Math 2201 Homework #6 Sections 14.3, 14.4, and 14.5 Due Wednesday, October 19 1. Find all possible first partial derivatives of f (x, y) = exy + x3 + y. 2. Find all the second partial derivatives of f (x, y) = xex + y 2 ey + x3 y 2 . Verify that Clairaut's theorem is valid in this case. 3. Suppose you know the following table of values of f . x\\y 0.8 1.0 1.2 2.9 4.3 3.8 3.4 3.0 4.6 4.0 3.7 3.1 4.8 4.4 3.9 For example f (0.8, 3.1) = 4.8. Estimate fx (1.0, 3.0) and fx (1.0, 3.1), and use your answers to estimate fxy (1.0, 3.0). 4. If 3 f (x, y) = sin (x + y 2 )exy , compute fx (, 0). (There is an easy way and a hard way; use the easy way.) 5. Find the equation of the tangent plane to z = xex2y 2 at the point (2, 1, 2). 6. Find the linearization L(x, y) of the function f (x, y) = arctan (xy 2 ) at the point (1, 1, 4 ). p 7. p If f (x, y) = x2 + y 2 , find the linearization at (3, 4, 5), and use this to estimate (3.2)2 + (3.9)2 . Compare your estimate to the actual answer. 8. A cell phone game measures your average speed based on computing your change in distance over a particular time interval; over a 20 second time interval it measures your distance as 600 feet. If it can measure time to within 0.1 seconds and distance to within 50 feet, estimate the maximum error in its speed measurement using differentials. 1 9. Use the Chain Rule to compute dz/dt if z = sin xey and x= t, y = t2 . 10. Use the Chain Rule to find z/s and z/t if x = st3 , z = arctan (xy), y = s2 t. 11. Suppose you know that k(t) = f (g(t), h(t)) for some functions f (x, y) and g(t) and h(t). (a) Given that g(1) = 2, h(1) = 4, g 0 (1) = 3, h0 (1) = 5, f (2, 4) = 7, fx (2, 4) = 2, and fy (2, 4) = 3, find k 0 (1). (b) Given that g(1) = 2, h(1) = 4, g 0 (1) = 3, f (2, 4) = 7, fx (2, 4) = 2, fy (2, 4) = 3, and k 0 (1) = 8, find h0 (1). 12. Write the Chain Rule formula for z/p if z = f (u, v) and u = g(p, q, r), 2 v = h(p, q, r), w = k(p, q, r)