Acceleration - mathematicalminds.net
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Derivative as a Rate of Change
Chapter 3 Section 4
Usually omit
instantaneous
Interpretation: The rate of change at which f is changing at the point x
Interpretation: Instantaneous rate are limits of average rates.
Example
• The area A of a circle is related to its diameter by
A
D
the equation
4
• How fast does the area change with respect to
the diameter when the diameter is 10 meters?
• The rate of change of the area with respect to
the diameter dA D 2 dA D 2 2 D D
2
dD 4
4 dD
4
2
• Thus, when D = 10 meters the area is changing
with respect to the diameter at the rate of
D
2
(10)
2
15.71 m 2 / m
Motion Along a Line
• Displacement of object over time
Δs = f(t + Δt) – f(t)
• Average velocity of object over time interval
vaverage
displacement s f (t t ) f (t )
travel time
t
t
Velocity
• Find the body’s velocity at the exact instant t
– How fast an object is moving along a horizontal line
– Direction of motion (increasing >0 decreasing <0)
Speed
• Rate of progress regardless of direction
Graph of velocity f ’(t)
Acceleration
• The rate at which a body’s velocity changes
– How quickly the body picks up or loses speed
– A sudden change in acceleration is called jerk
• Abrupt changes in acceleration
Example 1: Galileo Free Fall
• Galileo’s Free Fall Equation
1 2
s gt
2
s distance fallen
g is acceleration due to Earth’s gravity
(appx: 32 ft/sec2 or 9.8 m/sec2)
– Same constant acceleration
– No jerk
d d ds 1 2 d d
d
j gt gt 1 0
dt dt dt 2
dt dt
dt
Example 2: Free Fall Example
• How many meters does the ball fall in
the first 2 seconds?
1
s (9.8)t 2 4.9t 2
2
• Free Fall equation s = 4.9t2 in meters
s(2) – s(0) = 4.9(2)2 - 4.9(0)2 = 19.6 m
Example 2: Free Fall Example
• What is its velocity, speed and acceleration
when t = 2?
– Velocity = derivative of position at any time t
v(t ) s ' (t )
d
4.9t 2 9.8t
dt
– So at time t = 2, the velocity is
v(t ) s ' (t )
d
4.9t 2 9.8t 9.8(2) 19.6 m / sec
dt
Example 2: Free Fall Example
• What is its velocity, speed and acceleration
when t = 2?
– Velocity = derivative of position at any time t
v(t ) s ' (t )
d
4.9t 2 9.8t
dt
– So at time t = 2, the speed is
speed velocity 19.6 m / sec 19.6 m / sec
Example 2: Free Fall Example
• What is its velocity, speed and acceleration
when t = 2?
– Velocity = derivative of position at any time t
v(t ) s ' (t )
d
4.9t 2 9.8t
dt
– The acceleration at any time t
a (t ) v' (t ) s ' ' (t )
d
9.8t 9.8 m / sec 2
dt
– So at t = 2, acceleration is
(no air resistance)
a (t ) 9.8 m / sec 2
Derivatives of
Trigonometric Functions
Chapter 3 Section 5
Derivatives
Application: Simple Harmonic Motion
• Motion of an object/weight bobbing
freely up and down with no resistance
on an end of a spring
• Periodic, repeats motion
• A weight hanging from a spring is
stretched down 5 units beyond its rest
position and released at time t = 0 to
bob up and down. Its position at any
later time t is
s = 5 cos(t)
• What are its velocity and acceleration
at time t?
Application: Simple Harmonic Motion
• Its position at any later time t is s = 5 cos(t)
– Amp = 5
– Period = 2
• What are its velocity and acceleration at time t?
– Position:
– Velocity:
s = 5cos(t)
s’ = -5sin(t)
• Speed of weight is 0, when t = 0
– Acceleration: s’’ = -5 cos(t)
• Opposite of position value, gravity pulls down, spring pulls up
Chain Rule
Chapter 6 Section 6
Implicit Differentiation
Chapter 3 Section 7
Implicit Differentiation
• So far our functions have been y = f(x) in one variable
such as y = x2 + 3
– This is explicit differentiation
• Other types of functions
x2 + y2 = 25 or y2 – x = 0
• Implicit relation between the variables x and y
• Implicit Differentiation
– Differentiate both sides of the equation with respect to x,
treating y as a differentiable function of x (always put
dy/dx after derive y term)
– Collect the terms with dy/dx on one side of the equation
and solve for dy/dx
Circle Example
Folium of Descartes
• The curve was first proposed
by Descartes in 1638. Its claim to
fame lies in an incident in the
development of calculus.
• Descartes challenged Fermat to
find the tangent line to the curve
at an arbitrary point since Fermat
had recently discovered a
method for finding tangent lines.
• Fermat solved the problem
easily, something Descartes was
unable to do.
• Since the invention of calculus,
the slope of the tangent line can
be found easily using implicit
differentiation.
Folium of Descartes
• Find the slope of the folium
of Descartes
x 3 y 3 9 xy 0
• Show that the points (2,4)
and (4,2) lie on the curve
and find their slopes and
tangent line to curve
Folium of Descartes
• Show that the points (2,4) and (4,2) lie on the
curve and find their slopes and tangent line to
curve
3
3
x y 9xy 0
23 43 9(2)(4) 0
8 64 72 0
72 72 0
00
43 23 9(4)(2) 0
64 8 72 0
72 72 0
00
Folium of Descartes
• Show that the points (2,4) and (4,2) lie on the
curve and find their slopes and tangent line
to curve
– Find slope of curve by implicit differentiation by
finding dy/dx
x y 9xy 0
3
3
d 3 d 3 d
d
x
y 9 xy 0
dx
dx
dx
dx
dy dy
d
3x 3 y
9 x y ( x) 0
dx dx
dx
2
2
PRODUCT RULE
dy
dy
3x 3 y
9x 9 y 0
dx
dx
2
2
dy
dy
3x 3 y
9x 9 y 0
dx
dx
2
2
Factor out dy/dx
dy
3 y 2 9 x 3x 2 9 y 0
dx
dy
3 y 2 9 x 9 y 3x 2
dx
dy 9 y 3 x
2
dx 3 y 9 x
2
dy 3 y x
2
dx y 3 x
2
Divide out 3
dy 3 y x 2
2
dx y 3x
Evaluate at (2,4) and (4,2)
dy
dx
( 2, 4 )
3y x2
2
y 3x
( 2, 4 )
3(4) 2 2 12 4 8 4
2
4 3(2) 16 6 10 5
Slope at the point (2,4)
dy
dx
( 4, 2 )
3y x2
2
y 3x
( 4, 2 )
3(2) 4 2 6 16 10 5
2
2 3(4) 4 12 8 4
Slope at the point (4,2)
x3 y 3 9xy 0
dy
dx
( 2, 4 )
4
5
4
y 4 x 2
5
4
8
y4 x
5
5
4
8
y x 4
5
5
4
8 20
y x
5
5 5
4
12
y x
5
5
Find Tangents
y y1 m x x1
dy
dx
y2
( 4, 2 )
5
4
5
x 4
4
5
y 2 x 5
4
5
y x 3
4
4
12
y x
5
5
5
y x3
4
Folium of Descartes
• Can you find the slope of
the folium of Descartes
x 3 y 3 9 xy 0
• At what point other than
the origin does the folium
have a horizontal tangent?
– Can you find this?
Derivatives of Inverse Functions
and Logarithms
Chapter 3 Section 8
Examples 1 & 2