Answer:
slope = 7
Step-by-step explanation:
You have that the plane y=1 intersects the following surface:
[tex]z=x^5+2xy-y^5[/tex] (1)
To find the slope of the curve, you first replace y=1 in the equation (1):
[tex]z=x^5+2x(1)-(1)^5=x^5+2x-1[/tex] (2)
This is the generated curve when the plane y=1 intersect the surface z.
The slope of a function is given by the derivative of the function. Then, you calculate dz/dx in the equation (2):
[tex]\frac{dz}{dx}=5x^4+2[/tex]
The slope only depends of the value of x. The slope for the point P(1,1,2) is:
[tex]\frac{dz}{dx}_{x=1}=5(1)^4+2=7[/tex]
The value of the slope of the tangent line to point P is 7
The value of the slope of the tangent line is 7.
The given surface is, [tex]z=x^{5}+2xy-y^{5}[/tex]
The plane [tex]y=1[/tex] intersects the surface given surface.
[tex]z=x^{5}+2x-1[/tex]
To find the slope of tangent line differentiate above curve equation with respect to x.
[tex]\frac{dz}{dx}=5x^{4}+2[/tex]
Substitute point [tex](1,1,2)[/tex] in above equation.
[tex]\frac{dz}{dx}=5(1)^{4} +2=7[/tex]
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