Derivative as a Rate of Change
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Derivative as a Rate of Change
Chapter 3 Section 4
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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.
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Example• The area A of a circle is related to its diameter by
the equation• 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
• Thus, when D = 10 meters the area is changing with respect to the diameter at the rate of
2
4DA
2
2444
22 DDD
dD
dAD
dD
dA
mmD
/71.152
)10(
22
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Motion Along a Line
• Displacement of object over time Δs = f(t + Δt) – f(t)
• Average velocity of object over time interval
t
tfttf
t
svaverage
)()(
timetravel
nt displaceme
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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)
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Speed
• Rate of progress regardless of direction
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Graph of velocity f ’(t)
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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
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Example 1: Galileo Free Fall
• Galileo’s Free Fall Equations distance falleng is acceleration due to Earth’s gravity
(appx: 32 ft/sec2 or 9.8 m/sec2)– Same constant acceleration– No jerk
2
2
1gts
012
1 2
dt
dgt
dt
d
dt
dgt
dt
ds
dt
d
dt
dj
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Example 2: Free Fall Example
• How many meters does the ball fall in the first 2 seconds?
• Free Fall equation s = 4.9t2 in meters
s(2) – s(0) = 4.9(2)2 - 4.9(0)2 = 19.6 m
22 9.4)8.9(2
1tts
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Example 2: Free Fall Example
• What is its velocity, speed and acceleration when t = 2?– Velocity = derivative of position at any time t
– So at time t = 2, the velocity is
ttdt
dtstv 8.99.4)(')( 2
sec/6.19)2(8.98.99.4)(')( 2 mttdt
dtstv
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Example 2: Free Fall Example
• What is its velocity, speed and acceleration when t = 2?– Velocity = derivative of position at any time t
– So at time t = 2, the speed is
ttdt
dtstv 8.99.4)(')( 2
sec/6.19sec/6.19 mmvelocityspeed
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Example 2: Free Fall Example
• What is its velocity, speed and acceleration when t = 2?– Velocity = derivative of position at any time t
– The acceleration at any time t
– So at t = 2, acceleration is (no air resistance)
ttdt
dtstv 8.99.4)(')( 2
2sec/8.98.9)('')(')( mtdt
dtstvta
2sec/8.9)( mta
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Derivatives of Trigonometric Functions
Chapter 3 Section 5
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Derivatives
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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?
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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: s = 5cos(t)– Velocity: 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
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Chain Rule
Chapter 6 Section 6
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Implicit Differentiation
Chapter 3 Section 7
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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 functionsx2 + 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
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Circle Example
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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.
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Folium of Descartes• Find the slope of the 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
0933 xyyx
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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
0933 xyyx
0)4)(2(942 33 072648
07272 00
0)2)(4(924 33
072864 07272
00
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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
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0933 xyyx
0933
dx
dxy
dx
dy
dx
dx
dx
d
0)(933 22
x
dx
dy
dx
dyx
dx
dyyx
PRODUCT RULE
09933 22 ydx
dyx
dx
dyyx
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09933 22 ydx
dyx
dx
dyyx
Factor out dy/dx
09393 22 yxxydx
dy
22 3993 xyxydx
dy
xy
xy
dx
dy
93
392
2
xy
xy
dx
dy
3
32
2
Divide out 3
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xy
xy
dx
dy
3
32
2
Evaluate at (2,4) and (4,2)
5
4
10
8
616
412
)2(34
2)4(3
3
32
2
)4,2(2
2
)4,2(
xy
xy
dx
dy
Slope at the point (2,4)
4
5
8
10
124
166
)4(32
4)2(3
3
32
2
)2,4(2
2
)2,4(
xy
xy
dx
dy
Slope at the point (4,2)
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0933 xyyx
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Find Tangents5
4)4,2(
dx
dy
11 xxmyy
25
44 xy
5
8
5
44 xy
45
8
5
4 xy
5
20
5
8
5
4 xy
5
12
5
4 xy
4
5)2,4(
dx
dy
44
52 xy
54
52 xy
34
5 xy
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5
12
5
4 xy
34
5 xy
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Folium of Descartes• Can you find the slope of
the folium of Descartes
• At what point other than the origin does the folium have a horizontal tangent?– Can you find this?
0933 xyyx
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Derivatives of Inverse Functions and Logarithms
Chapter 3 Section 8
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Examples 1 & 2
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