Q. 2 - Calculus

Question

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.

Step-by-Step Solution

Answer

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

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

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

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,

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

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,

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

Part (c) Step 1. Write the equation.

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

Substitute $y=0$ in equation (i),

$$\begin{gathered} 3x^2+x=2 \\ x\left(3x+1\right)=2 \\ x=2,\frac{1}{3} \end{gathered}$$

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