Rates of change Example 1 Find the rate of change of the Area of a circle with respect to its radius Evaluate the rate of change of A at r 5 and r 10 If r is measured in inches and ID: 569350
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Slide1
3.4 Rates of changeSlide2
Rates of change
Example 1:
Find the rate of change of the Area of a circle with respect to its radius.
Evaluate
the rate of change of A at
r = 5
and
r = 10.
If
r
is measured in inches and
A
is measured in square inches, what units would be appropriate for dA/dr?
in2/in
These are units, so we do not cancel them!!!Slide3
Motion
Displacement
of an object is how far an object has moved over time.
Average velocity
is the slope of a displacement vs. time graph.Slide4
Motion
Instantaneous Velocity
is the velocity at a certain point.Instantaneous Velocity is the first derivative of the position function.
Speed is the absolute value of velocity.Slide5
Motion
Acceleration
is the rate at which a particle’s velocity changes.Measures how quickly the body picks up or loses speed.
2nd derivative of the position function!!!Slide6
Example
A particle moves along a line so that its position at any time
t ≥ 0 is given by the function
s(t) = 2t2 – 5t + 3 where s is measured in meters and t
is measured in seconds.
a.) Find the displacement of the particle during the first 2 seconds.
b.) Find the average velocity of the particle during the first 6 seconds
.Slide7
Example
A particle moves along a line so that its position at any time
t ≥ 0 is given by the function
s(t) = 2t2 – 5t + 3 where s is measured in meters and t
is measured in seconds.
c.) Find the instantaneous velocity of the particle at 6 seconds.
d.) Find the acceleration of the particle when
t = 6
.Slide8
Example
A particle moves along a line so that its position at any time
t ≥ 0 is given by the function
s(t) = 2t2 – 5t + 3 where s is measured in meters and t
is measured in seconds.
e.) When does
the particle change directions?Slide9
time
distance
acc pos
vel pos &
increasing
acc zero
vel pos &
constant
acc neg
vel pos &
decreasing
velocity
zeroacc negvel neg &decreasingacc zerovel neg &constantacc posvel neg &increasingacc zero,velocity zeroIt is important to understand the relationship between a position graph, velocity and acceleration:Slide10
Free-Fall
Gravitational
Constants:
Free-fall equation:
s is the position at any time t during the fall
g is the acceleration due to Earth’s gravity (gravitational constant)Slide11
Vertical motion
Example:
A dynamite blast propels a heavy rock straight up with a launch velocity of 160 ft/sec. It reaches a height of
s = 160t – 16t2 ft after t seconds.
a.) How high does the rock go?
Find when position = 0 and divide by 2 (symmetric path)
Since it takes 10 seconds
for the rock to hit the ground, it takes it 5 seconds to reach it max height. Slide12
Vertical motion
Example:
A dynamite blast propels a heavy rock straight up with a launch velocity of 160 ft/sec. It reaches a height of
s = 160t – 16t2 ft after t seconds.
a.) How high does the rock go?
Find when velocity = 0 (this is when
the rock changes direction)Slide13
Vertical motion
Example:
A dynamite blast propels a heavy rock straight up with a launch velocity of 160 ft/sec. It reaches a height of
s = 160t – 16t
2 ft after t seconds.
b.) What is the velocity
and speed of the rock when it is 256 ft above the ground on the way up?
At what time is the rock 256 ft above the ground on the way up?Slide14
Vertical motion
Example:
A dynamite blast propels a heavy rock straight up with a launch velocity of 160 ft/sec. It reaches a height of
s = 160t – 16t
2 ft after t seconds.
b.) What is the velocity
and speed of the rock when it is 256 ft above the ground on the way down?
At what time is the rock 256 ft above the ground on the way down?Slide15
Vertical motion
Example:
A dynamite blast propels a heavy rock straight up with a launch velocity of 160 ft/sec. It reaches a height of
s = 160t – 16t2 ft after t seconds.
c.) What is the acceleration
of the rock at any time t
at any time t during its flight?Slide16
from Economics:
Marginal cost
is the first derivative of the cost function, and represents an approximation of the cost of producing one more unit.
Marginal
revenue
is the first derivative of the revenue function
, and represents an approximation of the revenue of selling one more unit.