2415-2-review

2415-2-review - Cal III: Test 2 Review

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Unformatted text preview: Cal III: Test 2 Review Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Differentiate the function. 13 tj 18 1) r(t) = (4t2 - 6)i + A) rʹ(t) = 1) 12 t i + (8t) j 6 C) rʹ(t) = (8)i + B) rʹ(t) = (8t) i + D) rʹ(t) = (8t) i - 1 tj 3 12 tj 6 12 tj 6 If r(t) is the position vector of a particle in the plane at time t, find the indicated vector. 2) Find the velocity vector. r(t) = (cot t) i + (csc t)j A) v = (csc2 t)i + (cot t csc t)j B) v = (sec 2 t)i + (tan t sec t)j C) v = ( - sec 2 t)i - (tan t sec t)j D) v = ( - csc2 t)i - (cot t csc t)j 3) Find the velocity vector. 13 tj r(t) = (2t2 - 14)i + 18 A) v = 3) 12 t i + (4t) j 6 C) v = (4t) i + B) v = (4)i + 12 tj 6 D) v = (4t) i - 1 tj 3 12 tj 6 4) Find the acceleration vector. r(t) = (6 ln(4t))i + (3t3 )j 4) 6 A) a = i + 9tj t C) a = 2) 3 B) a = - t- 2 i + 18tj 2 6 -2 t i + 18tj t2 D) a = - 6 i + 18tj t2 Solve the initial value problem. 5) Differential Equation: dr = (cos t)i + (4t3 + 3)j dt 5) Initial Condition: r(0) = - 4 j A) r(t) = ( - sin t)i + (12t2 - 4)j B) r(t) = (sin t)i + (t4 )j D) r(t) = (sin t)i + (t4 + 3t - 4)j C) r(t) = (sin t)i + (t4 - 4)j 6) Differential Equation: dr 9 = (t + 3)7/2i + etj dt 2 6) Initial Condition: r(0) = 0 A) r(t) = [(t + 3)11/2 - 3 11/2]i + (et - 1)j B) r(t) = (t + 3)9 /2i + etj D) r(t) = [(t + 3)9 /2 + 3 9 /2]i + (et + 1)j C) r(t) = [(t + 3)9/2 - 3 9/2]i + (et - 1)j 1 Solve the problem. Unless stated otherwise, assume that the projectile flight is ideal, that the launch angle is measured from the horizontal, and that the projectile is launched from the origin over a horizontal surface 7) An ideal projectile is launched from the origin at an angle of α radians to the horizontal and an 7) initial speed of 100 ft/sec. Find the position function r(t) for this projectile. A) r(t) = (100t sin α)i + (100t cos α - 16t2 )j B) r(t) = (100t cos α)i + (100t sin α - 16t2 )j D) r(t) = (100t sin α - 16t2 )i + (100t cos α)j C) r(t) = (100t cos α - 32t2 )i + (100t sin α)j 8) A projectile is fired with an initial speed of 544 m/sec at an angle of 45°. What is the greatest height reached by the projectile? Round your answer to the nearest tenth. A) 7549.4 m B) 78.5 m C) 73,984.0 m D) 30,197.6 m Find the unit tangent vector of the given curve. 