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- Calculus volume 3
- Differentiation of functions of
- Partial derivatives
Key concepts
- A partial derivative is a derivative involving a function of more than one independent variable.
- To calculate a partial derivative with respect to a given variable, treat all the other variables as constants and use the usual differentiation rules.
- Higher-order partial derivatives can be calculated in the same way as higher-order derivatives.
Key equations
-
Partial derivative of
with respect to
-
Partial derivative of
with respect to
For the following exercises, calculate the partial derivative using the limit definitions only.
For the following exercises, calculate the sign of the partial derivative using the graph of the surface.
For the following exercises, calculate the partial derivatives.
Evaluate the partial derivatives at point
Given
find
and
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The area of a parallelogram with adjacent side lengths that are
and in which the angle between these two sides is
is given by the function
Find the rate of change of the area of the parallelogram with respect to the following:
- Side
a
- Side
b
-
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Express the volume of a right circular cylinder as a function of two variables:
- its radius
and its height
- Show that the rate of change of the volume of the cylinder with respect to its radius is the product of its circumference multiplied by its height.
- Show that the rate of change of the volume of the cylinder with respect to its height is equal to the area of the circular base.
a.
b.
c.
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Find the indicated higher-order partial derivatives.
Show that
is a solution of the differential equation
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Given
find all points on
at which
simultaneously.
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The function
gives the pressure at a point in a gas as a function of temperature
and volume
The letters
are constants. Find
and
and explain what these quantities represent.
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The law of cosines can be thought of as a function of three variables. Let
and
be two sides of any triangle where the angle
is the included angle between the two sides. Then,
gives the square of the third side of the triangle. Find
and
when
and
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Suppose the sides of a rectangle are changing with respect to time. The first side is changing at a rate of
in./sec whereas the second side is changing at the rate of
in/sec. How fast is the diagonal of the rectangle changing when the first side measures
in. and the second side measures
in.? (Round answer to three decimal places.)
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A
Cobb-Douglas production function is
where
represent the amount of labor and capital available. Let
and
Find
and
at these values, which represent the marginal productivity of labor and capital, respectively.
at
at
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The apparent temperature index is a measure of how the temperature feels, and it is based on two variables:
which is relative humidity, and
which is the air temperature.
Find
and
when
and
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Questions & Answers
if three forces F1.f2 .f3 act at a point on a Cartesian plane in the daigram .....so if the question says write down the x and y components ..... I really don't understand
hey , can you please explain oxidation reaction & redox ?
hey , can you please explain oxidation reaction and redox ?
Boitumelo
for grade 12 or grade 11?
Sibulele
advantages of electrons in a circuit
we're do you find electromagnetism past papers
Ntombifuthi
it is the force or component of the force that the surface exert on an object incontact with it and which acts perpendicular to the surface
Sihle
what is the half reaction of Potassium and chlorine
how to calculate coefficient of static friction
how to calculate static friction
Lisa
How to calculate a current
Tumelo
how to calculate the magnitude of horizontal component of the applied force
Mogano
How to calculate force
Monambi
a structure of a thermocouple used to measure inner temperature
a fixed gas of a mass is held at standard pressure temperature of 15 degrees Celsius .Calculate the temperature of the gas in Celsius if the pressure is changed to 2×10 to the power 4
How is energy being used in bonding?
a rate of change in velocity of an object whith respect to time
Khuthadzo
how can we find the moment of torque of a circular object
Kidist
Acceleration is a rate of change in velocity.
Justice
how to calculate tension by substitution
use fnet method. how many obects are being calculated ?
Khuthadzo
how to calculate acceleration and tension force
you use Fnet equals ma , newtoms second law formula
Masego
please help me with vectors in two dimensions
how to calculate normal force
Mulaudzi
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Source:
OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
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