To bake 100 of his favorite cookies, Mr. Wallis

A student was asked to find the equation of the tangent plane to the surface z=x3−y4 at the point (x,y)=(1,3). The student's ans

aliya0001 [1]

Answer:

C) The partial derivatives were not evaluated a the point.

D) The answer is not a linear function.

The correct equation for the tangent plane is $z = 241 + 3 x - 108 y$ or $3x-108y-z+241 = 0$

Step-by-step explanation:

The equation of the tangent plane to a surface given by the function $S=f(x, y)$ in a given point $(x_0, y_0, z_0)$ can be obtained using:

$z-z_0=f_x(x_0,y_0,z_0)(x-x_0)+f_y(x_0,y_0, z_0)(y-y_0)$ (1)

where $f_x(x_0, y_0, z_0)$ and $f_y(x_0, y_0, z_0)$ are the partial derivatives of $f(x,y)$ with respect to $x$ and $y$ respectively and evaluated at the point $(x_0, y_0, z_0)$ .

Therefore we need to find two missing inputs in our problem in order to use equation (1). The $z_0$ coordinate and the partial derivatives $f_x(x_0, y_0, z_0)$ and $f_y(x_0, y_0, z_0)$ . For $z_0$ just evaluating in the given function we obtain $z_0= -80$ and the partial derivatives are:

$\frac{\partial f(x,y)}{\partial x} \equiv f_x(x, y)= 3x^2 \\f_x(x_0, y_0) = f_x(1, 3) = 3$

$\frac{\partial f(x,y)}{\partial y} \equiv f_y(x, y)= -4y^3 \\f_y(x_0, y_0) = f_y(1, 3) = -108$

Now, substituting in (1)

$z-z_0=f_x(x_0,y_0,z_0)(x-x_0)+f_y(x_0,y_0, z_0)(y-y_0)\\z + 80 = 3x^2(x-1) - 4y^3(y-3)\\z = -80 + 3x^2(x-1) - 4y^3(y-3)$

Notice that until this point, we obtain the same equation as the student, however, we have not evaluated the partial derivatives and therefore this is not the equation of the plane and this is not a linear function because it contains the terms ( $x^3$ and $y^4$ )

For finding the right equation of the tangent plane, let's substitute the values of the partial derivatives evaluated at the given point:

$z = -80 + 3x^2(x-1) - 4y^3(y-3)\\z = -80 +3(x-1)-108(y-3)\\z = 241 + 3 x - 108 y$

or $3x-108y-z+241 = 0$

7 0

3 years ago

Answer choices 192m to the 2nd power, 25m to the 2nd power 177 ft to the 2nd power 242 inches to the second power 61 ft to the 2

Leona [35]

40m to the 2nd power when doing the math its 40m to the 2nd power and then you get ur answer :)

6 0

3 years ago

Read 2 more answers

Find the value c that will make 9x^2 +30x+c a perfect square trinomial

anzhelika [568]

Answer:

$c = 25$

Step-by-step explanation:

Given :-

A quadratic equation is given to us.
The equation is 9x² + 30x + c .

And we need to find out the value of c for which the given trinomial is a perfect square. On looking at the given expression all the terms are having positive signs before them .So we can rewrite it on the basis of ,

Identity :-

$\implies (a+b)^2=a^2+b^2+2ab$