Mathematics Engineering Calculus III Fall 13 Test #1

Transcription

1 Mathematics Engineering Calculus III Fall 13 Test #1 Instructor: Dr. Alexandra Shlapentokh (1) Which of the following statements is always true? (a) If x = f(t), y = g(t) and f (1) = 0, then dy/dx(1) = 0. (b) Not every function y = F (x) with x [a, b] can be rewritten in parametric form. (c) If x = f(t), y = g(t) are differentiable with continuous derivatives for t [1, 3], f (2) = 1, g (2) = 0 then dy/dx(1) = 0. (d) If x = f(t), y = g(t) are differentiable with continuous derivatives for t [1, 3], f (2) = 0, g (2) = 0 then dy/dx(1) 0. (2) Describe the curve defined by the following parametric equations x = 7t 4 + 1, y = 12t 2 + 2: (a) This is a parabola. (b) This is a circle. (c) This is a line. (d) The nature of the curve cannot be determined. (3) Describe the curve defined by the following parametric equations x = 2t 3 + 1, y = t 3 + 2: (a) This is a line. (b) This is a parabola. (c) This is a circle. (d) The nature of the curve cannot be determined. (4) Describe the curve defined by the following parametric equations x = a 2 cos 2t, y = b + 2 sin 2t: (a) This is a parabola. (b) This is a line. (c) This is a circle centered at (a, b) of radius 1. (d) The nature of the curve cannot be determined. (5) Consider a parametric curve given by x = sin(t/2) + 3, y = 3 cos(t/2) 1 for t [ 4π, 4π). How many times is the curve traced? (a) 1 (b) 3 (c) 7 (d) 5 1

2 (6) Suppose x = f(t), y = g(t), y = F (x), where all the functions involved are differentiable for t I for some interval I and the corresponding values of x. How can one compute dy/dx in this case? (a) dy/dx = dx/dt, provided dy/dt 0 dy/dt (b) dy/dx = dy dx dt dt (c) dy/dx = dy/dt, provided dx/dt 0 dx/dt (d) dy/dx = dy/dt + dx/dt (7) Let x = sin t, y = 2 cos t 3. Find dx/dy for all values of t where dy/dt is not zero. (a) cos t, t π + πk, where k is an integer (b) dx/dy cannot be determined from these data (c) tan t, t π/2 + πk, where k is an integer (d) cot t, t πk, where k is an integer (8) Let x = 1 + sin t, y = 1 cos t and assume that y = F (x) for t (0, π/2), x (1, 2). Determine which of the statements below is true. (a) F (x) is a decreasing function in the specified interval. (b) F (x) is an increasing function in the specified interval. (c) F (x) is an increasing function for t (0, π/4) and a decreasing function for t (π/4, π/2). (d) There is not enough information to determine whether F (x) is increasing or decreasing in the specified interval. (9) Let x = 1 + sin t, y = 1 cos t and assume that y = F (x) for t (0, π/2), x (1, 2). Determine which of the statements below is true. (a) F (x) concaves down in the specified interval. (b) F (x) concaves up in the specified interval. (c) F (x) concaves up for t (0, π/4) and down for t (π/4, π/2). (d) There is not enough information to determine whether F (x) is concaving up or down in the specified interval. (10) Let x = f(t), y = g(t) be equations describing a parametric curve for t [a, b]. Suppose f (t), g (t) are both continuous and non-zero for t [a, b] and f (t) > 0. In this case, for x [f(a), f(b)] it is the case that (a) d 2 y/dx 2 = ( f (t) g (t) ) 2. (b) d 2 y/dx 2 = h (t) f (t), where h(t) = g (t) f (t). (c) d 2 y/dx 2 = g (t) f (t). (d) d 2 y/dx 2 cannot be determined from f and g. 2

3 (11) Let x = cos t, y = 4 cos 2 t + 6 cos t + 1. Find d 2 y/dx 2 for all values of t where dx/dt is not zero. (a) 0 (b) d 2 y/dx 2 cannot be determined from these data (c) 8 (d) 8 (12) Let x = cos t, y = e t. Find all the points where the curve has a vertical tangent. (a) t = π/2 + πk, where k is any integer (b) t = πk, where k is any integer (c) t = 0 (d) There are no points with a horizontal tangent to the curve. (13) Let x = e t, y = sin t. Find the equation of the tangent line at the point (x(π/3), y(π/3)). (a) y 1 2 = 1 (x ln(π/3)) e π/3 3 3 (b) y 2 = ln(π/3) (x eπ/3 ) (c) y 3 2 = 1 2e (x π/3 eπ/3 ) (d) There is no tangent line at the point (x(π/3), y(π/3)). (14) Suppose x = f(t), y = g(t), y = F (x), where all the functions involved are differentiable for t [α, ], F (x) 0 and the curve is traced once. In this case the area under the curve and above the x-axis between x = x(α) and x = x() can be computed using the following formula: (a) (b) (c) (d) α g (t)f(t)dt g (t)f (t)dt g(t)f (t)dt g(t)f(t)dt (15) Find the area under the curve C and the x-axis between x(0) and x(π/2) if C is given by the following parametric equations: x = 3 sin(t) + 2, y = 2 cos(t). (a) 2π 1 (b) 3π/2 (c) π/4 + 2 (d) π/3 3

