School

University of California, Irvine **We aren't endorsed by this school

Course

CALCULUS 3A

Subject

Mathematics

Date

Oct 20, 2023

Pages

6

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1
Section 2.7
Derivatives and Rates of change
Recall the tangent line problem in section 2.1
We define the slope of the tangent line at
P
to be the limit:
?
??
= lim
𝑥→𝑎
?(?) − ?(𝑎)
? − 𝑎
Ex1: Find the equation of the tangent line
to
? = ?
2
𝑎? 𝑃(1,1)
There is another expression for the slope of a tangent line that is sometimes easier to use.
If we let
ℎ = ? − 𝑎 ?ℎ?? ? = 𝑎 + ℎ
.
Also,
? → 𝑎 ??𝑎?? ℎ → 0
?
??
=

2
Ex2: Find the equation of the tangent line to the hyperbola
? =
3
𝑥
𝑎? ? = 3
Velocity:
In general, suppose an object moves along a straight line according to an equation of
motion
s = f(t)
, where
?
is the displacement (directed distance) of the object from the
origin at time
?
. The function
?
that describes the motion is called the position function of
the object.
In the time interval from
? = 𝑎
to
? = 𝑎 + ℎ
the change in position is
?(𝑎 + ℎ) − ?(𝑎)
average velocity
=
displacement
time elapsed
=
f(a+h)−f(a)
h
Instantaneous velocity is the limit of average velocities:
𝒗(𝒂) = ?𝐢?
𝒉→𝟎
𝒇(𝒂+𝒉)−𝒇(𝒂)
𝒉
(
Instantaneous velocity at
𝒕 = 𝒂
)
(Speed: is the absolute value of velocity)
Note: if v >0 means the particle moves to the positive (right or up) direction
If v<0 means the particle moves to the negative (left or down) direction
If v=0 means the particle does not change its position.

3
Ex3: Suppose that a ball is dropped from the upper observation deck of a tower 450 m
above the ground.
(a) What is the velocity of the ball after 5 seconds?
(b) How fast is the ball traveling when it hits the ground?
Solution:
We will need the velocity both when t = 5 and when the ball hits the ground.
So it is efficient to find the velocity at a general time t, then evaluate it at the needed
values.
Recall from 2.1: The distance (in meters) fallen after t seconds is
?(?) = ?(?) = 4.9?
2
Now: instantaneous velocity at time t :
𝑣(?) =
?𝐢?
𝒉→𝟎
𝒇(𝒕+𝒉)−𝒇(𝒕)
𝒉

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