Calculus Volume 3 · free preview
Tangent Planes and Linear Approximations
The tangent plane to a surface at a point is built from the partial derivatives, z minus f(a, b) equals f_x times x minus a plus f_y times y minus b, the linear approximation whose differentials estimate small changes.
1. The tangent plane
The tangent plane
2. Linear approximation
Linear approximation
3. Differentials
Differentials
4. When the tangent plane fails
When the tangent plane fails
5. Tangent plane equation
The tangent plane to z = f(x, y) at (a, b) has which equation?
6. Normal vector
A normal vector to the tangent plane of z = f(x, y) is which vector?
7. Differential definition
The differential dz of z = f(x, y) equals which expression?
8. Error order
The tangent plane's approximation error is what order in the distance from the base point?
9. Approximation accuracy
The tangent plane approximates f(x, y) near (a, b) with error quadratic in the distance. Halving the distance does what to the error bound?
10. Paraboloid tangent value
The paraboloid z = x^2 + y^2 at (1, 1) has f = 2. As a number, what is f(1, 1)? Answer with a number.
11. Volume differential terms
The volume V = xyz differential has three terms. As a number, how many? Answer with a number.
12. Tangent plane for linear function
If f(x, y) is itself linear, f = 2x + 3y + 1, the tangent plane at any point is what?
13. Plane trace lines
The tangent plane contains the tangent lines of the two coordinate traces. As a number, how many traces' tangents does it contain? Answer with a number.
14. Differentiability hypothesis
The tangent plane's existence requires differentiability, stronger than the mere existence of partials. As one word, what is it? Answer in lowercase.
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