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Find the unit tangent vector of the given curve.r(t) = (6t cos t – 6 sin t)j + (6t sin t + 6 cos t)k

Use implicit differentiation to find the specified derivative at the given point.Find at the point (1, -1) for 5xy2 – 4x2y + 2x = 0.

Find the domain and range and describe the level curves for the function f(x,y).f(x, y) =

Find the curvature of the space curve.r(t) = (6t cos t – 6 sin t)j + (6t sin t + 6 cos t)k

Find the unit binormal vector B for the curve r(t)  r(t) = (6t cos t – 6 sin t)j + (6t sin t + 6 cos t)k

An ice skater is spinning on frictionless ice with her arms extended outward. She then pulls her arms in toward her body, reducing her moment of inertia. Her angular momentum is conserved, so as she reduces her moment of inertia, her angular velocity increases and she spins faster. Compared to her initial rotational kinetic energy, her final rotational kinetic energy is ______

A woman is enjoying a tropical drink while lying on a beach.  The acceleration due to gravity at her location is 9.78 m/s2. What gauge pressure must exist in the woman’s mouth if she is drinking through a straw extending 0.075 m above the surface of the drink? Assume the drink has a density of 1020 kg/m3.

Solve the initial value problem.Differential Equation: = i + (3t3 – 6t)j + kInitial Condition: r(0) = 2j + (ln 2)k

Find the limit if it exists

Write the equation for the plane.The plane through the points P(5, -8, 28) , Q(-3, -6, 60) and R(-1, -8, 64).