Math 242, Spring 2019, Assignment 7

Do not imagine that mathematics is hard and crabbed and repulsive to commmon sense. It is merely the etherealization of common sense.

- Lord Kelvin

Read:

1. Section 14.4.
2. Section 14.5.
3. Section 14.6.

Carefully define the following terms, then give one example and one non-example of each:

1. Tangent plane (to the surface $z=f(x,y)$; see the first paragraph on page 928).
2. Linearization (of a function $f(x,y)$ at the point $(a,b)$; see page 930).
3. Total differential (of a function $f$; see page 932).
4. Directional derivative (of a function $f(x,y)$ with respect to a unit vector $\vec u=\left\langle a,b\right\rangle$; see page 947).
5. Gradient.

Carefully state the following theorems (you need not prove them):

1. Equation of the tangent plane (see page 928).
2. Formula for the linearization (equation 3 on page 930).
3. The Chain Rule (see page 940).
4. Formulas for implicit differentiation (equations 6 and 7 on pages 942 and 943).
5. Formula for the directional derivative (equation 3 on page 948).
6. Formula relating the directional derivative to the gradient (equation 9 on page 950).

Solve the following problems:

1. Section 14.4, problems 1, 3, 11, 13, 17, 19, 21, 25, 29, 31, and 33.
2. Section 14.5, problems 1, 3, 7, 9, 13, 21, 23, 27, 31, 39, and 40.
3. Section 14.6, problems 1, 5, 7, 9, and 21.

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