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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