Chapter 12: Q. 2 (page 963)

Construct examples of the thing(s) described in
the following.
Try to find examples that are different than
any in the reading.
(a) A function z = f(x, y) for which ∇f(0, 0) = 0 but f is
not differentiable at (0, 0).
(b) A function z = f(x, y) for which ∇f(0, 0) = 0 for every
point in R2.
(c) A function z = f(x, y) and a unit vector u such that
Du f(0, 0) = ∇f(0, 0) · u.

Short Answer

Expert verified

Part (a): The three required functions are ${(x^{4} + 2)}^{\frac{3}{2}}, {(\sqrt{3 x^{2} + 3})}^{3}, {(3 (\sqrt{x + 2}))}^{2}$ .

Part (b): The required equation is $x^{2} - 3 y^{2} = 16$ .

Part (c): The required implicit function is $3 x^{2} + 2 y^{3} + x + y = 2$ .

Step by step solution

Part (a) Step 1. Consider the three functions.

Three functions that we could not have differentiated before learning the chain rule are of the type of composition are given below,

${(x^{4} + 2)}^{\frac{3}{2}}, {(\sqrt{3 x^{2} + 3})}^{3}, {(3 (\sqrt{x + 2}))}^{2}$

Part (b) Step 1. Write the equation.

The equations which have more than one value of y for each x value, and which are not differentiable in simple way, those equations are called y is an implicit function of x.

The required equation is given below,

$x^{2} - 3 y^{2} = 16$

Part (c) Step 1. Write the equation.

The graph of an implicit function with three horizonal lines and two vertical lines is $3 x^{2} + 2 y^{3} + x + y = 2 . . . . . . (i)$ .

Substitute $y = 0$ in equation (i),

$3 x^{2} + x = 2 x (3 x + 1) = 2 x = 2, \frac{1}{3}$

Thus is can be written $(2, 0), (\frac{1}{3}, 0)$ are the points on the graph of equation (i).

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Short Answer

Step by step solution

Part (a) Step 1. Consider the three functions.

Part (b) Step 1. Write the equation.

Part (c) Step 1. Write the equation.

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