F=X3-2xy + xy 3+ 3y2, thenfxx
Answers
Answer:
NICHA DIY GYA DOX KO KHOLA US MA EK CHAPTER HA US CHAPTER MA SA SANGYA,SARVARAN AUR VISHSAN SHABD LIKHA.
NICHA DIY GYA DOX KO KHOLA US MA EK CHAPTER HA US CHAPTER MA SA SANGYA,SARVARAN AUR VISHSAN SHABD LIKHA.
Step-by-step explanation:
Answer:
SOLUTIONS TO PRACTICE TEST 1
1. In the xy-plane, the curve with parametric equations x = cost and
y = sin t, 0 ≤ t ≤ π, has length
(A) 3
(B) π
(C) 3π
(D) 3/2
(E) π/2
Solution: The velocity is given by ˙x = − sin t and ˙y = cost. The speed
is given by [(− sin t)
2 + (cost)
2
]
1/2 = [1]1/2 = 1. The length of the
curve is then Z π
0
1 dt = π. Answer: (B)
Alternate solution: The curve is a semicircle of radius 1, so its length
is π. Answer: (B)
2. Which of the following is an equation of the line tangent to the
graph of y = x + e
x at x = 0.
(A) y = x
(B) y = x + 1
(C) y = x + 2
(D) y = 2x
(E) y = 2x + 1
Solution: The slope is [(d/dx)(x + e
x
)]x:→0 = [1 + e
x
]x:→0 = 1 + e
0 = 2.
The y-coordinate of the point of tangency is [x + e
x
]x:→0 = 0 + e
0 = 1,
so the point of tangency is (0, 1). An equation of the line is therefore
y − 1 = 2(x − 0), or y = 2x + 1. Answer: (E)
Date: Printout date: November 18, 2015.
1
2 SOLUTIONS TO PRACTICE TEST 1
3. If V and W are 2-dimensional subspaces of R
4
, what are the possible
dimensions of V ∩ W?
(A) 1 only
(B) 2 only
(C) 0 and 1 only
(D) 0, 1 and 2 only
(E) 0, 1, 2, 3 and 4
Solution: The minimum possible dimension of V + W is
max{dim V, dim W} = max{2, 2} = 2.
The maximum possible dimension of V + W is
min{ (dim V ) + (dim W), dim R
4
} = min{2 + 2, 4} = 4.
So the set of possible dimensions of V + W is {2, 3, 4}. As
(dim V ) + (dim W) = (dim (V + W)) + (dim (V ∩ W)),
we see that 4 = 2 + 2 = (dim (V + W)) + (dim (V ∩ W)), so
dim (V ∩ W) = 4 − (dim (V + W)).
Then the set of possible dimensions of V ∩ W is
{ 4 − 2 , 4 − 3 , 4 − 4 } = {2, 1, 0