Is the function f (z) = z analytic on C? Explain your answer. Let f (z) be an analytic function on a domain D. Suppose that f (z) is a purely imaginary for all z ∈ D. Show that f is constant on D.

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Complex Analysis Questions

Instructions: Do not open this exam until instructed to do so. You will have 50 minutes to complete the exam. Please print your name and student ID number above. You may not use calculators, books, notes, or any other material to help you. Make sure your phone is silenced and stowed where you cannot see it. You may use any available space on the exam for scratch work. If you need more scratch paper, please ask one of the proctors. You must show your work to receive credit. This exam contains 8 pages (including this cover page) and 5 questions.
Total of points is 50.

1. (a) (4 points) What is the image of the following curves under w = z2?

  • (I) y = 1
  • (II) y = x + 1

   (b) (5 points) Find the equation of the fractional linear transformation mapping 0 to 1, 1 to 1 + i and ∞ to 2.

 

2. Write the following expressions in polar coordinates z = reiθ. (The expressions are not always single numbers.)

  • (a) (3 points) 1−i√2 1+i
  • (b) (3 points) 5 p1 − i√3
  • (c) (3 points) (−1)2+3i

3. Let u(x, y) = x2 − y2 − x + y.

  • (a) (6 points) Show that u(x, y) is a harmonic function.
  • (b) (6 points) Find the harmonic conjugate v(x, y) of u(x, y).


4. (a) (5 points) Is the function f (z) = z analytic on C? Explain your answer.
     (b) (5 points) Let f (z) be an analytic function on a domain D. Suppose that f (z) is a purely imaginary for all z ∈ D. Show that f is constant on D.

5. (a) (5 points) Write Logz = u(r, θ) + iv(r, θ), where z = reiθ. Find the functions u(r, θ) and v(r, θ).
(b) (5 points) Verify that the functions u(r, θ) and v(r, θ) you found in part(a) satisfy the polar form of Cauchy-Riemann equations:
∂u
∂r = 1
r
∂v
∂θ
∂u
∂θ = −r ∂v
∂r .

Find all functions f: R → R that satisfy the following conditions: f(x) is a continuous and differentiable function throughout the domain of real numbers.

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Find all functions

Find all functions f: R → R that satisfy the following conditions:

  1. f(x) is a continuous and differentiable function throughout the domain of real numbers.
  2. f(0) = 1 and f(1) = e (where e is the base of the natural logarithm).
  3. f(x)f(y) = f(xy) + f(x + y) for all real numbers x and y.
  4. f'(x) = f(x) for all real numbers x.

Calculate the expected sterling receipts in three months using a money market hedge and recommend whether a forward market hedge or a money market should be used.

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Finances

1.Answer and show working out with the following maths questions:

Assume the following information

  Quoted price
Spot rate of Euro 0.80 USD
90 day forward rate of Euro 0.79 USD
90 day European interest rate 4%
90 day U.S interest rate 2.5

Given the information above, discuss if covered interest arbitrage worthwhile for an US investor who has US dollars to invest (assume the investor invests $1,000,000) Explain with calculations.

 

2. Read the following information to answer the questions d and e

  • Newton Co is a UK based multinational company that has the following expected transactions
  • One month Receipt of $240,000
  • One month Receipt of $140,000
  • Three month Receipt of  $300,000
  • Newton Co’s finance manager has collected the information below
  • Spot rate ($ per £): 1.7820 + 0.0002
  • One month forward rate ($ per £): 1.7829 + 0.0003
  • Three months forward ($ per £): 1.7846 + 0.0004
  • Money market rates for Newton Co:
  Borrowing Deposit
One year sterling interest rate 4.9% 4.6%
One year dollar interest rate 5.4% 5.1%

Assume that it is 1 April now

  1. d) Calculate the expected sterling receipts in one month and in three months using the forward market.
  2. e) Calculate the expected sterling receipts in three months using a money market hedge and recommend whether a forward market hedge or a money market should be used.

 

3.Complete the following question

  • The Gamma Co. have contracted to purchase steel from Beta Co. in the Netherlands at a cost of EUR 20,000,000
  • Delta bank in Amsterdam quotes USD/EUR as 0.8325 and GBP/USD as 1.3343
  • Alfa Bank in UK FX quotation for GBP/EUR is 1.1267
  • 1 year money market interest rates for EUR are 2% and for GBP 4% (per annum)
  • Annualised inflation in the Eurozone is 1% and in the UK it is 3.5%

Calculate the cost in GBP to Gamma Co of conducting the spot rate FX transaction with Delta Bank or Alfa Bank and which would they use?

