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- Derivatives
- Derivatives as rates of change
Suppose that the profit obtained from the sale of
fish-fry dinners is given by
Use the marginal profit function to estimate the profit from the sale of the 101st fish-fry dinner.
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Key concepts
- Using
it is possible to estimate
given
and
- The rate of change of position is velocity, and the rate of change of velocity is acceleration. Speed is the absolute value, or magnitude, of velocity.
- The population growth rate and the present population can be used to predict the size of a future population.
- Marginal cost, marginal revenue, and marginal profit functions can be used to predict, respectively, the cost of producing one more item, the revenue obtained by selling one more item, and the profit obtained by producing and selling one more item.
For the following exercises, the given functions represent the position of a particle traveling along a horizontal line.
- Find the velocity and acceleration functions.
- Determine the time intervals when the object is slowing down or speeding up.
a.
b. speeds up
slows down
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A rocket is fired vertically upward from the ground. The distance
in feet that the rocket travels from the ground after
seconds is given by
- Find the velocity of the rocket 3 seconds after being fired.
- Find the acceleration of the rocket 3 seconds after being fired.
a.
b.
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A ball is thrown downward with a speed of 8 ft/s from the top of a 64-foot-tall building. After
t seconds, its height above the ground is given by
- Determine how long it takes for the ball to hit the ground.
- Determine the velocity of the ball when it hits the ground.
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The position function
represents the position of the back of a car backing out of a driveway and then driving in a straight line, where
is in feet and
is in seconds. In this case,
represents the time at which the back of the car is at the garage door, so
is the starting position of the car, 4 feet inside the garage.
- Determine the velocity of the car when
- Determine the velocity of the car when
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The position of a hummingbird flying along a straight line in
seconds is given by
meters.
- Determine the velocity of the bird at
sec.
- Determine the acceleration of the bird at
sec.
- Determine the acceleration of the bird when the velocity equals 0.
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A potato is launched vertically upward with an initial velocity of 100 ft/s from a potato gun at the top of an 85-foot-tall building. The distance in feet that the potato travels from the ground after
seconds is given by
- Find the velocity of the potato after
and
- Find the speed of the potato at 0.5 s and 5.75 s.
- Determine when the potato reaches its maximum height.
- Find the acceleration of the potato at 0.5 s and 1.5 s.
- Determine how long the potato is in the air.
- Determine the velocity of the potato upon hitting the ground.
a. 84 ft/s, −84 ft/s b. 84 ft/s c.
d.
in both cases e.
f.
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Source:
OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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