Q 39. - Calculus

Question

In Exercises $37-42$, use the partial derivatives of $f\left(x,y\right)=x^y$ and the point $\left(e,3\right)$ specified to

$\left(a\right)$ find the equation of the line tangent to the surface defined by the function in the $x$ direction,

$\left(b\right)$ find the equation of the line tangent to the surface defined by the function in the $y$ direction, and

$\left(c\right)$ find the equation of the plane containing the lines you found in parts $\left(a\right)$ and $\left(b\right)$.

Step-by-Step Solution

Answer

Part $\left(a\right)$, The equation of the line tangent to the surface defined by the function in the $x$ direction is

$x=e+t, y=3, z=e^3+3e^{2}t$

Part $\left(b\right)$, The equation of the line tangent to the surface defined by the function in the $y$ direction is

$x=e, y=3+t, z=e^3+e^{3}t$

Part $\left(c\right)$,  The equation of the plane containing the lines you found in parts $\left(a\right)$ and $\left(b\right)$ is

$3e^{2}x+e^{3}y-z=5e^3$

Part (a) Step 1. Explanation of part a , Finding equation of tangent in x direction

The line tangent to the surface at $\left(a,b,f\left(a,b\right)\right)$ in the $x$ direction is given by the parametric equation

$x=a+t, y=b, z=f\left(a,b\right)+f_x\left(a,b\right)t$

Now we have function $f\left(x,y\right)=x^y$ and point $\left(e,3\right)$

So $a=e, b=3, f\left(a,b\right)=e^3, f_x\left(a,b\right)=3e^2$

Therefore, equation of tangent in $x$ direction is

$x=e+t, y=3, z=e^3+3e^{2}t$

Part (b) Step 1. Explanation of part b , Finding equation of tangent in y direction

The line tangent to the surface at $\left(a,b,f\left(a,b\right)\right)$ in the $y$ direction is given by the parametric equation

$x=a, y=b+t, z=f\left(a,b\right)+f_y\left(a,b\right)t$

Now we have function $f\left(x,y\right)=x^y$ and point $\left(e,3\right)$

So $a=e, b=3, f\left(a,b\right)=e^3, f_y\left(a,b\right)=e^3$

Therefore, equation of tangent in $y$ direction is

$x=e, y=3+t, z=e^3+e^{3}t$

Part (c) Step 1. Explanation of part c , Finding equation of plane

The equation of plane containing the given lines and point is

$f_x\left(a,b\right)\left(x-a\right)+f_y\left(a,b\right)\left(y-b\right)=z-f\left(a,b\right)$

$\Rightarrow 3e^2\left(x-e\right)+e^3\left(y-3\right)=z-e^3$

$\Rightarrow 3e^{2}x-3e^3+e^{3}y-3e^3=z-e^3$

$\Rightarrow 3e^{2}x+e^{3}y-z=5e^3$