Answer:
slope of tangent line: 10
Step-by-step explanation:
To find the slope of the tangent line to the curve of intersection at the point [tex] P(1, 1, 4) [/tex], we need to first find the curve of intersection by substituting [tex] y = 1 [/tex] into the equation of the surface [tex] z = x^6 + 4xy - y^5 [/tex].
Substitute [tex] y = 1 [/tex] into [tex] z = x^6 + 4xy - y^5 [/tex]:
[tex] z = x^6 + 4x(1) - 1^5 [/tex]
[tex] z = x^6 + 4x - 1 [/tex]
So, the curve of intersection is represented by the equation:
[tex] z = x^6 + 4x - 1 [/tex].
Now, to find the slope of the tangent line at the point [tex] P(1, 1, 4) [/tex], we need to find the derivative of [tex] z [/tex] with respect to [tex] x [/tex], and evaluate it at [tex] x = 1 [/tex].
Taking the derivative of [tex] z = x^6 + 4x - 1 [/tex] with respect to [tex] x [/tex]:
[tex] \dfrac{dz}{dx} = 6x^5 + 4 [/tex]
Evaluate the derivative at [tex] x = 1 [/tex]:
[tex] \dfrac{dz}{dx} \bigg|_{x=1} = 6(1)^5 + 4 \\\\ = 6 + 4 \\\\ = 10 [/tex]
So, at [tex] P(1, 1, 4) [/tex], the slope of the tangent line to the curve of the intersection is:
[tex] \Large \boxed{\boxed{10}} [/tex].