If Relative Purchasing Power Parity held what would you expect the GBP/EUR exchange rate to be in 6 months’ time? ( base calculations on a spot GBP/EUR rate of 1.1267

 

 

4. Answer and show calculations for the following questions below :

A British company is considering investing in Malaysia due to cheaper labour rates and proximity to raw materials used in production

Capital investment in the project would be Malaysian Ringgit (MYR) 11,000,000. The project would last for 4 years producing annual net operating cash flows of MYR 4 million per year

At the end of the project the assets would be sold at an estimated value of MYR 2 million

In calculating the return the firm will use a GBP weighted average cost of capital (WACC) of 8%

The GBP/MYR exchange rate is 5.5000 and is expected to remain stable through the end of Year 1 but is expected to weaken to 5.75 by the end of year 2, 6.00 in year 3 and 6.25 in year 4

Calculate the expected GBP NVP of the USA project showing all workings out?

 The company assesses the probability of NPV calculated in the above answer to be 40% and calculates an alternative scenario forecasting an NPV return of £800,000 with a 60% probability. What is the combined weighted expected return?

Use Part 2 of the Fundamental Theorem of Calculus to evaluate the integral or explain why it does not exist: ∫ 𝑠𝑒𝑐2 𝑡 𝜋/4 0dt

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10 calculus questions

Test 5 (Ch 5)
1. Express the limit as a definite integral on the given interval:
𝑙𝑖𝑚
𝑛→∞ ∑ 𝑥𝑖 𝑠𝑖𝑛 𝑥𝑖 Δ𝑥𝑛
𝑖=1 [0, 𝜋]

2. Express the integral as a limit of the Riemann sums. Do not evaluate the limit:
∫ 𝑥
1 + 𝑥5 𝑑𝑥
8
1

3. Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function:
𝑔(𝑦) = ∫ 𝑡2 𝑠𝑖𝑛 𝑡
𝑦
2dt

4. Use Part 2 of the Fundamental Theorem of Calculus to evaluate the integral or explain why it does not exist:
∫ 𝑠𝑒𝑐2 𝑡
𝜋/4
0dt

5. Find the general indefinite integral: ∫(1 − 3𝑡)(5 + 𝑡2)𝑑𝑡

6. Evaluate the integral:
∫ (10𝑥 + 𝑒𝑥)𝑑𝑥
0
−1

7. Evaluate the indefinite integral:
∫ (𝑙𝑛 𝑥)
𝑥
3
𝑑𝑥

8. Evaluate the indefinite integral: ∫ 𝑒𝑥 √1 + 𝑒𝑥𝑑𝑥

9. Evaluate the definite integral, if it exists:
∫ (𝑥 − 1)9𝑑𝑥
2
0

10. Find most general anti–derivative of the function:
𝑓(𝑢) = 𝑢4+𝑢√𝑢
𝑢2

Let a, b and q be positive integers. Prove that a and b have the same remainder when divided by q if and only if a − b is a multiple of q.

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Discrete Math Final Spring 2023

   Yoram Sagher

Your solutions should be submitted through Canvas in .pdf in 150 minutes.

1. Find all x so that |x 1| 3|x 2| 7|x 4| 16

2. Prove that if r1 an r2 are rational numbers and r1 r2 then there are irrational numbers , x1 and x2 so that r1 x1 x2 r2

3. Let a, b and q be positive integers. Prove that a and b have the same remainder when divided by q if and only if a − b is a multiple of q

4. Prove that 447 is an irrational number

5. Let a and b be positive integers, we denote by gcda, b the largest integer that divides both a and b. Prove that if k ≤ a b is an integer, then gcda, b gcda − kb, b

6. Find integers x, y do that x 1001 y 385 gcd1001, 385.

7. Prove that if p is a prime number and x, y, z are positive integers and p divides x y z then it divides at least one of x, y, z.

8. We denote by a ∨ b the larger of the two numbers, a and b, and by a ∧ b the smaller of the two numbers. Prove that
a ∨ b a b
2 |a − b|
2 and a ∧ b a b
2 − |a − b|
2 .

