Urgent 2

20 Jun Urgent 2

Zhejiang University of Technology MATH 011: Calculus I

Mid Term # 2 Max Marks: 30

Question 1 [10 marks] Choose the correct answer:

(i) The area of the surface generated by revolving the curve y = 2 √ x, 1 ≤ x ≤ 2 about the x axis is:

(a) 8π3 (3 √

3− 2 √

2)

(b) π √

2 (c) π/2 (d) 61π1728

(ii) The area of the surface generated by revolving y = tanx, 0 ≤ x ≤ π/4 about the x axis is:

(a) 2π ∫ π/4 0

tanx √

1 + sec4 xdx

(b) 2π ∫ π/4 0

x2 √

1 + sec4 xdx

(c) 2π ∫ π/4 0

sec2 xdx

(d) 2π ∫ π/4 0

secx tanxdx

(iii) The lateral surface area of the cone generated by revolving the line segment y = x2 , 0 ≤ x ≤ 4 about the x axis is:

(a) 2 √

5 (b) 4π (c) 4π

√ 5

(d) π/4

(iv) The curve x = y3/4, 0 ≤ y ≤ 1 is revolved about the y axis. The resulting surface area is:

(a) π2 (b) π2

√ 5

(c) 49π3 (d) π9 (

√ 8− 1)

(v) Which of the following functions is a solution of the differential equation y′′ + y = sinx?

(a) y = sinx (b) y = cosx (c) y = 12x sinx (d) y = − 12x cosx

(vi) For what values of P is a population modeled by the differential equation dPdt = 1.2P (1− P

4200 ) increasing?

(a) 0 < P < 4200 (b) P > 4200 (c) P = 0

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(d) P = 4200

(vii) Which of the following functions is a solution of the differential equation tdydt = y + t 2 sin t?

(a) y = t cos t (b) y = t sin t (c) y = −t cos t− t (d) y = t cos t− t sin t

(viii) Which of the following differential equations is linear?

(a) y′ + x √ y = x2

(b) y′ − x = y tanx (c) u2et = t+

√ tdudt

(d) dRdt + t cosR = e −t

(ix) The parametric equations x = sinh t, y = cosh t describe a particle moving along

(a) a parabola (b) an ellipse (c) the upper branch of a hyperbola (d) the lower branch of a hyperbola

(x) The motion of the particle described by the parametric equations x = 5 sin t, y = 2 cos t takes place on

(a) a parabola (b) an ellipse (c) the upper branch of a hyperbola (d) the lower branch of a hyperbola

(xi) A point P (r, θ) give by the polar coordinates (−2,−π/3) may also be represented by

(a) (−3, π) (b) (2, π/3) (c) (2, 2π/3) (d) (2, 7π/3)

(xii) A point P (r, θ) give by the polar coordinates (−2, π/3) may also be represented by

(a) (2,−2π/3) (b) (−2, 2π/3) (c) (2,−π/3) (d) (2, π/3)

(xiii) The nth term of the infinite sequence 1,−4, 9,−16, 25, … is

(a) (−1)n+1;n ≥ 1 (b) n2 − 1;n ≥ 1 (c) (−1)n+1n2;n ≥ 1

2

(d) 1+(−1) n+1

2 ;n ≥ 1

(xiv) The nth term of the infinite sequence 1, 5, 9, 13, 17, … is

(a) 4n− 3;n ≥ 1 (b) n− 4;n ≥ 1 (c) 3n+2n! ;n ≥ 1 (d) n

3

5n+1 ;n ≥ 1

(xv) The first four terms of the infinite sequence an = 1−n n2 are

(a) 0,−1/4,−2/9,−3/16 (b) 1,−1/3, 1/5,−1/7 (c) 1/2, 1/2, 1/2, 1/2 (d) none of the above

