For the function
$$f(x, y, z) := 7x^3 + 2y^3 + 7z^3 + 12xy + 6yz + 12xz + 9$$
find the gradient at the point (1, 1, 1):
$$\nabla f(1, 1, 1) = \langle$$
Find the directional derivative of f at the point (1, 1, 1) in the direction of
the vector (10, 5, 10):
$$D_u f(1, 1, 1) = $$
Consider the limit
$$\lim_{(x,y) \to (0,0)} \frac{3xy^6}{x^4 + y^8}$$
Select the correct answer:
The limit exists.
The limit does not exist.
dz
Find
where z = 6 sin(xy), x = te9s, and y = 4t9 - 5s10
dt
$$\frac{dz}{dt} = 6(4t^9 - 5s^{10}) cos(te^{9s} (4t^9 - 5s^{10}))(e^{9s}) + 6(te^{9s}) cos(te^{9s} (4t^9 - 5s^{10}))(4t^9)$$
$$\frac{dz}{dt} = 6 cos(te^{9s} (4t^9 - 5s^{10}))(e^{9s}) + 6 cos(te^{9s} (4t^9 - 5s^{10}))(36t^8)$$
$$\frac{dz}{dt} = 6(4t^9 - 5s^{10}) cos(te^{9s} (4t^9 - 5s^{10}))(e^{9s}) + 6(te^{9s}) cos(te^{9s} (4t^9 - 5s^{10}))(36t^8)$$
$$\frac{dz}{dt} = 6(4t^9 - 5s^{10}) cos(te^{9s} (4t^9 - 5s^{10}))(te^{9s}) + 6(te^{9s}) cos(te^{9s} (4t^9 - 5s^{10}))(36t^8)$$