4 (16) Suppose a parametric curve C is given by equations x = f(t), y = g(t), α t, where f, g have continuous derivatives and C is traversed just once as t increases from α to. In this case which of the formulas below compute the length of C? (dx/dt) (a) 2 α (dy/dt) dt 2 (b) (dx/dt)2 (dy/dt) 2 dt (c) (d) α (dx/dt)2 (dy/dt) 2 dt (dx/dt)2 + (dy/dt) 2 dt (17) Find the length of the curve given by x = sin t, y = cos t + 1, 0 t π/2. (a) 2 (b) 1 (c) π (d) π/2 (18) Suppose a parametric curve C is given by equations x = f(t), y = g(t), α t, where f, g have continuous derivatives, C is traversed just once as t increases from α to and g(t) 0. In this case which of the formulas below compute the surface area of the figure obtained by rotating C around x-axis? (a) (b) (c) (d) α 2πy (dx/dt) 2 (dy/dt) 2 dt y (dx/dt) 2 + (dy/dt) 2 dt (dx/dt)2 + (dy/dt) 2 dt 2πy (dx/dt) 2 + (dy/dt) 2 dt (19) Compute the are of the surface obtained by rotating around the x-axis the curve given by the following equations: x(t) = 9 2 t2 + 5, y = 6t 2 + 3, 0 t 1. (a) 50π (b) 400π (c) 200π (d) 90π (20) If a point has polar coordinates (r, θ), where r 0 and θ 0, then which of the following are also polar coordinates of this point: (a) (r, θ + π) (b) ( r, θ + 15π) (c) ( r, θ 4π) (d) (r, θ 3π) 4

5 (21) If a point has polar coordinates (r, θ), then its Cartesian coordinates are (a) impossible to determine from the polar coordinates. (b) (r cos 2 θ, r sin θ). (c) (r sin θ, r cos θ). (d) (r cos 2 θ, r sin 2 θ). (22) If a point has polar coordinates (r, θ) and Cartesian coordinates (x, y), then (a) r 2 = x 2 y 2, tan θ = y x (b) r 2 = 2x 2 + y 2, tan θ = x y (c) r 2 = x 2 + y 2, cos θ = y x (d) r 2 = y 2 x 4, tan θ = y x (23) Convert polar coordinates ( 6, π/4) to Cartesian coordinates. (a) ( 3 2, 3 2) (b) (3/4, 3/4) (c) ( 3 3/2, 3/2) (d) (3 3/2, 1/2) (24) Convert cartesian coordinates (3,3) to polar coordinates (a) (2 2, π/4) (b) (2 2, 3π/4) (c) (4, π/4) (d) (3 2, π/4) (25) Determine the values of the parameter for which the the following curve concaves up: x = t 3 + 4t, y = t 3 7. (a) The curve always concaves up. (b) t > 0 (c) t < 0 (d) The curve never concaves up. (26) Let C be the curve defined by x(t) = 2 sin 2 t 13t, y(t) = 4 cos 2 t+2t 3 +10t. Find all the points where this curve has a vertical tangent line. (Hint: 2 sin t cos t = sin 2t.) (a) t = πk/3, k Z. (b) t = π/2 + πk/2, k Z. (c) There is no point with a vertical tangent. (d) t = π + 2πk, k Z. (27) Let C be the curve defined by x(t) = t 2 + 2t + 4, y(t) = tan t. Find all the points where this curve has a vertical tangent line. (a) t = π/2 + πk, k Z. (b) t = 2 5

6 (c) t = 1 (d) t = π + 2πk, k Z. (28) Let C be a polar curve defined by an equation r = f(θ) where f has a continuous derivative at every point of the of the arc corresponding to θ [α, ], with α < < π. Then the sector area bounded by the graph and lines θ = α and θ = can be computed by the formula (a) A = 1 2 α r2 θdθ (b) A = 1 3 α r4 dθ (c) A = 1 2 α r2 dθ (d) A = α r2 dθ (29) Let C be a polar curve defined by an equation r = 3e θ/2. Compute the area of the sector bounded by the curve and lines θ = 0 and θ = 2. (a) 1 2 e (b) e 2 1 (c) 1 3 e2 1 (d) 9 2 e

7 Key 1c, 2a, 3a, 4e, 5a, 6c, 7d, 8b, 9b, 10b, 11d, 12b, 13c, 14c, 15b, 16d, 17d, 18d, 19d, 20b, 21e, 22e, 23a, 24d, 25c, 26c, 27c, 28c 29d. 7