All Calculus 3 Resources
Example Questions
Example Question #31 : Applications Of Partial Derivatives
Find the slope of the function at the point
To consider finding the slope, let's discuss the topic of the gradient.
For a function , the gradient is the sum of the derivatives with respect to each variable, multiplied by a directional vector:
It is essentially the slope of a multi-dimensional function at any given point.
The approach to take with this problem is to simply take the derivatives one at a time. When deriving for one particular variable, treat the other variables as constant.
Looking at at the point
x:
y:
Example Question #11 : Gradient Vector, Tangent Planes, And Normal Lines
Find the slope of the function at the point
To consider finding the slope, let's discuss the topic of the gradient.
For a function , the gradient is the sum of the derivatives with respect to each variable, multiplied by a directional vector:
It is essentially the slope of a multi-dimensional function at any given point.
Knowledge of the following derivative rules will be necessary:
Trigonometric derivative:
Note that u may represent large functions, and not just individual variables!
The approach to take with this problem is to simply take the derivatives one at a time. When deriving for one particular variable, treat the other variables as constant.
Looking at at the point
x:
y:
Example Question #12 : Gradient Vector, Tangent Planes, And Normal Lines
Find the slope of the function at the point
To consider finding the slope, let's discuss the topic of the gradient.
For a function , the gradient is the sum of the derivatives with respect to each variable, multiplied by a directional vector:
It is essentially the slope of a multi-dimensional function at any given point.
Knowledge of the following derivative rules will be necessary:
Trigonometric derivative:
Note that u may represent large functions, and not just individual variables!
The approach to take with this problem is to simply take the derivatives one at a time. When deriving for one particular variable, treat the other variables as constant.
Looking at at the point
x:
y:
Example Question #13 : Gradient Vector, Tangent Planes, And Normal Lines
Find the equation of the tangent plane to at .
The definition of a tangent plane of a surface is given by
.
Therefore, we need to first find , given below as
Noting that
,
we can write our complete expression for the tangent line as
.
Example Question #15 : Gradient Vector, Tangent Planes, And Normal Lines
Calculate if and
By definition . Therefore,
, so we will need to find the partial derivatives of , shown below as
Therefore,
Example Question #41 : Applications Of Partial Derivatives
Calculate given
By definition,
, where are the respective components of .
Therefore, we need to calculate the above terms, shown as
Therefore,
.
Example Question #17 : Gradient Vector, Tangent Planes, And Normal Lines
Find , where
The gradient vector of f, , is equal to .
So, we must find the partial derivatives of the function with respect to x, y, and z, keeping the other variables constant for each partial derivative:
The derivatives were found using the following rules:
, , ,
Plugging this in to a vector, we get
Example Question #18 : Gradient Vector, Tangent Planes, And Normal Lines
Find , where f is the following function:
The gradient of a function is given by
To find the given partial derivative of the function, we must treat the other variable(s) as constants.
Now, we find the partial derivatives:
The derivatives were found using the following rules:
, , ,
Example Question #14 : Gradient Vector, Tangent Planes, And Normal Lines
Find of the following function:
To find the given partial derivative of the function, we must treat the other variable(s) as constants. For higher order partial derivatives, we work from left to right for the given variables.
To start, we must find the partial derivative with respect to y:
The following rules were used to find the derivative:
,
Next, we find the derivative of the above function above with respect to x:
The rule used is stated above.
Finally, find the partial derivative of the above function with respect to z:
The rule used is already stated above.
Example Question #20 : Gradient Vector, Tangent Planes, And Normal Lines
Find of the given function:
The gradient of a function is given by
So, we must find the partial derivatives. To find the given partial derivative of the function, we must treat the other variable(s) as constants.
The derivatives were found using the following rules:
, , ,