# Several Calculus Questions

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I have several calculus questions that I need help with. You MUST show your work and therefore must have a thorough understanding of calculus concepts and theories. You must NOT simply get answers from other online sites such as Coursehero etc. All work will be checked for accuracy so do not just copy answers you find online.

Borys_S
School: New York University

1. β4π=1(β1)π (π + 11) = β12 + 13 β 14 + 15 = π.

2.

π₯2

π¦2

+ 11 = 1. 36 > 11 so it is a horizontal ellipse. The center is at the origin.
36

π = β36 = 6, π = β11, π = βπ2 β π 2 = 5, thus the foci are (βπ, π), (π, π).

2π¦ β π₯ = 5,
From the first equation π₯ = 2π¦ β 5, substitute it to the second:
π₯ + π¦ 2 β 25 = 0.
(2π¦ β 5)2 + π¦ 2 β 25 = 0, 5π¦ 2 β 20π¦ = 5π¦(π¦ β 4) = 0, π¦ = 0 ππ π¦ = 4.
So π₯ = β5 ππ π₯ = 3 and the answers are (βπ, π), (π, π).
3. {

2

4. lim(5π₯ β 3) = 5 lim π₯ β 3 = 5 β 5 β 3 = ππ.
π₯β5

π₯β5

4 β1 3 12
0 β17 β21 140
5. (1 4 6 β32) β |π1 = π1 β 4π2 , π3 = π3 β 5π2 | β (1
4
6
β32).
5 3 9 20
0 β17 β21 180
We see that the first and the second rows make a contradiction. So no solutions.

1 1
6. (1 β1
4 1

1 β5
1 1
1 β5
3 β1) β |π2 = π2 β π1 , π3 = π3 β 4π1 | β (0 β2 2
4 )β
0 β3 β3 18
1 β2
1 1 1 β5
1 1 1 β5
1
1
β |π2 = β π2 , π3 = β π3 | β (0 1 β1 β2) β |π3 = π3 β π1 | β (0 1 β1 β2) β
2
3
0 1 1 β6
0 0 2 β4
1
1
1
β5
1
1
1
β5
1
β |π3 = π3 | β (0 1 β1 β2) β |π2 = π2 + π3 | β (0 1 0 β4) β
2
0 0 1 β2
0 0 1 β2
1 0 0 1
β |π1 = π1 β π2 β π3 | β (0 1 0 β4), which means π = π, π = βπ, π = βπ.
0 0 1 β2

7. First, we know π₯ β₯ 0, π¦ β₯ 0. At least 336 calories means ππ + ππ β₯ πππ.
[these inequalities give the first quadrant except the triangle (0,42), (0,0), (48,0)]

8. π1 : 12 =

1β2
2

π2 : 12 + 42 =

, 1 = 1, which is true.
2β(24β6β1)
2

π3 : 12 + 42 + 72 =

, 17 = 17, which is true.

3β(6β9β9β1)
2

, 66 =

3β44
2

= 66, which is true.

9. ππ : βππ=1 π(π + 1) =

π(π+1)(π+2)

(π+1)(π+2)(π+3)

3

3

, so ππ+1 : βπ+1
π=1 π(π + 1) =

.

[I am not sure what is "simplify ππ+1 "]

10. Denote π₯ the number of pounds of breakfast tea and π¦ of afternoon tea.
Then the constraints are:
1
1
2
1
π₯ β₯ 0, π¦ β₯ 0,
π₯ + π¦ β€ 45,
π₯ + π¦ β€ 70.
3
2
3
2
The profit functions ...

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