Linear Regression

Find the linear speed 𝑣𝑣 of a point on the tread of a tire of radius 18 cm, rotating 35 times per min.
11) Assume 𝐶𝐶 = 90∘
. Solve right triangle 𝐴𝐴𝐴𝐴𝐴𝐴, if 𝑏𝑏 = 35.6 ft and 𝑐𝑐 = 61.7 ft.
12) Solve the right triangle ABC with 𝑐𝑐 = 12.7 𝑖𝑖𝑖𝑖, 𝐴𝐴 = 34°30′. Assume C is the right angle.
13) Radar stations 𝐴𝐴 and 𝐵𝐵 are on an east-west line, 5.7 km apart. Station 𝐴𝐴 detects a plane at 𝐶𝐶, on a bearing
of 42∘
. Station 𝐵𝐵 simultaneously detects the same plane, on a bearing of 312∘
. Find the distance from 𝐵𝐵
to 𝐶𝐶. Answer must include UNITS.
14) A ship leaves port and sails on a bearing of N 47° E for 3.5 hours. It then turns and sails on a bearing
of S 43° E for 4 hours. If the ship’s rate is 22 knots (nautical miles per hour), find the distance that the
ship is from the port. Round to the nearest whole number.
15) The angle of depression from the top of a building to a point on the ground is 67∘
. How far is the point
on the ground from the top of the building if the building is 194 m high? Answer must include UNITS.
16) From a window 62.0 ft above the street, the angle of elevation to the top of the building across the street
is 56.0∘
and the angle of depression to the base of this building is 13.0∘
. Find the height of the building
across the street. Answer must include UNITS.
17) You need to find the height of a building. From a given point on the ground, you find that the angle of
elevation to the top of the building is 74.2∘
. You then walk back 35 ft. From the second point, the angle
of elevation to the top of the building is 51.8∘
. Find the height of the building.
Answer must include UNITS.
18) Graph 𝑦𝑦 = 4sin(𝑥𝑥 − 𝜋𝜋) + 3 over a two-period interval. Label 5 key points in one period based on the
question.
19) Graph 𝑦𝑦 = −5cos(4𝑥𝑥 + 2𝜋𝜋) over a two-period interval. Label 5 key points in one period based on the
question.

Problem 9.
For A ⊆ Σ
∗ and n ∈ N, we define the n
th slice of A to be the language
An = {y ∈ Σ
∗
|<n, y> ∈ A} ,
where <n, y> = <sn, y> and s0, s1, . . . is the standard enumeration of Σ∗
.
Let C and D be classes of languages.
1. C parametrizes D (or C is universal for D) if there exists A ∈ C such that
D = {An|n ∈ N}.
2. D is C-countable if there exists A ∈ C such that D ⊆ {An|n ∈ N}.
(a) Prove: A class D of languages is countable if and only if D is P(Σ∗
)-countable.
(b) Prove that DEC is not DEC-countable.
Problem 10.
(a) Assume that C and D are sets of languages and g : C
onto −−→ D. Prove: if C is countable,
then D is countable.
(b) Prove: if C is a countable set of languages, then ∃C and ∀C are countable.
Problem 11. Prove that the class of countable languages (defined as CTBL in class) is a
σ-ideal on P(Σ∗
).
1
Problem 12 Prove all the inclusions in the infinite diagram.
∆0
1
⊇
Σ
0
1
⊆
Π0
1
⊆
⊇
∆0
2
⊇
Σ
0
2
⊆
Π0
2
⊆
⊇
∆0
3
⊇
Σ
0
3
⊆
Π0
3
⊆
⊇ .
.
.
Problem 13. Prove that there is a function g : N → N with the following properties.
(i) g is nondecreasing, i.e., g(n) ≤ g(n + 1) holds for all n ∈ N.
(ii) g is unbounded, i.e., for every m ∈ N there exists n ∈ N such that g(n) > m.
(iii) For every computable, nondecreasing, unbounded function f : N → N, f(n) > g(n)
holds for all but finitely many n ∈ N.
Problem 14. Prove that a partial function f : ⊆ Σ
∗ → Σ
∗
is computable if and only if its
graph
Gf = {<x, f(x)> | x ∈ dom f}
is c.e.
Problem 15. Let A ⊆ Σ
∗ be c.e., and let B be an infinite decidable subset of A. Prove: If
A is undecidable, then A B is undecidable.
Problem 16. Let A = L(U) be the universal c.e. language defined in class lectures, and let
B ⊆ Σ
∗
. Prove: If A ≤m B and Σ∗ A ≤m B, then B is neither c.e. nor co-c.e.

Tag: Linear Regression

Linear regression