Consider the function F(x, y, z) = 2x^2 + y^2 +3z^2 and the points P(1, 2, -1) and Q(-1, 1,1). ...

Question:

Consider the function F(x, y, z) = {eq}2x^2 + y^2 +3z^2 {/eq} and the points P(1, 2, -1) and Q(-1, 1,1).

(a) Compute the directional derivative of F(x, y, z) at P in the direction from P to Q.

(b) Find the tangent plane to the ellipsoid {eq}2x^2 + y^2 +3z^2 {/eq} = 9 at the point P.

Let z = f(x, y) = {eq}1/3x^3 - 3x + 2xy + Y^2 - 4 {/eq}. Suppose there exit differentiable functions g(t) = 2, h(0) = - g'(0) = 3, and h'(0) = 2. Compute the derivative dz/dt|_t=0.

Application of Gradient to find Directional Derivatives and Equation of Tangent Plane and Normal Line:

The directional derivative of f(x, y, z) at P in the direction of {eq}\vec v {/eq} is {eq}D_{\hat v} f=\nabla f(P) \cdot \hat v, {/eq} where {eq}\nabla f(P) {/eq} denotes the gradient of f at the point P and {eq}\hat v=\frac{\vec v}{|\vec v|} {/eq} is the unit vector.

The normal vector to the surface is found out using gradient. This normal being orthogonal to the plane will also be orthogonal to any vector in the required plane. So we find a vector lying in the required plane and substitute their dot product to be zero which gives the required equation of the tangent plane.

{eq}\displaystyle F(x, \, y, \, z) \, = 2x^2 + y^2 +3z^2 {/eq}

Differentiate partially with respect to x, y and z, we get:

{eq}\displaystyle \frac{\partial F}{\partial x}=4x\\ \displaystyle \frac{\partial F}{\partial y}=2y\\ \displaystyle \frac{\partial F}{\partial z}=6z {/eq}

Evaluating the partial derivatives at the point P(1, 2, -1), we get:

{eq}\displaystyle \frac{\partial F}{\partial x}=4\\ \displaystyle \frac{\partial F}{\partial y}=4\\ \displaystyle \frac{\partial F}{\partial z}=-6 {/eq}

The normal or gradient {eq}\, \nabla \, F(P) \, {/eq} of {eq}\, F \, {/eq} at {eq}\, P {/eq} is given as:

{eq}\begin{align} \displaystyle \, \nabla \, F \,&=\frac{\partial F}{\partial x} \vec i+\frac{\partial F}{\partial y} \vec j+\frac{\partial F}{\partial z} \vec k\\ \displaystyle &=4x \vec i+2y \vec j+6z \vec k\\ \displaystyle \, \nabla \, F(P) \,&=4 \vec i + 4 \vec j - 6 \vec k \end{align} {/eq}

{eq}\displaystyle \vec u=(-1-1) \hat i+(1-2) \hat j+(1-(-1)) \hat k=-2 \hat i- \hat j+2 \hat k\\ \displaystyle |\vec u|=\sqrt{(-2)^2+(-1)^2+2^2}=\sqrt{4+1+4}=3\\ \displaystyle \hat u=\frac{\vec u}{|\vec u|}=\frac{-2 \hat i- \hat j+2 \hat k}{3} {/eq}

The directional derivative of {eq}F {/eq} at P in the direction of {eq}\vec u {/eq} is:

{eq}\begin{align} \displaystyle D_{\vec u}F &=\nabla F(P) \cdot \hat u\\ \displaystyle &= \left ( 4 \vec i + 4 \vec j - 6 \vec k \right ) \cdot \left ( \frac{-2 \hat i- \hat j+2 \hat k}{3} \right )\\ \displaystyle &= \frac{-2(4)-4-6(2)}{3} \\ \displaystyle &= \frac{-8-4-12}{3} \\ \displaystyle &= \frac{-24}{3} \\ &=-8 \end{align} {/eq}

Therefore, the directional derivative {eq}D_{\vec u}F (P)=\color{blue}{-8}. {/eq}

b) Let {eq}(x,y,z) {/eq} be an arbitrary point on the required tangent plane. Also, {eq}P(1, \ 2, \ -1) {/eq} lies on the tangent plane. Therefore, the vector joining these points {eq}(x-1)i+(y-2)j+(z+1)k {/eq} lies on the required plane.

This vector and the normal vector to the surface will be perpendicular to each other. Hence, their dot product will be zero, which gives us the equation of the required tangent plane.

{eq}\displaystyle \left ((x-1) \vec i+(y-2) \vec j+(z+1) \vec k \right ) \cdot \left (4 \vec i + 4 \vec j - 6 \vec k \right )=0\\ \displaystyle \Rightarrow 4(x-1)+4(y-2)-6(z+1)=0\\ \displaystyle \Rightarrow 4x+4y-6z-4-8-6=0\\ \displaystyle \Rightarrow 4x+4y-6z-18=0\\ \displaystyle \Rightarrow 2x+2y-3z=9 {/eq}

Hence, an equation of the tangent plane to the ellipsoid {eq}\, F(x, \, y, \, z) \, = \, 9 \, {/eq} at {eq}\, P {/eq} is {eq}\color{blue}{2x+2y-3z=9}. {/eq}

c) Given:

{eq}x(0)=2\\ x'(0)=-3\\ y(0)=3\\ y'(0)=2 {/eq}.

We have

{eq}\displaystyle z=\frac 1 3 x^3-3x+2xy+y^2-4\\ \displaystyle \frac{\partial z}{\partial x}=x^2-3+2y\\ \displaystyle \frac{\partial z}{\partial y}=2x+2y {/eq}

Finding the value of the derivatives at t=0, we get

We have {eq}\displaystyle \frac{\partial z}{\partial x}|_{t=0}=2^2-3+2(3)=7\\ \displaystyle \frac{\partial z}{\partial y}|_{t=0}=2(2+3)=10\\ {/eq}

Using Chain Rule, we have

{eq}\begin{align} \displaystyle \frac{d z}{d t} &=\frac{\partial z}{\partial x}\frac{dx}{dt}+\frac{\partial z}{\partial y}\frac{dy}{dt}\\ \displaystyle \frac{d z}{d t}|_{t=0} &=\frac{\partial z}{\partial x}|_{t=0} \frac{dx}{dt}|_{t=0}+\frac{\partial z}{\partial y}|_{t=0} \frac{dy}{dt}|_{t=0}\\ \displaystyle &=7(-3)+10 (2)\\ \displaystyle &=-21+20\\ &=-1 \end{align} {/eq}

Therefore, we get {eq}\frac{d z}{d t} =\color{blue}{-1}. {/eq}