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Quiz 7: Two variable optimisation of surfaces

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Question 1

Find the equation of the tangent to the curve f(x) = xsin2x at π x = -- 4 .

a)
y = πx 4
  b)
y = x
c)
π- y = 4x - 1
  d)
π y = x+ 1 - -- 4

 

Not correct. Choice (a) is false.
Your answer is correct.
′ f (x ) = sin2x + xcos2x, f′(π-) = 1, 4π π f(--) = -. 4 4
So with π π y = x+ b,4 = 4 + b, b = 0.
therefore the equation of the tangent is y = x.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.

Question 2

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.

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Your answer 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 2 d-P-= 2 > 0 da2 for all a.
Not correct. Choice (d) is false.

Question 3

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)
√ - 12 3 cm √ - × 6 3 cm 8 × 3 cm
c)
√ - 12 32 cm √- × 6 32 cm √ - × 4 3 2 cm
d)
None of the above

 

Your answer 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-= 8x- 17228 dx x . Hence, dS- dx = 0 , when 3 8x = 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 2 d-S-= 8+ 3456 > 0 dx2 x3 at x = 6.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.

Question 4

Evaluate f(x,y) = e3xcosy at (2, π) 4 .

a)
π 3π- f(2,4) = e 4 cos 2
  b)
f(2, π-) = √1-e6 4 2
c)
π 2 f(2,--) = √-e6 4 2
  d)
f(2, π-) = e32π 4

 

Not correct. Choice (a) is false.
Your answer is correct.
When x = 2 and y = π 4 , e3xcosy = e6cos π-= 1√-e6 4 2 .
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.

Question 5

Evaluate f(x,y) = sin2xy + cos3xy at π (1,6) .

a)
√- π 3 f(1,6) = 2--
  b)
f(1, π-) = 1 6 2
c)
√ - f(1, π-) =-3+ 1 6 2
  d)
π 3 f(1,6-) = 2

 

Your answer is correct.
f(1,π6) = sinπ3 + cosπ2 = √- -32- + 0 = √ - -23.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.

Question 6

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)

 

Not correct. Choice (a) is false.
Your answer is correct.
When x = y = 1, 2 z = 1+ 1- (3) = - 7 .
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.

Question 7

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)

 

Your answer is correct.
When x = y = 2, z = 16 + 2 -√ - 4 = 16.
Not correct. Choice (b) is false.
Note that (-1,-1) is not in the domain of the surface.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.

Question 8

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.

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Your answer 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.
Not correct. Choice (d) is false.

Question 9

Which response below most accurately describes the intersection of the surface x2 y2 z = 9- - 4- and the plane y = 4 ?

a)
An ellipse, x2 y2 36 - 16 = 1 , in the plane z = 4.
b)
A circle radius 2, x2 +y2 = 4 , in the plane z = 4.
c)
A parabola x2 z = 9 - 1 , in the plane y = 4.
d)
A parabola z = x2 - 4 9 , in the plane y = 4.

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Your answer is correct.
The curve of intersection of the surface and the plane is given by 2 z = x - 16 9 4 . i.e x2 z = -- - 4 9 which is a parabola in the plane y = 4.

Question 10

Which response below most accurately describes the intersection of the surface 2 2 z = 2x - 2x- y - 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.

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Your answer is correct.
x - y = 0 ⇒ y = x so z = 2x2 - 2x - x2 - 2x +3 = x2 - 4x + 3 .
Not correct. Choice (d) is false.
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