All Calculus 3 Resources
Example Questions
Example Question #91 : Applications Of Partial Derivatives
Find the equation to the tangent plane to the following function at the point :
The equation of the tangent plane is given by
So, we must find the partial derivatives for the function evaluated at the point given. To find the given partial derivative of the function, we must treat the other variable(s) as constants.
The partial derivatives are
, ,
Evaluated at the given point, the partial derivatives are
, ,
Plugging this into the formula above, we get
which simplifies to
Example Question #71 : Gradient Vector, Tangent Planes, And Normal Lines
Find for the following function:
The gradient vector of a function is given by
written in standard form as
To find the given partial derivative of the function, we must treat the other variable(s) as constants.
The partial derivatives are
Example Question #92 : Applications Of Partial Derivatives
Find the gradient vector of the following function:
The gradient vector is given by
To find the given partial derivative of the function, we must treat the other variable(s) as constants.
The partial derivatives are
, ,
Example Question #74 : Gradient Vector, Tangent Planes, And Normal Lines
Find the equation of the plane tangent to the following function at :
The equation of the tangent plane is given by
So, we must find the partial derivatives for the function evaluated at the point given. To find the given partial derivative of the function, we must treat the other variable(s) as constants.
The partial derivatives are
, ,
Evaluated at the given point, the partial derivatives are
, ,
Plugging these into the formula above, we get
which simplifies to
Example Question #75 : Gradient Vector, Tangent Planes, And Normal Lines
Find the equation of the tangent plane to the following function at the point :
The equation of the tangent plane is given by
So, we must find the partial derivatives for the function evaluated at the point given. To find the given partial derivative of the function, we must treat the other variable(s) as constants.
The partial derivatives are
, ,
Evaluated at the given point, the partial derivatives are
, ,
Putting this into the equation above, we get
which simplified becomes
Example Question #72 : Gradient Vector, Tangent Planes, And Normal Lines
Find the gradient vector for the following function:
The gradient vector of a function is given by
To find the given partial derivative of the function, we must treat the other variable(s) as constants.
The partial derivatives are
, ,
Example Question #71 : Gradient Vector, Tangent Planes, And Normal Lines
Find the equation of the plane that passes through the points , and
Step 1:
Let
Using these three points we will find two vectors and . [You can find PQ and QR too]
Step 2:
We are required to find a perpendicular (normal) vector to both and . So we need to take their cross product
We have found our normal vector.
Step 3: We will use the following formula to find the final answer
Example Question #71 : Gradient Vector, Tangent Planes, And Normal Lines
Find the gradient vector for
Suppose that
then
taking the respective partial derivatives and putting them into order as stated in the formula above yields
Example Question #73 : Gradient Vector, Tangent Planes, And Normal Lines
Find the gradient vector for the following function:
The gradient vector of a function is given by
To find the given partial derivative of the function, we must treat the other variable(s) as constants.
The partial derivatives are
, ,
Example Question #74 : Gradient Vector, Tangent Planes, And Normal Lines
Find the gradient vector of the following function:
The gradient vector of a function is given by
To find the given partial derivative of the function, we must treat the other variable(s) as constants.
The partial derivatives are
, ,