑓′ 𝑥 𝑦′ 𝑥 𝑦′

𝑑𝑦

𝑑𝑥

𝑑

𝑑𝑥𝑓 𝑥

𝑑𝑓

𝑑𝑥

Newton

Leibniz

EXAMPLES:

Find the slope of the line tangent to 𝑓 𝑥 = 𝑥3 − 1at 𝑥 = 2

Find the slope of the line tangent to 𝑔 𝑥 = 𝑥2 − 𝑥at 𝑥 = 0 and 𝑥 = 1

EXAMPLES:

Find the equation of the tangent line to 𝑓 𝑥 =1

𝑥−2

at 𝑥 = 3.

Find the equation of the tangent line to 𝑔 𝑥 = 2 − 𝑥at 𝑥 = 1.

CLASSWORK & HOMEWORK

MATH JOURNAL: Summarize what you’ve learned today

CLASSWORK: DEFINITION OF DERIVATIVES – On a separate

sheet of paper, find the values indicated below, using the definition

of the derivative:

a) Find the slope of the line tangent to 𝑦 = 2𝑥2 at 𝑥 = 2

b) Find the slope of the line tangent to 𝑓 𝑥 = 𝑥 + 6 at 𝑥 = 3

c) Find the equation of the tangent line to 𝑦 𝑥 = 𝑥2 − 𝑥 at 𝑥 = −1

d) Find the equation of the tangent line to 𝑔 𝑥 =1

𝑥+3at 𝑥 = 1

Homework: Pg. 103-104, #1-32