Midterm Practice II for Math 10C Disclaimer: There may be problems on the midterm that do not appear on this practice midterm. Conversely, there may be problems on this practice midterm that are not on the midterm. It is highly recommended that you complete the homework problems before taking this practice midterm. Click here for answers. 1. Find an equation of the plane that contains the points (1, 1, 0), (0, 1, 1), and (2, 0, 1). 2. Let f (x, y ) = ax + by + c be a linear function such that f (1, 2) = 3 and the trace of its graph in the yz -plane is the line z = y + 1. Find a, b, and c. 3. Find the linear function f (x, y ) whose graph is a plane perpendicular to the vector v = 2i + 3j k and its trace in the xz -plane is the line z = 2x + 3. 4. Evaluate the expression a (b c) for a = 2i k , b = 2i, c = j + k . 5. Find the of area the triangle ABC , where A = (1, 0, 0), B = (0, 2, 1), C = (1, 0, 2). 6. Let ABCD be a parallelogram in the 3-space such that AB = i 2j , AD = j + k , and AB + AD = AC . Find the area of ABCD. 7. Assume f (1, 2) = 10, Let g (x, y ) = xf (x, y ) 3xy . (a) Find g (1, 2), gx (1, 2) and gy (1, 2). (b) Estimate g (0.9, 2) and g (1, 2.1). 8. Compute the partial derivatives fx and fy for f (x, y ) = 2xy sin(x2 + y ). 9. Evaluate f (1, 2) and x f (1, 2) y f (1, 2) = 5, x f (1, 2) = 2. y
for f (x, y ) = 2x2 3xy + y 2 .
Answers 1. x + 2y z = 1. 2. a = 4, b = 1, c = 1. 3. z = 2x + 3y + 3. 4. 2. 5. 5. 6. 6. 7. (a) g (1, 2) = 4, gx (1, 2) = 9, gy (1, 2) = 5; (b) g (0.9, 2) 3.1, g (1, 2.1) 3.5. 8. fx = 2y sin(x2 + y ) + 4x2 y cos(x2 + y ), fy = 2x sin(x2 + y ) + 2xy cos(x2 + y ). 9. f f (1, 2) = 2, (1, 2) = 1. x y