All Calculus 3 Resources
Example Questions
Example Question #1 : Applications Of Partial Derivatives
Find the linear approximation to at .
The question is really asking for a tangent plane, so lets first find partial derivatives and then plug in the point.
,
,
,
Remember that we need to build the linear approximation general equation which is as follows.
Example Question #1 : Tangent Planes And Linear Approximations
Find the tangent plane to the function at the point .
To find the equation of the tangent plane, we use the formula
.
Taking partial derivatives, we have
Substituting our values into these, we get
Substituting our point into , and partial derivative values in the formula we get
.
Example Question #3 : Tangent Planes And Linear Approximations
Find the Linear Approximation to at .
None of the Above
We are just asking for the equation of the tangent plane:
Step 1: Find
Step 2: Take the partial derivative of with respect with (x,y):
Step 3: Evaluate the partial derivative of x at
Step 4: Take the partial derivative of with respect to :
Step 5: Evaluate the partial derivative at
.
Step 6: Convert (x,y) back into binomials:
Step 7: Write the equation of the tangent line:
Example Question #3 : Tangent Planes And Linear Approximations
Find the equation of the plane tangent to at the point .
To find the equation of the tangent plane, we find: and evaluate at the point given. , , and . Evaluating at the point gets us . We then plug these values into the formula for the tangent plane: . We then get . The equation of the plane then becomes, through algebra,
Example Question #5 : Tangent Planes And Linear Approximations
Find the equation of the plane tangent to at the point
To find the equation of the tangent plane, we find: and evaluate at the point given. , , and . Evaluating at the point gets us . We then plug these values into the formula for the tangent plane: . We then get . The equation of the plane then becomes, through algebra,
Example Question #6 : Tangent Planes And Linear Approximations
Find the equation of the tangent plane to at the point
To find the equation of the tangent plane, we need 5 things:
Using the equation of the tangent plane
, we get
Through algebraic manipulation to get z by itself, we get
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