e3 - Exam 1ESP Show all work 1)(20 points) Find the angle...

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Unformatted text preview: Exam 1ESP Show all work 1)(20 points) Find the angle between x - y + z = 3 and the xy plane. i-j+ i+j+ 2)(20 points) Do the vectors a = k and b = k and c = 3 k i-j+ all lie in the same plane? 3)(20 points) Find the plane containing both = (1, 3, 0) + t(-1, 2, 1) r and x-1 =3-y =z 2 4)(20 points) A line through P (1, 1, 2) is parallel to the planes x + y - z = 1 and 2x + y + z = 0. Find scalar parametric equations for the line. 5)(20 points) A plane through P (1, 1, 2) is parallel to the lines =(1 + t) + (2 - t) + tk r i j and x - 1 = 2y = 1 - z What is the equation for the plane? b) On the surface in a), sketch the trace y = 1. What kind of curve is it? 1)(20 points) Let f (x, y) = y 2 + x2 . c) Find the directional derivative of f at P (0, 1), in the direction i. a) Sketch the surface. d) Referring back to the pictures in a) and b), discuss why the dib) On the surface in a), sketch the trace y = 1. What kind of curve rectional derivative has the value it does. is it? c) Find the directional y, z) = xy + xz at yz; let in the direction i. 2)(10 points) Let f (x, derivative of f + P (0, 1), d)= t, y = s,back to + t. pictures inchain rule for f /s andthe dix Referring z = s the State the a) and b), discuss why use it rectional derivative has the value it does. to compute f /s. 2)(10 points) Show (x, y, z)(x, xy + xz + 2 + let) satisfies Laplace's 5)(15 points) Let f that f = y) = ln(x yz; y 2 x = t, y = s, z = s + t. State the chain rule for f /s and use it equation: Exam 2b 2 to compute f /s. f 2f + =0 x2 y 2 2 2 1)(15 points) Show that f 5)(15 points) Let f (x, y) = (x,y + x ln(x2 + y 2 ) satisfies Laplace's y) = . equation: the surface. a) Sketch 2f 2 2b Exam f f b) Find the directional derivative of = at P (1, 1), in the direction + 0 2 xExam2 2 y - i j. c) Referring Let to the picture in 2 . 1)(15 points) back f (x, y) = y 2 + xa), discuss why the directional derivative the surface. it does. a) Sketch has the value 1)(10 points) Let f (x, y) = xy 2 - yx2 . Find all the second b) Find the directional derivative of at P derivatives ofLet f (x, y) = x3 + y 3 +fxy; let(1, 1), in the direction f. 2)(10 Exam 2 - points) i j. x = cos , y = sin . State the chain rule for f / and use it to c) Referring/. to the y) = in - 2 . Find the directional compute f back 2)(10 points) Let f (x,picture x2 a), ydiscuss why the directional derivative has the value it does. 2 2 derivative of f Let the point P (1,- yx .the direction from P 1)(10 points) at f (x, y) = xy 1) in Find all the second to the origin. f . derivatives ofLet f (x, y) = x3 + y 3 + xy; let 2)(10 points) y = sin . of f the chain rule and use the compute f /. derivative Stateat the point P (1, 1) in it to direction from P to the origin. 2)(20 points) Let f (x, y) = (y 2 - x2 )/xy; let x = cos ; y = sin . State the chain rule and use it to compute f /. x = cos , y = sin . State the chain rule for f / and use it to 2)(20 f /. Let (x, y) = (y 2 y . Find let directional compute 2)(10 points) Let f f (x, y)= x2 - -2 x2 )/xy; the x = cos ; a) Sketch the surface. 1)(35 points) Let f (x, y)The 2End2 . =y -x b) Sketch the trace y = 1. What kind of curve is it? 1)(35 points) Let f (x, y) = y 2 - x2 . c) Find the gradient at P (1, 1). a) Sketch the surface. d) Referring back to the pictures in a) and b), discuss why b) Sketch the trace y = 1. What kind of curve is it? the gradient points the direction it does. c) Find the gradient at P (1, 1). e) Find a direction to move, starting at P (1, 1), so the surface d) Referring back to the pictures in in that direction. why does not change height if you move a) and b), discuss the gradient points the direction it does. e) Find a direction to move,= f (x, y) = y/(x +1), so the surface mixed partials are equal, if z starting at P (1, y). does not change height if you move in that direction. 1)(10 points) Find all four second derivatives, and check that the 2)(10 points) Let z = f (x, y) = (x - y)2 . Let x = r cos t, y = t cos r. State the chain rule for z/r and use it to compute z/r. 4)(15 points) Let z = f (x, y) = x2 + y 2 . Let P = P (1, 1) a) Sketch the surface b) Sketch the trace z = f (x, 1), both in the xz plane and on the surface. c) Compute f /x(P ) Referring back to b), explain why it has the value it does. 5)(25 points) Let z = f (x, y) = 1 - (x2 + y 2 ). Let P = P (0, .5) a) Sketch the surface b) Sketch the trace z = f (x, .5), in the xz plane and on the surface. c) Sketch the trace z = f (0, y), in the yz plane and on the surface. d) Compute f (P ). Referring back to b) and c), explain why it points the direction it does. Quiz 3a 25 minutes Show All Work 1)(60 points) Let r(t) = 1 i - t1 j. 2 t a) Sketch the curve for t > 0. b) Locate r(1) on the curve. c) Compute r (t). d) Plot r (1) with its tail at r(1). 2)(40 points) Let r(t) = t3 i - t2 j, 0 t 1. Find the length of the curve. Quiz 3b 25 minutes Show All Work 1)(60 points) Let r(t) = et i - e2t j. a) Sketch the curve for all t. b) Locate r(0) on the curve. c) Compute r (t). d) Plot r (0) with its tail at r(0). 2)(40 points) Let r(t) = (t - 1)2 i - (t - 1)3 j, 1 t 2. Find the length of the curve. Quiz 3c 25 minutes Show All Work 1)(60 points) Let r(t) = ti - tj. a) Sketch the curve for t > 0. b) Locate r(1) on the curve. c) Compute r (t). d) Plot r (1) with its tail at r(1). 2)(40 points) Let r(t) = ti - t3 j, 0 t 1. Find the length of the curve. Quiz 3 ESP 25 minutes Show All Work 1)(60 points) Let r(t) = ti - tj. a) Sketch the curve for t > 0. b) Locate r(1) on the curve. c) Compute r (t). d) Plot r (1) with its tail at r(1). 2)(40 points) Let r(t) = ti - t3 j, 0 t 1. Find the length of the curve. Quiz 3a 15 minutes Show All Work 1)(40 points) Sketch the curve r(t) = (cos2 t)i - (sin2 t)j. 2)(60 points) Find vector parametric and symmetric equations for the line through the points P (1, 2, 1), Q(2, 1, 2). Quiz 3b 15 minutes Show All Work 2 2 2)(60 points) Find vector parametric equations for the line in the plane y = 2x + 2. Quiz 3c 15 minutes Show All Work 1)(40 points) Sketch the curve r(t) = cos t, - cos t . 1)(40 points) Sketch the curve r(t) = e-t i - e-t j. 2)(60 points) A line is pependicular to the xz plane. It passes through P (1, 2, 3). Find vector and scalar parametric equations for it. ESP Quiz 3 15 minutes Show All Work 1)(40 points) Sketch the curve r(t) = e-t i - e-t j. 2)(60 points) A line is pependicular to the xz plane. It passes through P (1, 2, 3). Find vector and scalar parametric equations for it. ...
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