A right circular cylinder has a radius of 50 cm and a

Question

1. Find the equation of the tangent plane at the given point.

z = ln(x³ + 1) + y² at the point (0, 8, 64).

2. Find the equation of the tangent plane at the given point.

 z = 3e^y + x + x⁴+ 5 at the given point (2, 0, 26)

3. Find the local linearization of the function f(x, y) = x³y at the point (5, 1).

4. For the differentiable function h(x, y), we are told that h(700, 100) = 400 and h_x(700, 100) = 11 and h_y(700, 100) = 08. Estimate h(706, 98).

5. (a) Find the equation of the plane tangent to the graph of f(x, y) = x¹¹e^xy at (1, 0).  

(b) Find the linear approximation of f(x, y) for (x, y) near (1, 0).

(c) Find the differential of f at the point (1, 0).

6. (a) Find the differential of g(u, v) = u² + uv

(b) Use your answer to part (a) to estimate the change in g as you move from (4, 5) to (4.1, 5.2).

7. A right circular cylinder has a radius of 50 cm and a height of 110 cm. Use differentials to estimates the change in volume of the cylinder if its height and radius are both increased by 2 cm.