It was estimated that in 2013, the Greenland Ice Sheet and its outlying ice caps were losing mass at a rate of about -393 Gt/y (gigatonnes per year).1 Suppose that in 2023, plans are put into place to gradually improve the rate of polar ice melt over the next 10 years, so that by 2033, the rate of ice melt in Greenland will be -93 Gt/y. Assuming that there are no further interventions after 2033, graph a function that represents the rate of ice melt from 2013 to 2043. Based on this graph, graph a function that could represent the total amount of ice in Greenland from 2013 to 2043.
In: Advanced Math
1. Prove ├(∀x)A ∨ (∀x)B → (∀x)(A ∨ B).
2. Use the ∃ elimination technique —and ping-pong if/where needed— to show ├(∃x)A → (∃x)(B → A).
Appreciate you.
In: Advanced Math
(2) More inclusion-exclusion counting: How many bit strings of length 15 have bits 1, 2, and 3 equal to 101, or have bits 12, 13, 14, and 15 equal to 1001 or have bits 3, 4, 5, and 6 equal to 1010? (Number bits from left to right. In other words, bit #1 is the left most bit and bit #15 is the right most bit.) Hint: The fact that the third bit appears in two of the required patterns means some special care will be needed to get the count correct.
In: Advanced Math
For which values of λ does the system of equations
(λ − 2)x + y = 0
x + (λ − 2)y = 0
have nontrivial solutions? (That is, solutions other than x = y = 0.) For each such λ find a nontrivial solution.
In: Advanced Math
Given f(x) = sin^2(x) and g(x) = sin^4(x) (give exact answers for all parts): a) Plot the functions on the x-interval [0, π]. Find the volume when the region enclosed by the curve and the x-axis is rotated about the line x = π. b) Find the area of the region. 1?b Aa and y = 1? b 1(f(x)2 −g(x)2)dx. Find the x-coordinate of the center of Aa2 mass of the region. In a print statement, explain why this answer makes sense based on the graph in part a). d) When the region rotates about the line x = π, how far does the center of mass travel? Multiply this value by the area. What do you notice when you compare your answer to part a)?
Python
In: Advanced Math
Define error function and complementary error function, mathematically. Explain how these functions can be tabulated.
In: Advanced Math
For each system described below say as much as possible about
each system’s solution
set. Note which theorems you are using to reach your
conclusions.
(a) A consistent system of 8 equations in 5 variables.
(b) A consistent system of 5 equations in 8 variables.
(c) A system of 4 equations in 9 variables.
(d) A system with 15 equations in 35 variables.
(e) A system with 8 equations and 5 variables. The reduced
row-echelon form of the
augmented matrix of the system has 6 pivot columns.
In: Advanced Math
Use LU decomposition to solve the following system of equations (show your work). Do not use a pivoting strategy, and check your results by using the matrix inverse to show that [A][A]-1= [I].
8x+ 2y−z=10
- 2x+4y+z=5
3x−y+ 6z=7
In: Advanced Math
Using a power series methodology, obtain the general solution (form u = c1u1 + c2u2 + f(x)) to the equation u” + 4u = x.
In: Advanced Math
Consider a homogeneous system of linear equations with m equations and n variables.
(i) Prove that this system is consistent.
(ii) Prove that if m < n then the system has infinitely many solutions. Hint: Use r (the number of pivot columns) of the augmented matrix.
In: Advanced Math
A voltage of 10 sin(3t) volts is impressed on a series circuit containing a 20Ω resistor, 10^-3 H inductor and a 1μF capacitor. Obtain expressions for the charge q on the capacitor and current I in the circuit if q=0=I at t=0.
In: Advanced Math
Scenario 7.2 - Gulab Greatness
Historical demand for gulab jamun from a sweet stall on Commercial
Road is as displayed in the table.
Month | Demand (orders) |
January | 66,753 |
February | 67,686 |
March | 68,641 |
April | 68,979 |
May | 69,278 |
June | 69,577 |
July | 69,602 |
August | 70,348 |
September | 70,806 |
October | 71,011 |
November | 71,819 |
December | 72,752 |
What is the trend component of Holt's model for period 0?
642 |
||
462 |
||
66,768 |
||
78,666 |
In: Advanced Math
Key information:
Box dimension: 12” x 10” x 6” O.D.
Pallet dimensions: 48” x 42” x 6”
Constraints: (1) No pallet overhang
(2) Pallet unit load <= 48” high, including the pallet, for storage in a rack system
Trailer dimensions: 53’ long x 8’ 6” wide (b/w the hinges) x 9’ high
Deliverables:
In: Advanced Math
Let V = R^2×2 be the vector space of 2-by-2 matrices with real
entries over
the scalar field R. We can define a function L on V by
L : V is sent to V
L = A maps to A^T ,
so that L is the “transpose operator.” The inner product of two
matrices B in R^n×n and C in R^n×n is usually defined to be
<B,C> := trace (BC^T) ,
and we will use this as our inner product on V . Thus when we talk
about
elements B,C in V being orthogonal, it means that <B,C> :=
trace (BC^T) = 0.
Problem 1.
1. First show that L is linear, so that L in B (V ).
2. Now choose a basis for the vector space V = R^2×2, and find the
matrix of
L with respect to your basis.
In: Advanced Math
Consider the initial value problem
y′ = 18x − 3y, y(0) = 2
(a) Solve it as a linear 1st order ODE with the method of the integrating factor.
(b) Solve it using a substitution method.
(c) Solve it using the Laplace transform.
In: Advanced Math