Midterm I Past Exams UNKNOWN YEAR PDF

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Math 251 : Midterm 2

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1. [10 points] The velocity of a particle moving along a helical path is given by the vector v(t) = 4 sin 2t i + 3 j − 4 cos 2t k ,

t ≥ 0.

(a) At time t = 0, the particle passes through the point P (0, 1, 0). Find the position vector r(t) for the particle.

(b) Reparametrize the curve with respect to arc length measured from P (0, 1, 0) in the direction of increasing t.

(c) Find the tangent vector T and principal normal vector N to the helical curve.

Math 251 : Midterm 2

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2. [8 points] The position vector r(t) of a particle moving in space is given as a function of time t by r(t) = ht2 , 34 t3/2 , ti. (a) Find the velocity and acceleration vectors, v(t) and a(t), and the speed v(t) of the particle as a function of t.

(b) Determine the curvature κ of the curve traced out by the particle at time t = 1 (using any method of your choice).

Math 251 : Midterm 2

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3. [8 points] (a) Let w = y + f (x2 − y 2 ), where f is a differentiable function of one variable. Show that ∂w ∂w +x y = x. ∂x ∂y

(b) The design of a certain cylindrical tube specifies a diameter of r = 10 cm and a height of h = 30 cm. However, after construction, the measurements are found to be slightly incorrect: the radius is 1 mm too low, while the height is 5 mm greater than that given in the design specifications. Does the constructed tube have a larger or a smaller volume than originally designed, and by how much? Use differentials to estimate the error in the volume of the cylindrical tube (recall: the volume of a cylinder of radius r and height h is V = πr 2 h).

Math 251 : Midterm 2

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4. [6 points] For which values of (x, y) is the following function continuous? Explain and justify your reasoning carefully. p   6xy 5 − x2 − 4y 2 (x, y) 6= (0, 0), f (x, y) = 3x2 + 2y 2  0 (x, y) = (0, 0).

5. [8 points] Let z = f (x, y), where f has continuous second-order partial derivatives, and x = r cos θ, y = r sin θ. (a) Find

∂z ∂z . and ∂θ ∂r

(b) Find

∂ 2z . ∂r ∂θ

Math 251 : Midterm 2

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6. [10 points] Let F (x, y, z) = x2 + y 3 + z 4 −

2xy , and let P be the point P(1, 2, 1). z

(a) Find the directional derivative of F (x, y, z) at P(2, −1, 1) in the direction of the vector v = h1, 3, −2i.

(b) Consider the level surface F (x, y, z) = x2 + y 3 + z 4 −

2xy =8, z

which contains the point P(2, −1, 1); on this surface, z = z(x, y) is defined implicitly as a function of x and y near P. ∂z ∂z and Compute at P. ∂x ∂y

(c) Find the tangent plane to the level surface F (x, y, z) = 8 at P(2, −1, 1)....