Analyttic Geometry and Matrix Midterm Exam November 6, 2015
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1. (a) (6 points) Find the acute angle between the vectors u = (2, −1) and v = (3, 1).
(b) (6 points) Find the cross product a × b of a = (1, 1, −1) and b = (1, 2, 3).
2. (a) (6 points) Find parametric equations for the line through the points (0,12, 1) and (2, 1, −3).
(b) (6 points) Find parametric equations for the line of the intersection of the planes x + 2y + 3z = 1 and x − y + z = 1.
Analytic Geometry and Matrix Midterm Exam (Continued) November 6, 2015 3. (a) (6 points) Find an equation of the plane through the point (1, 2, 3) and perpendicular to the vector
(−2, 1, 5).
(b) (6 points) Find the distance from the point (1, −2, 4) to the plane 3x + 2y + 6z = 5.
4. (a) (6 points) Let R = {(r, θ ) | r ≥ 0, π /4 ≤ θ ≤ 3π /4} be a region in the xy-plane. Sketch R.
(b) (6 points) Find a Cartesian equation for the curve represented by the polar equation r = −2 cos θ .
Analytic Geometry and Matrix Midterm Exam (Continued) November 6, 2015 5. Eliminate the parameter t to find a Cartesian equation of the curve.
(a) (6 points) x = e2t− 1, y= et.
(b) (6 points) x = 3 + 2 cost, y= 1 + 2 sint, π /2 ≤ t ≤ 3π /2.
6. (a) (6 points) Let f (x) = 3x2− 2 sin x. Find the derivative d f dx.
(b) (8 points) Let y = c cost + t2sint, where c is a constant. Find the derivative dy dt.
Analytic Geometry and Matrix Midterm Exam (Continued) November 6, 2015 7. Let f : R3→ R be a function defined by f (x, y, z) = x2+ 3y2+ z2+ 2x + 2z − 23.
(a) (6 points) Find the gradient vector grad f = ∂ f
∂ x,∂ f
∂ y,∂ f
∂ z of f at the point (x, y, z).
(b) (8 points) Find an equation of the tangent plane to the surface S = {(x, y, z) ∈ R3| f (x, y, z) = 0}
at the point P = (2, 2, 1).
8. (a) (6 points) Find a Cartesian equation of the tangent to the curve x = t4+t, y= t2+t, at t = −1.
(b) (6 points) Find the Euclidean coordinates of the points on the curve x = t3− 3t, y= 2t3− 1, where the tangent is horizontal or vertical.
