Mathematics 2153-02
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 0 (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 0 (2) = 1, g 0 (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 0 (2) = 0, g 0 (2) = 0 then dy/dx(1) 6= 0.
(e) None of the above
(2) Describe the curve defined by the following parametric equations x = 7t4 + 1,
y = 12t2 + 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.
(e) None of the above
(3) Describe the curve defined by the following parametric equations x = 2t3 + 1,
y = −t3 + 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.
(e) None of the above
(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.
(e) None of the above
(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
(e) None of the above
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(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?
dx/dt
(a) dy/dx =
, provided dy/dt 6= 0
dy/dt
dy dx
(b) dy/dx =
dt dt
dy/dt
, provided dx/dt 6= 0
(c) dy/dx =
dx/dt
(d) dy/dx = dy/dt + dx/dt
(e) None of the above
(7) Let x = 1 + 2 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 6= π + πk, where k is an integer
(b) dx/dy cannot be determined from these data
(c) − tan t, t 6= π/2 + πk, where k is an integer
(d) − cot t, t 6= πk, where k is an integer
(e) None of the above
(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.
(e) None of the above
(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.
(e) None of the above
(10) Let x = f (t), y = g(t) be equations describing a parametric curve for t ∈ [a, b].
Suppose f 0 (t), g 0 (t) are both continuous and non-zero for t ∈ [a, b] and f 0 (t) > 0. In
this case, for x ∈ [f (a), f (b)] it is the case that
0 2
f (t)
2
2
(a) d y/dx =
.
g 0 (t)
0
h0 (t)
(b) d2 y/dx2 = 0 , where h(t) = fg 0(t)
.
(t)
f (t)
g 00 (t)
(c) d2 y/dx2 = 00 .
f (t)
2
2
(d) d y/dx cannot be determined from f and g.
2
(e) None of the above
(11) Let x = cos t, y = 4 cos2 t + 6 cos t + 1. Find d2 y/dx2 for all values of t where dx/dt
is not zero.
(a) 0
(b) d2 y/dx2 cannot be determined from these data
(c) −8
(d) 8
(e) None of the above
(12) Let x = cos t, y = et . 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.
(e) None of the above
(13) Let x = et , y = sin t. Find the equation of the tangent line at the point (x(π/3), y(π/3)).
1
1
(a) y − = −π/3 (x − ln(π/3))
2√ e √
3
3
=
(x − eπ/3 )
(b) y −
2
ln(π/3)
√
1
3
(c) y −
= π/3 (x − eπ/3 )
2
2e
(d) There is no tangent line at the point (x(π/3), y(π/3)).
(e) None of the above
(14) Suppose x = f (t), y = g(t), y = F (x), where all the functions involved are differentiable for t ∈ [α, β], 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(β) can be
computed using the following formula:
Z β
(a)
g 0 (t)f (t)dt
Zαβ
(b)
g 0 (t)f 0 (t)dt
Zαβ
g(t)f 0 (t)dt
(c)
Zαβ
(d)
g(t)f (t)dt
α
(e) None of the above
(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
(e) None of the above
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(16) Suppose a parametric curve C is given by equations x = f (t), y = g(t), α ≤ t ≤ β,
where f, g have continuous derivatives and C is traversed just once as t increases
from α tosβ. In this case which of the formulas below compute the length of C?
Z β
(dx/dt)2
(a)
dt
(dy/dt)2
α
Z βp
(dx/dt)2 (dy/dt)2 dt
(b)
Zαβ p
(dx/dt)2 − (dy/dt)2 dt
(c)
α
Z βp
(d)
(dx/dt)2 + (dy/dt)2 dt
α
(e) None of the above
(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
(e) None of the above
(18) Suppose a parametric curve C is given by equations x = f (t), y = g(t), α ≤ t ≤ β,
where f, g have continuous derivatives, C is traversed just once as t increases from
α to β 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?
Z β
p
2πy (dx/dt)2 − (dy/dt)2 dt
(a)
Zαβ p
(b)
y (dx/dt)2 + (dy/dt)2 dt
α
Z βp
(c)
(dx/dt)2 + (dy/dt)2 dt
Zαβ
p
2πy (dx/dt)2 + (dy/dt)2 dt
(d)
α
(e) None of the above
(19) Compute the are of the surface obtained by rotating around the x-axis the curve
given by the following equations: x(t) = 29 t2 + 5, y = 6t2 + 3, 0 ≤ t ≤ 1.
(a) 50π
(b) 400π
(c) 200π
(d) 90π
(e) None of the above
(20) If a point has polar coordinates (r, θ), where r 6= 0 and θ 6= 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)
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(e) None of the above
(21) If a point has polar coordinates (r, θ), then its Cartesian coordinates are
(a) impossible to determine from the polar coordinates.
(b) (r cos2 θ, −r sin θ).
(c) (r sin θ, r cos θ).
(d) (r cos2 θ, r sin2 θ).
(e) None of the above
(22) If a point has polar coordinates (r, θ) and Cartesian coordinates (x, y), then
y
(a) r2 = x2 − y 2 , tan θ =
x
x
(b) r2 = 2x2 + y 2 , tan θ =
y
y
2
2
2
(c) r = x + y , cos θ =
x
y
(d) r2 = y 2 − x4 , tan θ =
x
(e) None of the above
(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)
(e) None of the above
(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)
(e) None of the above
(25) Determine the values of the parameter for which the the following curve concaves
up: x = t3 + 4t, y = −t3 − 7.
(a) The curve always concaves up.
(b) t > 0
(c) t < 0
(d) The curve never concaves up.
(e) None of the above
(26) Let C be the curve defined by x(t) = 2 sin2 t − 13t, y(t) = 4 cos2 t + 2t3 + 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.
(e) None of the above
(27) Let C be the curve defined by x(t) = t2 + 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
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(c) t = −1
(d) t = π + 2πk, k ∈ Z.
(e) None of the above
(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, β], with α < β <
π. Then the sector area bounded by the graph and lines θ = α and θ = β can be
computed by the formula
Rβ
(a) A = 12 α r2 θdθ
Rβ
(b) A = 31 α r4 dθ
Rβ
(c) A = 21 α r2 dθ
Rβ
(d) A = α r2 dθ
(e) None of the above
(29) Let C be a polar curve defined by an equation r = 3eθ/2 . Compute the area of the
sector bounded by the curve and lines θ = 0 and θ = 2.
(a) 21 e2 + 21
(b) e2 − 1
(c) 13 e2 − 1
(d) 29 e2 − 29
(e) None of the above
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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.
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