Reference no: EM132827820

CALCULUS

A. Complete one of the following applied projects or a similar project in a topic/area that you are most interested in. These projects can be done individually or in a team of two. Teams of two completing one the textbook projects are required to ask and answer an additional question related to the topic of their project. The deliverables for this project consist in 1) a written report in scientific style, and 2) an oral presentation.

Question 1. Building a Better Roller Coaster (p.182) ← requires CAS, plotting software

Question 2. Where Should a Pilot Start Descent? (p.208) ← requires plotting software

Question 3. Controlling Red Blood Cell Loss During Surgery (p.244)

Question 4. The Calculus of Rainbows (p.285)

Question 5. The Shape of a Can (p.343) ← requires plotting software

Question 6. Planes and Birds: Minimizing Energy (p.344)

Question 7. The Gini Index (p.436) ← requires CAS and reading ahead - feel free to contact me with questions

B. Solve any three of the "FQ" questions below. This is an individual project. The deliverable for this project is a written report.

FQ1. Find all continuous functions f : R → R such that f (0) = 0 and for all x ∈ R

f (2x) ≥ x + f (x) and f (3x) ≤ 2x + f (x).

FQ2. Show that if f : (0, ∞) → R is continuous and f (2x) = f (3^x) for all x ∈ R, then f is a constant.

FQ3. Let p(x) be a polynomial function of degree n with simple roots x₁, . . . , x_n. Show that Σ_i=1ⁿp"(x_i)/p'(x_i) = 0.

FQ4. Assume f : [a, b] → R is (n + 1)-times differentiable on (a, b), n > 2 and there exits x₀ ∈ (a, b) such that f (x₀) = f'(x₀) = f"(x₀) = 0 and f⁽ⁿ⁺¹⁾(x0) ≠ 0. Show that

lim_x→x0f(x)/f'(x). (f"(x)/f"(x) + 1/x-x0) = 1.

FQ5. Let p(x) = a_mx^m + a_m-1x^m-1 +1 .... + a₁x + a₀, a_i > 0, i = 1, . . . , m and let A_n and G_n be the arithmetic and the geometric mean of the numbers p(1), p(2), . . . , p(n). Show that

lim  A_n/G_n = e^m/(m + 1).
n→∞

FQ6. If p is a prime number and a > 0 is an integer divisible by p, then 1 + Σ^p-1_i=1(a + i)^p-1 is divisible by p.

FQ7. Let f : [a, b] → [a, b] such that f (f (x)) = x and f (x) ≠ x for all x ∈ [a, b]. Show that f is discontinuous at infinitely many points. (Hint: try proving it by contradiction)

Attachment:- Selected Problems Calculus.rar