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MATH1011 Quizzes

Quiz 7: Two variable optimisation of surfaces
Question 1 Questions
Find the equation of the tangent to the curve f(x) = xsin2x at x = π 4 .
a)
y = π 4 x
 b)
y = x
c)
y = π 4 x 1
 d)
y = x + 1 π 4

Choice (a) is incorrect
Choice (b) is correct!
f(x) = sin2x + xcos2x, f(π 4 ) = 1, f(π 4 ) = π 4 . So with y = x + b, π 4 = π 4 + b, b = 0.
therefore the equation of the tangent is y = x.
Choice (c) is incorrect
Choice (d) is incorrect
Find two numbers whose difference is 20 and whose product is minimal.
a)
1, 21
 b)
19, -1
c)
10, -10
 d)
None of these.

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
Let the two numbers be a,b where a > b.
Then a b = 20. Let the product of the numbers be P.
P = ab = a(a 20) = a2 20a, dP da = 2a 20, dP da = 0 when a = 10 and b = 10. This is clearly a minimum value for the product because d2P da2 = 2 > 0 for all a.
Choice (d) is incorrect
Imagine constructing a closed steel box with volume 576 cm3 and with its base twice as long as it is wide. The steel costs $40 per square metre. Determine the dimensions of the box that will minimise the cost of construction.
PIC
a)
12 cm × 6 cm × 8 cm
b)
123 cm ×63 cm ×8 3 cm
c)
1223 cm ×623 cm ×423 cm
d)
None of the above

Choice (a) is correct!
The volume of the box is 2x2h = 576, so h = 288 x2 . The surface area of the box is = S = 4x2 + 1728 x . Therefore dS dx = 8x 1728 x2 . Hence, dS dx = 0, when 8x3 = 1728; that is, when x = 6. Consequently, the dimensions of the box are 12 cm × 6 cm × 8 cm. Note that S has a minimum at x = 6 since d2S dx2 = 8 + 3456 x3 > 0 at x = 6.
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Evaluate f(x,y) = e3x cosy at (2, π 4 ).
a)
f(2, π 4 ) = e3π 4 cos2
 b)
f(2, π 4 ) = 1 2e6
c)
f(2, π 4 ) = 2 2e6
 d)
f(2, π 4 ) = e3π 2

Choice (a) is incorrect
Choice (b) is correct!
When x = 2 and y = π 4 , e3x cosy = e6 cos π 4 = 1 2e6.
Choice (c) is incorrect
Choice (d) is incorrect
Evaluate f(x,y) = sin2xy + cos3xy at (1, π 6 ).
a)
f(1, π 6 ) = 3 2
 b)
f(1, π 6 ) = 1 2
c)
f(1, π 6 ) = 3 2 + 1
 d)
f(1, π 6 ) = 3 2

Choice (a) is correct!
f(1, π 6 ) = sin π 3 + cos π 2 = 3 2 + 0 = 3 2 .
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Which point below is on the surface z = x2 + xy (4 y)2 ?
a)
(1,1,27)
 b)
(1,1,7)
c)
(1,2,5)
 d)
(1,2,37)

Choice (a) is incorrect
Choice (b) is correct!
When x = y = 1, z = 1 + 1 (3)2 = 7.
Choice (c) is incorrect
Choice (d) is incorrect
Which point below is on the surface z = 2x2y + x x + y ?
a)
A(2,2,16)
 b)
B(1,1,1)
c)
C(2,2,14)
 d)
D(1,1,1)

Choice (a) is correct!
When x = y = 2, z = 16 + 2 4 = 16.
Choice (b) is incorrect
Note that (1,1) is not in the domain of the surface.
Choice (c) is incorrect
Choice (d) is incorrect
Which response below most accurately describes the intersection of the surface
z = 9 x2 y2
and the plane z = 2 ?
a)
A circle of radius 5 in the plane z = 2.
b)
A circle of radius 5 in the xy plane.
c)
A circle of radius 5 in the plane z = 2.
d)
A circle of radius 5 in the xy plane.

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
The curve of intersection of the surface and the plane is given by 9 x2 y2 = 2. i.e x2 + y2 = 5. This is a circle of radius 5 in the plane z = 2.
Choice (d) is incorrect
Which response below most accurately describes the intersection of the surface z = x2 9 y2 4 and the plane y = 4 ?
a)
An ellipse, x2 36 y2 16 = 1, in the plane z = 4.
b)
A circle radius 2, x2 + y2 = 4, in the plane z = 4.
c)
A parabola z = x2 9 1, in the plane y = 4.
d)
A parabola z = x2 9 4, in the plane y = 4.

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
The curve of intersection of the surface and the plane is given by z = x2 9 16 4 . i.e z = x2 9 4 which is a parabola in the plane y = 4.
Which response below most accurately describes the intersection of the surface z = 2x2 2x y2 2y + 3 and the plane x y = 0?
a)
The parabola z = 3y2 + 3 in the plane x y = 0.
b)
The parabola z = 3x2 + 3 in the plane x y = 0.
c)
The parabola z = x2 4x + 3 in the plane x y = 0.
d)
The intersection cannot be determined.

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
x y = 0 y = x so z = 2x2 2x x2 2x + 3 = x2 4x + 3.
Choice (d) is incorrect