MATH 3C - SECTION 3.3 HOMEWORK Laney College, Spring 2023 Fred Bourgoin 1. Find an equation for the plane tangent to z = e" + r + 22 +6 at the point (1, 0,9). 2. Find an equation for the plane tangent to x y' + z - 40 =0 at x = 2, y = 3. 3. Find the local linearization of the function f(r, y) = aly at the point (3, 1). 4. Find the differential of f(x, y) = vx2 + y at the point (1, 2). Use it to estimate f(1.04, 1.98). 5. A right circular cylinder has a radius of 50 cm and a height of 100 cm. Use differentials to estimate the change in volume of the cylinder if its height and radius are both increased by 1 cm. MATH 3C - SECTION 3.4 HOMEWORK Laney College, Spring 2023 Fred Bourgoin 1. Find the gradient of z = rev. 2. Find the gradient of f(x, y) = xy + 7xy at the point (1, 2). 3. Find the gradient of f(x, y) = vtana + y at the point (0, 1). 4. Find the directional derivative fa(1, 2) for f(x, y) = ry + 3 with u = $7+ $3. 5. Find the rate of change of f(x, y) = x2 + y? at the point (1, 2) in the direction of the vector 1 = 0.6 + 0.8j. 6. Let f(x, y) = 3xy + 1. (a) What is the rate of change of f at the point (2, 3) in the direction of 7 = 37-3? (b) What is the direction of maximum rate of change of f at (2, 3)? (c) What is the maximum rate of change of f at (2, 3)? MATH 3C - SECTION 3.5 HOMEWORK Laney College, Spring 2023 Fred Bourgoin Find the gradient of the given function. 1. f(x, y, 2) = 12 5. f(x, y, z) = yz2/(1 + 22) 2. f(x, y, 2) = 12 +13-24 6. f(x, y, z) = 1/(12 + + 22) 3. f(x, y, z) = ertytz 7. f(x, y, z) = Vx'ty' + 22 4. f(x, y, z) = cos(r + y) + sin(y + z) 8. f(x, y, z) = re' sin z