(xvi) The first four terms of the infinite sequence an = 2n

2n+1 are

(a) 0,−1/4,−2/9,−3/16 (b) 1,−1/3, 1/5,−1/7 (c) 1/2, 1/2, 1/2, 1/2 (d) none of the above

(xvii) Given −→a = 3̂i+ 4ĵ − k̂ and −→ b = î− ĵ + 3k̂, −→a •

−→ b is

(a) 23 (b) 0 (c) 13 (d) −4

(xviii) −→a and −→ b are two non-zero vectors. The vector −→r = (

−→ b • −→ b )−→a − (−→a •

−→ b ) −→ b is

(a) perpendicular to −→ b

(b) parallel to −→a (c) anti-parallel to −→a (d) parallel to

−→ b

(xix) The work done by a constant force of 40N , applied at an angle of 60◦ to the horizontal, results in a displacement of 3m. The work done is

(a) 60J (b) 90J (c) 30J (d) 120J

(xx) −→u = (2, 0,−1) and −→v = (1, 4, 7). Their cross product is

(a) (14,−15, 22) (b) (4,−15, 8)

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(c) (3,−1, 0) (d) (−6, 2, 0)

Question 2 [4 marks]

(a) [2 marks] Find the exact lengths of the following curves by setting up and evaluating the corresponding definite integral.

(i) y = 1 + 6×3/2, 0 ≤ x ≤ 1 (ii) y = ln(secx), 0 ≤ x ≤ π/4

(b) [2 marks] Find the centroid of the region bounded by the curves y = x3 − x and y = x2 − 1.

Question 3 [4 marks]

(a) [2 marks] For what values of r does the function y = erx satisfies the differential equation 2y′′+ y′− y = 0? If r1 and r2 are the values of r that you found, show that every member of the family of functions y = aer1x + ber2x is also a solution.

(b) [2 marks] Solve the following linear differential equations:

(i) xy′ + y = √ x

(ii) xy′ − 2y = x2, x > 0

Question 4 [4 marks]

(a) [2 marks] Eliminate the parameter to find a Cartesian equation of the following curves. Sketch the curve in the xy plane and indicate with an arrow the direction in which the curve is traced as the parameter increases.

(i) x = sin 12θ, y = cos 1 2θ,−π ≤ θ ≤ π

(ii) x = sin t, y = csc t, 0 < t < π/2 (iii) x = e2t, y = t+ 1 (iv) x = cos(π − t), y = sin(π − t), 0 ≤ t ≤ π (b) [2 marks] Identify the curve by finding a Cartesian equation for it: (i) r cos θ = 2 (ii) r2 = 5 (iii) r = 2 cos θ (iv) r2 cos 2θ = 1 Question 5 [4 marks] (a) [2 marks] Determine if the following sequences converge: (i) xn = sin2 n√ n (ii) xn = 2n+1 en (iii) xn = n3 10n2+1 (iv) xn = tan−1 n n 4 (b) [2 marks] Determine if the following series converge using either the integral test or the ratio and root test: (i) ∑∞ n=1 cos(nπ) (ii) ∑∞ n=1 3n n(2n+1) (iii) ∑∞ n=1 3n n! (iv) ∑∞ n=1( 3n+2 2n−1 ) n Question 6 [4 marks] (a) [1 mark] Find the point on the plane x− y+ z = 0 that is closest to the point P : (2, 3, 0). Also find the distance from the point P to the plane. (b) [3 marks] (i) Find a vector parametric form of the plane in 3-space that passes through the points (3, 2, 1), (0, 0, 2), and (1,−2, 3). (ii) Write the plane r = (3, 2, 1) + s(−3,−2, 1) + t(−2,−4, 2) in the scalar form ax + by + cz = d. (iii) Find a parametric equation for the line in 3-space through the point (2, 5, 0) and that is perpendicular to the plane in (ii). (iv) Find the point on the plane that is closest to (2, 5, 0). (v) Find the distance from the point (2, 5, 0) to the plane. 5