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Unformatted text preview: at is perpendicular to both vectors. Thus Now we have a Normal vector to the tangent plane and also the point
Thus the equation of the Tangent Plane:
The Normal Line to the surface at is in the tangent plane. Approximations and Differentials
Review for a function of ONE variable:
Increment of y Increment of x Differential of x The tangent line at x0 is: [from the point-slope form: where
The INCREMENT can be approximated by can be referred to as or for sufficiently small and is called the Differential of y. So Rearrange this to get: X0 X0 + X Now for a function of TWO variables: The tangent plane at is: Solving the equation for z and relabeling z as L(x,y) and z0 as f(x0 , y0): So . Differential of y Thus The INCREMENT can be approximated by can also be referred to as . for sufficiently small can be referred to as . can be referred to as . Thus we have is the (total) differential of z. This can be represented in a more compact form:
Where is called the GRADIENT of f and is the vector-valued function Notice the similarity to the ONE variable case: VS TWO variable case: Example:
Use the total differential to estimate the value of
Let x0 = 3 and y0 = 4 Then Calculate the Relative Error: .04 and - .05 and . DIFFERENTIABILITY
What does it mean for a function to be differentiable? That Does Continuity Imply Differentiability? NO. If exists. has a sharp corner then does not exist.
Limit does not exist The curve needs to be smooth. Meaning a tangent line will approximate the curve well for small
Does a function need to be continuous at the point to be differentiable? YES
Limit does not exist
What happens if the slope of the tangent line is vertical? Does the derivative exist there?
does not exist
What does it mean for a function
to be differentiable? We have looked at slicing the surface with a plane
parallel to the xz-plane and the yz-plane. This gave us trace curves which we found the slope of the tangent line which
we defined as our partial derivatives. The problem is there are infinitely many ve...
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This note was uploaded on 03/05/2014 for the course MATH 101c taught by Professor Loukianoff,v during the Spring '08 term at Ohlone.
- Spring '08