9. Prove that if p 2 is a prime number then 2p−1 − 1 is a multiple of p.

10. Prove that if p is a prime number and n a positive integer then n p − n is divisible by p.

11. Prove that
∑
k1
n
k n
k n2n−1

12. Prove that
∑
k1
n
−1k k n
k 0.

13. Prove that
n − 1
k − 1 n − 1
k n
k

14. Prove that

n − 2

k − 2 2 n − 2
k − 1 n − 2
k n
k

15. Let a 0 and b 0. loga b is the number so that aloga b b Prove that  loga blogb a 1

Create a developmentally appropriate 1-week instructional math unit for a concept area of your choice, focusing on one age/grade level from pre-K to Grade 3.

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Wk 5 – Signature Assignment: Instructional Math Unit [due Mon]

Assignment Content

  1. Create a developmentally appropriate 1-week instructional math unit for a concept area of your choice, focusing on one age/grade level from pre-K to Grade 3. Use this templae to help you include all requirements.

ECH 416 Week 5 Signature Assignment.docx

Include the following components in your instructional unit: Part 1: Unit Plan Provide the following information for your unit plan:

  • Title
  • Focus (i.e., content area and grade level)
  • Objectives

Describe the development of the unit by answering the following:

  • How is the content of the unit appropriate for the age/grade level chosen?
  • How will specific learning needs be addressed throughout the unit?
  • Which current learning theories influenced instructional decisions in the unit?
  • Which strategies for children’s motivation and engagement will be utilized in the unit?
  • How will formative and summative assessment be used in the unit to plan, evaluate, and strengthen instruction and to promote continuous learning?
  • How could children’s families be involved in the unit?
  • Consider the “family connection” in each lesson.
  • Be sure to reflect families’ language differences and cultural and/or ethnic diversity.

Part 2: Resources Toolkit Create an annotated bibliography of 5 books and resources that support your unit. Follow the guidelines below:

  • Books and resources should:
  • Include 2 targeted for teacher use and 3 targeted for student and family use
  • Comprise of at least 3 multimedia materials, including web-based resources
  • Be aligned with and support unit objectives
  • Account for the diversity of your learners, including English learners, varying math proficiency levels, and students with exceptional needs
  • Each annotation should be block indented and begin with the reference. All annotations should be in alphabetical order.
  • Explicitly connect the selected book or resource to the unit in 3 to 4 sentences.

Part 3: Lesson Plans Create 3 developmentally appropriate lesson plans for your unit. Do the following in each lesson plan:

  • Create objectives that:
  • Reflect various levels of thinking, including Bloom’s taxonomy or depth of knowledge
  • Are specific, measurable, and observable
  • State what children should know and be able to do
  • Align to content standards
  • Are age/grade-level and content-area appropriate
  • List materials used in the lesson, including any necessary handouts, resources, and learning tools.
  • Identify materials that can be used to develop children’s critical-thinking and problem-solving skills.
  • Detail an instructional sequence that is based on accurate math content and concepts that sets the stage for learning through lesson delivery and guided practice. In the sequence, ensure:
  • Activities are appropriate for the needs of young children who are culturally diverse and differentiated for those with exceptional learning needs (e.g., English learners, learning disabilities, gifted/talented).
  • Activities are meaningful for young children by connecting learning to prior knowledge, to the community, and to real-world experiences.
  • Activities foster young children’s appreciation and engagement in subject matter content and align to objectives and core concepts.
  • Activities incorporate both children’s and teacher’s use of technology.
  • A closure is provided
  • Teacher wrap-up
  • Final check for understanding
  • Time for independent practice (independently or small groups)
  • Include assessments:
  • One formative assessment (check for understanding) for each lesson that aligns to the lesson objective(s)
  • One summative assessment for the unit (assesses all unit objectives)
  • Analysis of assessment data and expected next steps for instruction and student support

The options available on a particular model of a car are four interior colors, seven exterior colors, three types of seats, five types of engines, and two types of radios. How many different possibilities are available to the customer?

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MATH 3013 – Discrete Mathematics

Directions:
Show the significant steps of your work clearly for ALL problems. You may receive zero or reduced points for insufficient work.