9) r(t) = 4t3 i - 3t3 j + 12t3 k 9) 4 3 12 A) T = i - j - k 13 13 13 C) T = 8) 4 3 12 B) T = i + j + k 13 13 13 12 3 4 i - j + k 13 13 13 D) T = 10) r(t) = (7 - 2t)i + (2t - 4)j + (9 + t)k 2 1 2 A) T = i - j - k 3 3 3 4 12 3 i - j + k 169 169 169 10) 2 2 1 B) T = - i + j + k 9 9 9 2 1 2 C) T = - i + j + k 3 3 3 2 2 1 D) T = i - j - k 9 9 9 Find the arc length parameter along the curve from the point where t = 0 by evaluating s = ∫ t |v(τ)| dτ. 0 11) r(t) = (5cos t)i + (5sin t)j + 8tk 89 A) t B) 3 17t 2 11) C) 12) r(t) = (1 + 3t) i + (1 + 7t) j + (5 - 5t) k B) 83t A) 74t 114 t D) 89t 12) C) Find the length of the indicated portion of the curve. 13) r(t) = (2cos t)i + (2sin t)j + 5tk, 0 ≤ t ≤ π/2 29t 33t A) π B) π 4 2 58t D) 34t 13) C) D) 7t π 2 C) 14) r(t) = (1 + 2t) i + (1 + 5t) j + (3 - 3t) k, - 1 ≤ t ≤ 0 A) 38 B) 13 29t π 2 34 D) 29 14) Find the principal unit normal vector N for the curve r(t). 15) r(t) = (6 + t)i + (9 + ln(cos t)) k, - π/2 < t < π/2 A) N = ( - cos t)i - (tan t)k C) N = (cos t)i + (sin t)k 15) B) N = ( - cos t)i - (ln(cos t))k D) N = ( - sin t)i - (cos t)k 2 16) r(t) = (t2 + 1)j + (2t - 8)k 1 t A) N = - j + k (t2 + 1)3 (t2 + 1)3 C) N = 16) 1 t B) N = - j + k t2 + 1 t2 + 1 1 t j - k 2 + 1 2 + 1 t t D) N = 1 t j - k 2 + 1)3 2 + 1)3 (t (t Find the curvature of the space curve. 5 5 17) r(t) = 12ti + 6 + 9 cos t j + 3 + 9 sin k 9 9 A) κ = 25 117 B) κ = 17) 25 1521 C) κ = 18) r(t) = - 4 i + (4 + 2t)j + (t2 + 2)k 1 A) κ = 2 + 1)3/2 2(t C) κ = 5 13 D) κ = 5 169 18) B) κ = - 1 2 t2 + 1 D) κ = 1 2(t2 + 1)3/2 1 2 + 1)3/2 (t For the curve r(t), write the acceleration in the form a TT + a NN. 5 5 19) r(t) = 6 sin t + 2 i + 6 cos t - 10 j + 12tk 6 6 A) a = 25 T 6 19) B) a = 25T + 25N C) a = T + 25N 20) r(t) = (t - 2)i + (ln(sec t) - 5)j + 4 k, - π/2 < t < π/2 A) a = (sec 2 t)T + (cos t)N D) a = 25 N 6 20) B) a = (cos t)T + (cos t)N C) a = (csc t)T + (sec t)N D) a = (sec t tan t)T + (sec t)N Compute rʹʹ(t). 21) r(t) = (5 cos t) i + (8 sin t)j A) rʹʹ(t) = (5 cos t) i + (8 sin t)j C) rʹʹ(t) = ( - 5 sin t)i + ( - 8 cos t) j 21) B) rʹʹ(t) = (5 sin t)i + (8 cos t) j D) rʹʹ(t) = ( - 5 cos t) i + ( - 8 sin t)j Evaluate the integral. 3 4t 1 22) i - 3t2 j + k dt 1 + t (1 + t2 )2 0 9 9 B) 2 i - 27j + k A) 2 i + 27j + k 10 10 ∫ 23) ∫ 1 10ti + 15t2 j - 0 7 A) 5 i - 5 j - k 4 6 (1 + t)4 22) 9 C) 2 i - 27j + k 5 9 D) 1 i - 27j + k 5 23) k dt 7 C) 5 i + 5 j - k 4 B) 10i - 10j + 2 k 3 D) 10i + 10j - 2 k Find the domain of the function of two variables. 24) f(x, y) = 4 - x2 - y2 24) A) {(x, y)| x2 - y2 ≥ 4} C) {(x, y)|x2 - y2 ≤ 4} 25) f(x, y) = B) {(x, y)| x2 + y2 ≤ 4} D) {(x, y)| x2 + y2 ≥ 4} 1 25) y - 3x2 A) {(x, y)| y ≥ - 3x2 } C) {(x, y)| y ≠ - 3x2 } B) {(x, y)| y ≠ 3x2 } D) {(x, y)| y ≥ 3x2 } Describe the domain of the function of three or more varables. 