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(Unit 2.0) 24.7 Applied Maximum and Minimum Problems
1. Determine the quantity to be maximized or minimized.
2. Draw a sketch.
3. Write an equation for the quantity in terms of another variable of
the problem.
4. Take the derivative of the function.
5. Set the derivative equal to zero, and solve the resulting equation.
6. Check as to whether the value makes the function a max or a min.
(for example, the 2nd derivative test)
7. Make sure you’ve answered the question asked.
Examples
1. An automobile manufacturer, in testing a new engine on one of its
new models, found that the efficiency 𝑒 (in %) of the engine as a
function of the speed 𝑠 (in km/h) of the car was given by
𝑒 = 0.768𝑠 − 0.00004𝑠 3 . What is the maximum efficiency of the engine?
2. Find the number that exceeds its square by the greatest amount.
3. A rectangular corral is to be enclosed with 1600 ft of fencing. Find
the maximum possible area of the corral.
4. The height (in ft) of a flare shot upward from the ground is given by
𝑠 = 112𝑡 − 16.0𝑡 2 , where 𝑡 is the time (in s). What is the greatest
height to which the flare goes?
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5. A small oil refinery estimates that its daily profit 𝑃 (in dollars) from
refining 𝑥 barrels of oil is 𝑃 = 8𝑥 − 0.02𝑥 2 . How many barrels should be
refined for maximum daily profit, and what is the maximum profit?
6. The power output 𝑃 of a battery of voltage 𝐸 and internal resistance
𝑅 is 𝑃 = 𝐸𝐼 − 𝑅𝐼 2 , where 𝐼 is the current. Find the current for which
the power is a maximum.
7. In 2002 the projected U.S. Social Security Fund assets 𝑆 (in dollars
× 1012) was given by 𝑆 = −0.00074𝑡 3 + 0.020𝑡 2 + 1.1, where 𝑡 is the
number of years after 2002. What is the maximum projected fund
value, and in what year will it occur?
8. If an airplane is moving at velocity 𝑣, the drag 𝐷 on the plane is 𝐷 =
𝑎𝑣 2 +
𝑏
𝑣2
, where 𝑎 and 𝑏 are positive constants. Find the value(s) of 𝑣
for which the drag is the least.
**challenging! 9. If a resistance 𝑅 and inductance 𝐿 are in parallel with
a capacitance 𝐶, the impedance 𝑍 = √
𝑅 2 +𝜔2 𝐿2
𝜔2
𝐶 2 𝑅 2 +(𝜔2 𝐿𝐶−1)2
, where 𝜔 is the
angular frequency of the circuit impedance. For what value(s) of 𝐶 is 𝑍
a maximum, if 𝑅 and 𝐿 are constant?
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24.3 Curvilinear Motion
Vectors are often used to describe the velocity and acceleration of an
object in a plane along a specified path. Such motion is called
curvilinear motion. It is common to express the x and y coordinates of
an object’s position separately as functions of time. Equations given in
this form are said to be in parametric form. The third variable, t, is
called the parameter.
𝑣𝑥 =
𝑎𝑥 =
𝑑𝑥
𝑑𝑡
,
𝑑𝑣𝑥
𝑑𝑡
𝑣𝑦 =
=
𝑑2𝑥
𝑑𝑡 2
,
𝑑𝑦
𝑑𝑡
,
𝑎𝑦 =
𝑣 = √𝑣𝑥2 + 𝑣𝑦2 ,
𝑑𝑣𝑦
𝑑𝑡
=
𝑑2𝑦
𝑑𝑡 2
𝑡𝑎𝑛𝜃 =
, 𝑎 = √𝑎𝑥2 + 𝑎𝑦2 ,
𝑣𝑦
𝑣𝑥
𝑡𝑎𝑛𝜃 =
𝑎𝑦
𝑎𝑥
Examples
1. If the horizontal distance 𝑥 that an object has moved is given by 𝑥 =
3𝑡 2 and the vertical distance 𝑦 is given by 𝑦 = 1 − 𝑡 2 , find the resultant
velocity when 𝑡 = 2.
2. Find the velocity and direction of motion when 𝑡 = 2 of an object
moving such that its x and y coordinates of position are given by 𝑥 =
1 + 2𝑡 and 𝑦 = 𝑡 2 − 3𝑡.
3. Find the magnitude and direction of the acceleration when 𝑡 = 2 for
an object that is moving such that its x and y coordinates of position
are given by the parametric equations 𝑥 = 𝑡 3 and 𝑦 = 1 − 𝑡 2 .
4. In a physics experiment, a small sphere is constrained to move along
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a parabolic path described by 𝑦 = 𝑥 2 . If the horizontal velocity 𝑣𝑥 is
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constant at 6.00 cm/s, find the velocity at the point (2.00, 1.33).
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5. The x and y coordinates of a moving particle are given by the
indicated parametric equations, find the magnitude and direction of
the velocity for the specific value of 𝑡.
𝑥=
5𝑡
,
2𝑡 + 1
𝑦 = 0.1(𝑡 2 + 𝑡),
𝑡=2
6. The x and y coordinates of a moving particle are given by the
indicated parametric equations, find the magnitude and direction of
the velocity for the specific value of 𝑡.
𝑥 = 𝑡(2𝑡 + 1)2 ,
𝑦=
6
√4𝑡 + 3
,
𝑡 = 0.5
7. The water from a fire hose follows a path described by
𝑦 = 2.0 + 0.80𝑥 − 0.20𝑥 2 (units are in meters). If 𝑣𝑥 is constant at 5.0
m/s, find the resultant velocity at the point (4.0, 2.0).
8. A ship is moving around an island on a route described by
𝑦 = 3𝑥 2 − 0.2𝑥 3 . If 𝑣𝑥 = 1.2 km/h, find the velocity of the ship where
𝑥 = 3.5 km.
9. The x and y coordinates of a moving particle are given by the
indicated parametric equations, find the magnitude and direction of
the acceleration for the specific value of 𝑡.
𝑥 = √1 + 2𝑡 ,
𝑦 = 𝑡 − 𝑡2 ,
𝑡=4