Calculus III Exam 2 Sample Questions 2010

Exam 2 Sample Questions Be sure to look over your old quizzes and homework as well. For limits, we will provide a graph and contours. No calculators will be allowed for this exam. 1. True or False, and explain: (a) There exists a function f with continuous second partial derivatives such that fx (x, y) = x + y 2 fy = x − y 2 (b) The function f below is continuous at the origin. 2xy , x2 +2y 2 ( f (x, y) = 0 if (x, y) 6= (0, 0) if (x, y) = (0, 0) (c) If ~r(t) is a differentiable vector function, then (f) Different parameterizations of the same curve result in identical tangent vectors at a given point on the curve. (g) If ~u(t) and ~v (t) are differentiable vector functions, then d [~u(t) × ~v (t)] = ~u 0 (t) × ~v 0 (t) dt (h) If fx (a, b) and fy (a, b) both exist, then f is differentiable at (a, b). d |~r(t)| = |~r 0 (t)| dt 2 (e) We can always use the formula: ∇f (a, b)·~u to compute the directional derivative at (a, b) in the direction of ~u. 2 (d) If z = 1 − x − y , then the linearization of z at (1, 1) is L(x, y) = −2x(x − 1) − 2y(y − 1) (i) At a given point on a curve (x(t0 ), y(t0 ), z0 (t)), the osculating plane through that point is the plane through (x(t0 ), y(t0 ), z(t0 )) with nor~ 0 ). mal vector is B(t 2. Show that, if |~r(t)| is a constant, then ~r 0 (t) is orthogonal to ~r(t). (HINT: Differentiate |~r(t)|2 = k) 3. Reparameterize the curve with respect to arc length measuring from t = 0 in the direction of increasing t: r = 2ti + (1 − 3t)j + (5 + 4t)k 4. Is it possible for the directional derivative to exist for every unit vector ~u at some point (a, b), but f is still not differentiable there? √ Consider the function f (x, y) = 3 x2 y. Show that the directional derivative exists at the origin (by letting ~u = hcos(θ), sin(θ)i and using the definition), BUT, f is not differentiable at the origin (because if it were, we could use ∇f · ~u to compute D~u f ). 5. If f (x, y) = sin(2x + 3y), then find the linearization of f at (−3, 2). 1 6. The radius of a right circular cone is increasing at a rate of 3.5 inches per second while its height is decreasing at a rate of 4.3 inches per second. At what rate is the volume changing when the radius is 160 inches and the height is 200 inches? (V = 13 πr2 h) 7. Find the (total) differential of the function: v = y cos(xy) √ 8. Find the maximum rate of change of f (x, y) = x2 y + y at the point (2, 1), and the direction in which it occurs. 9. Find an expression for d [u(t) · (v(t) × w(t)] dt 10. Use the definition of the partial derivative to compute fx (x, y), if f (x, y) = x . x+y 2 11. The curves below intersect at the origin. Find the angle of intersection to the nearest degree: ~r1 (t) = ht, t2 , t9 i ~r2 (t) = hsin(t), sin(5t), ti 12. Find three positive numbers whose sum is 100 and whose product is a maximum. 13. Find the equation of the tangent plane and normal line to the given surface at the specified point: x2 + 2y 2 − 3z 2 = 3 (2, −1, 1) 14. If z = x2 − y 2 , x = w + 4t, y = w2 − 5t + 4, w = r2 − 5u, t = 3r + 5u, find ∂z/∂r. 15. If x2 + y 2 + z 2 = 3xyz and we treat z as an implicit function of z, then find ∂z/∂x and ∂z/∂y. 16. If a(t) = −10k and v(0) = i + j − k, r(0) = 2i + 3j, find the velocity and position vector functions. √ 17. Find the equation of the normal line through the level curve 4 = 5x − 4y at (4, 1) using a gradient. 18. Find all points at which the direction of fastest change in the function f (x, y) = x2 + y 2 − 2x − 4y is ~i + ~j. 19. Find the vectors T and N if r(t) = hcos(t), sin(t), ln(cos(t))i at the point (1, 0, 0). 20. Find and classify the critical points for f (x, y) = 4 + x3 + y 3 − 3xy. 21. Let z = x2 + y 2 . (a) Draw the level curves for z = 2, 4, 6. (b) Calculate the gradient at (2, 1). 2 (c) Plot the gradient vector you computed in the previous problem, along with the earlier level curves. (d) Find the equation of the tangent line to the curve at (2, 1). (e) Find the equation of the normal line to the curve at (2, 1). 22. Consider the surface where z is implicitly defined by: x3 + y 3 + z 3 = 9 − 6xyz (a) Find zx and zy if x = 1 and y = 1. (b) Find the equation of the tangent plane at (1, 1, 1). 23. Find parametric equations of the tangent line at the point (−2, 2, 4) to the curve of intersection of the surface z = 2x2 − y 2 and z = 4. (Hint: In which direction should the tangent line go?) 24. Find and classify the critical points: f (x, y) = x3 − 3x + y 4 − 2y 2 25. Find the curve of intersection between the plane y + z = 3 and x2 + y 2 = 5 (in parametric form). 26. Find parametric equations for the tangent line to the curve of intersection of the paraboloid z = x2 + y 2 and the ellipsoid 4x2 + y 2 + z 2 = 9 at (−1, 1, 2). 27. The following table gives numerical values of z = g(x, y). Use these to estimate gx (2, 5) and gy (2, 5) by taking an average. Also estimate the directional derivative of g at (2, 5) in the direction of h1, 1i Side Remark: I would give you numbers that would be easier to work with on the exam, since you wouldn’t have a calculator. Go ahead and use a calculator for this problem. x = 1.5 x = 2 x = 2.5 y = 5.2 16.0 17.2 18.4 y = 5.0 20.0 21.2 22.3 y = 4.8 24.2 25.3 26.6 √ 28. If P = u2 + v 2 + w2 , where u = xey , v = yex and w = exy , then find Px , Py when x = 0, y = 2. √ 29. Find the rate of change of f (x, y) = xy at P (2, 8) in the direction of Q(5, 4). 30. Find the points on the surface of y 2 = 9 + xz that are closest to the origin. 3 31. The figure below shows level curves for the temperature z = T (x, y) on a square plate. First, estimate the values of both partial derivatives of T at (3, 2), then for all of the second partial derivatives, just estimate whether they are positive or negative. 32. The figure below shows the level curves of z = h(x, y). Find whether each is positive or negative: (i) hx (50, 30) (ii) hy (50, 30) (iii) hxx (50, 30) 33. Using the previous graph, if we make a path from the center of the number “8” always going in the direction of −∇h, draw the result. 4