26) f(x, y, z) = x2 + y2 + z 2 - 100 26) A) The set of all points on and outside the sphere x2 + y2 + z 2 = 100. B) The set of all points inside the sphere x2 + y2 + z 2 = 100. C) The set of all points on and inside the sphere x2 + y2 + z 2 = 100. D) The set of all points outside the sphere x2 + y2 + z 2 = 100. 27) f(x, y, z) = 576 - 36x2 - 16y2 - 576z 2 27) x2 y2 A) The set of all points on and outside the ellipsoid + + z 2 = 1. 36 16 B) The set of all points on and inside the ellipsoid x2 y2 + + z 2 = 1. 36 16 C) The set of all points on and inside the ellipsoid x2 y2 + + z 2 = 1. 16 36 D) The set of all points on and outside the ellipsoid x2 y2 + + z 2 = 1. 16 36 Find the domain and range and describe the level curves for the function f(x,y). 1 28) f(x, y) = 2 + 7y2 6x A) Domain: all points in the xy‐plane except (0, 0); range: all real numbers; level curves: ellipses 6x2 + 7y2 = c B) Domain: all points in the xy‐plane; range: real numbers > 0; level curves: ellipses 6x2 + 7y2 = c C) Domain: all points in the xy‐plane except (0, 0); range: real numbers > 0; level curves: ellipses 6x2 + 7y2 = c D) Domain: all points in the xy‐plane; range: all real numbers; level curves: ellipses 6x2 + 7y2 = c 4 28) 29) f(x, y) = 36 - x2 - y2 29) A) Domain: all points in the xy‐plane satisfying x2 + y2 ≤ 36; range: real numbers 0 ≤ z ≤ 6; level curves: circles with centers at (0, 0) and radii r, 0 < r ≤ 6 B) Domain: all points in the xy‐plane; range: real numbers 0 ≤ z ≤ 6; level curves: circles with centers at (0, 0) and radii r, 0 < r ≤ 6 C) Domain: all points in the xy‐plane; range: all real numbers; level curves: circles with centers at (0, 0) D) Domain: all points in the xy‐plane satisfying x2 + y2 = 36; range: real numbers 0 ≤ z ≤ 6; level curves: circles with centers at (0, 0) and radii r, 0 < r ≤ 6 Find the limit. 30) 7x2 + 10y2 + 5 lim (x, y) → (0, 0) 7x2 - 10y2 + 9 A) - 1 31) lim (x, y) → (0, 1) B) 1 C) 5 9 D) No limit y2 sin x x A) ∞ 32) 30) 31) B) 1 C) 0 D) No limit lim x ln y (x, y) → (1, 2) A) 2 32) B) ln (2) - 1 C) ln 2 D) No limit At what points is the given function continuous? xy 33) f(x, y) = x + y 33) A) All (x, y) B) All (x, y) such that x ≠ y C) All (x, y) such that x ≠ - y D) All (x, y) ≠ (0, 0) 34) f(x, y) = 8x + 2y A) All (x, y) C) All (x, y) such that 8x + 2y ≥ 0 34) B) All (x, y) such that x + y ≥ 0 D) All (x, y) such that 8x + 2y ≠ 0 Find all the first order partial derivatives for the following function. 