1. Find an explicit formula for the recurrence relation

  • (a) an = −3an−1 + 11an−2 + 3an−3 + 10an−4
  • (b) an = 13an−1 − 57an−2 + 99an−3 − 54an−4

2. Solve the recurrence relation subject to the initial conditions

  • (a) wn = −10wn−1−25wn−2, and w0 = 5, w1 = −30
  • (b) −3sn = 4sn−1 − 4sn−2, and s0 = −1, s1 = 2

3. Assume that (4 − t2)Σ∞ n=2n(n − 1)antn−2 = −2Σ∞ n=0antn for all t.

  • (a) Show that the coefficients an is given by the recurrence relation an+2 = −(n − 2)an 4(n + 2) , for n ≥ 0.
  • (b) If a0 = −2, a1 = 6, find a2, a3, and a4
4. The options available on a particular model of a car are four interior colors, seven exterior colors, three types of seats, five types of engines, and two types of radios. How many different possibilities are available to the customer?

5. How many different car licensed plates can be constructed if the licenses contain four letters followed by three digits if

  • (a) repetitions are allowed?
  • (b) repetitions are not allowed?

6. How many strings can be formed by ordering the letters ”SUBBOOKKEEPER”.

7. Two dice are rolled simultaneously. How many out- comes give a sum
  • (a) of 2?
  • (b) less than 9?
  • (c) greater than or equal to 5?

8. In how many ways can we select a committee of three men and five women from a group of seven distinct men and nine distinct women.?

Each face of a cube is painted red or blue, with the colors chosen randomly and independently. The probability that the cube contains at least one vertex such that the three faces of the cube sharing that vertex are all painted the same color is p/q. Find this fraction.

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Probability

Each face of a cube is painted red or blue, with the colors chosen randomly and independently. The probability that the cube contains at least one vertex such that the three faces of the cube sharing that vertex are all painted the same color is p/q. Find this fraction.

Find the area of the region inside lemniscate r2 = 2 sin 2θ and outside the circle r = 1 . Graph the curves in one coordinate plane and find all the intersection points first.

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MATH 220 – Written Homework 4

Instructions: There are 8 problems. Try to show key steps. Correct solution without showing proper work may receive no credit.

1. (10 pts) Find the radius and the interval of convergence for following the power series. Show all needed justification.
(a)
∞
∑
k=0
k10(2x − 4)k
10k .
2. (10 pts) Compute the limit using Taylor series. lim
x→0
−x − ln(1 − x)
x2
1

3. Consider the the function f (x) = ln(1 + x)

(a) (8 pts) Find the Maclaurin series (Which is the Taylor series centered at 0). Show all details.
(b) (3 pts) Use the series for ln(1 + x) to find the Maclaurin series for the function ln(1 + x2)
(c) (9 pts) Estimate the value of the integral

∫ 0.4
0
ln(1 + x2) dx with error at most 10−4 using Taylor series.

4. Complete the following

(a) (8 pts) Find parametric equations for the line segment starting at P(−1, −3) and ending at (6, −16)
(b) (2 pts) Eliminate the parameter t of the parametric equation you found in (a) to obtain an equation in xy.

5. Consider the parametric equation x = sint, y = cost
(a) (3 pts)Find dy dx in terms of t
(b) (7 pts)Find the equation of tangent line at the point t = π/4

6. Complete the following.

(a) (4 pts) Express the point with polar coordinates P(2, 7π 4 ) in Cartesian coordinates.
(b) (4 pts) Express the point with Cartesian coordinates P(1, √3) in polar coordinates in two different ways.
(c) (7 pts) Graph the polar equation r = f (θ ) = 4 + 4 cos θ

7. (10 pts) Find the area of the region inside lemniscate r2 = 2 sin 2θ and outside the circle r = 1 . Graph the curves in one coordinate plane and find all the intersection points first.
8. Complete the following. Sketch the graph for each problems.

(a) (5 pts) Find the equation of the parabola with vertex (0, 0) symmetric about the x-axis and passes through the point (2, −3). Specify the location of the focus and the equation of directrix, and graph the parabola.
(b) (5 pts) Find the equation of the ellipse centered at the origin with its foci on the x-axis, a major axis of length 8, and a minor axis of length 6. Graph the ellipse, label the coordinates of vertices and the foci.
(c) (5 pts) Find the equation of the hyperbola centered at the origin with vertices V1 and V2 at (±5, 0) and foci F1 and F2 at (±7, 0). Find the equations of the asymptotes and graph the hyperbola.