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Example Questions
Example Question #1 :Tangent Planes And Linear Approximations
Find the linear approximation toat.
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 #2 :Tangent Planes And Linear Approximations
Find the tangent plane to the functionat the point.
为了找到切平面的方程,我们使用the formula
.
Taking partial derivatives, we have
Substituting ourvalues 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 toat.
None of the Above
We are just asking for the equation of the tangent plane:
Step 1: Find
Step 2: Take the partial derivative ofwith respect with (x,y):
Step 3: Evaluate the partial derivative of x at
Step 4: Take the partial derivative ofwith 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 #4 :Tangent Planes And Linear Approximations
Find the equation of the plane tangent toat the point.
To find the equation of the tangent plane, we find:and evaluateat the point given.,, and. Evaluatingat the pointgets 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 toat the point
To find the equation of the tangent plane, we find:and evaluateat the point given.,, and. Evaluatingat the pointgets 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 toat the point
To find the equation of the tangent plane, we need 5 things:
使用切平面的方程
, we get
Through algebraic manipulation to get z by itself, we get
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