35) f(x, y) = (4x3 y4 + 9)2 ∂f ∂f = 24x2 y4 (4x3 y4 + 9); = 32x3 y3 (4x3 y4 + 9) A) ∂y ∂x B) ∂f ∂f = 12x2 y4 ; = 16x3 y3 ∂y ∂x C) D) ∂f ∂f = 32x3 y3 (4x3 y4 + 9); = 24x2 y4 (4x3 y4 + 9) ∂y ∂x ∂f ∂f = 2(4x3 y4 + 9); = 2(4x3 y4 + 9) ∂y ∂x 5 35) 36) f(x, y) = 6x - 5y2 - 10 ∂f ∂f = 6; = - 10y A) ∂y ∂x C) 36) ∂f ∂f B) = - 4; = - 10y - 10 ∂x ∂y ∂f ∂f = 6x; = - 10y ∂y ∂x D) ∂f ∂f = - 10y; = 6 ∂x ∂y Find all the second order partial derivatives of the given function. 37) f(x, y) = x2 + y - ex+ y A) ∂2 f ∂2 f ∂2 f ∂2 f = 1 - ex+ y; = - ex+ y; = = - ex+ y 2 2 ∂y∂x ∂x∂y ∂y ∂x B) ∂2 f ∂2 f ∂2 f ∂2 f = 2 - y2 ex+ y; = - x2 ex+ y; = = - y2 ex+ y ∂y∂x ∂x∂y ∂y2 ∂x2 C) ∂2 f ∂2 f ∂2 f ∂2 f = 2 + ex+ y; = ex+ y; = = ex+ y ∂y∂x ∂x∂y ∂y2 ∂x2 D) 37) ∂2 f ∂2 f ∂2 f ∂2 f = 2 - ex+ y; = - ex+ y; = = - ex+ y ∂y∂x ∂x∂y ∂y2 ∂x2 Find all the first order partial derivatives for the following function. 38) f(x, y, z) = x2 y + y2 z + xz 2 38) ∂f ∂f ∂f A) = 2xy + z 2 ; = x2 + 2yz; = y2 + 2xz ∂x ∂y ∂z B) ∂f ∂f ∂f = 2xy; = x2 + 2yz; = y2 + 2xz ∂y ∂z ∂x C) ∂f ∂f ∂f = 2xy + z 2 ; = x2 + yz; = y2 + xz ∂y ∂z ∂x D) ∂f ∂f ∂f = 2y + z 2 ; = x2 + 2z; = y2 + 2x ∂y ∂z ∂x 39) f(x, y, z) = e(sin (x) + yz) ∂f ∂f ∂f A) = cos(x) e(sin (x) + yz); = ze(sin (x) + yz); = ye(sin (x) + yz) ∂x ∂y ∂z B) ∂f ∂f ∂f = e(sin (x) + yz); = ze(sin (x) + yz); = ye(sin (x) + yz) ∂y ∂z ∂x C) ∂f ∂f ∂f = cos(x)e(sin (x) + yz); = e(sin (x) + yz); = e(sin (x) + yz) ∂y ∂z ∂x D) ∂f ∂f ∂f = cos(x) + yz e(sin (x) + yz); = ze(sin (x) + yz); = ye(sin (x) + yz) ∂y ∂z ∂x 6 39) 40) f(x, y, z) = cos x sin2 yz ∂f ∂f ∂f = - sin x sin2 yz; = z cos x sin yz cos yz; = y cos x sin yz cos yz A) ∂y ∂z ∂x B) ∂f ∂f ∂f = - sin x sin2 yz; = 2z cos x sin yz cos yz ; = 2y cos x sin yz cos yz ∂y ∂z ∂x C) ∂f ∂f ∂f = sin x sin2 yz; = - cos x sin yz cos yz; = - cos x sin yz cos yz ∂y ∂z ∂x D) ∂f ∂f ∂f = sin x sin2 yz; = - 2z cos x sin yz cos yz; = - 2y cos x sin yz cos yz ∂y ∂z ∂x 7 40) Answer Key Testname: 2415‐2‐REVIEW 1) B 2) D 3) C 4) D 5) D 6) C 7) B 8) A 9) C 10) C 11) D 12) B 13) C 14) A 15) D 16) C 17) B 18) A 19) D 20) D 21) D 22) C 23) C 24) B 25) B 26) A 27) D 28) C 29) A 30) C 31) B 32) C 33) C 34) C 35) A 36) A 37) D 38) A 39) A 40) B 8 ...
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This note was uploaded on 10/25/2011 for the course ECON 2301 taught by Professor Staff during the Spring '08 term at HCCS.

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