ECN115. Problem Set I.
数学exam代考 where the symbol · denotes the product of the two functions, the symbol ◦ denotes the composition of the two functions, and x ∈ D.
Problem A1.
Assume that functions f , g, h : D → D, where D ⊆ R, are differen- tiable for all x ∈ D. Using the product rule (Theorem L4.5 from Lecture 4) and the chain rule (Theorem L4.6 from Lecture 4), find the following derivative:
where the symbol · denotes the product of the two functions, the symbol ◦ denotes the composition of the two functions, and x ∈ D. In other words, express the derivative of the function (f · g) ◦ h at an arbitrary point of its domain in terms of values of functions f , g, h and their derivatives.
Problem A2. Use your formula from Problem A1 for D = (0, ∞), 
to calculate the derivative of the following function:
at point x > 0. Alternatively, you can use a direct calculation. Try to simplify your answer if possible.
Problem B. In this problem, we are going to derive the formula for the Bino- mial coefficients. Recall that the Binomial coefficients are coefficients of the terms 
in the expression
in the expression
B1. Prove that for any natural n and natural k such that 1 ≤ k ≤ n, the following formula holds:
Hint: Use the fact that
(x + y)n+1 = (x + y)n · (x + y)
Substitute the Binomial formula in the right hand side and in the left hand side of the above equation and consider the coefficient of the term xn+1−k · yk .
B2. Argue why
B3. Use the principle of mathematic induction and your previous calculations to prove that
Problem C1. Consider a numerical function f : D → R, and suppose that the setsX1, X2 are such that X1 ⊆ D, X2 ⊆ D. Prove the following proposition:
where for any two real numbers a, b, the term max{a, b} denotes the maximum of these two real numbers.
Problem C2. Consider a numerical function f : D → R, and suppose that the sets X1, X2, …, Xn are such that for any i ∈ {1, 2, …, n}, there exists 
Prove that
there exists
and moreover,
where
X1 ∪ X2 ∪ … ∪ Xn = {x ∈ R | there is at least one i ∈ {1, 2, …, n} such that x ∈ Xi }
Thus, X1 ∪ X2 ∪ … ∪ Xn is the union of the sets Xi with i = 1, 2, …, n.
Problem C3 (Bonus problem. You can get the full mark for the Problem Set I without solving this problem. However, if you solve it, you can increase your mark in case if it is not a maximum possible mark). Suppose that we want to generalise results from problems C1-C2 to the case when the number of sets Xi is infinite. For instance, suppose that i takes the values of all natural numbers. The notion of the union of all these sets is well-defined:
Y = X1 ∪ X2 ∪ … ∪ Xn ∪ … = {x ∈ R | there is at least one i ∈ N such that x ∈ Xi }
Thus, the union of all sets Xi (which we denote by Y for brevity) is the set of real numbers x such that each x is an element of at least one of the sets Xi for i ∈ N. Is the following proposition true?
Proposition C3: if for all i ∈ N, there exists 
, then there exists 
If Proposition C3 is true, prove it. If not, find a counterexample (that is, sets Xi ,i ∈ N such that exists for each i, but does not exist).
Problem D1 (you can skip this problem if you solve problem D2—just substi- tute the corresponding values of the parameter b). Consider a polynomial func-tion f : R → R given by f (x) = 2x3 + 3x2 − 12x. Find the maximum of the function f and the set of its maximisers over the following sets X: 数学exam代考
- X = [−3,3].
- X = [−3,2].
- X = [−3,−2.5].
- X = [−3, −3] = {−3}.
Problem D2. Consider a polynomial function f : R → R given by f (x) = 2x3 +3x2 − 12x. Find the maximum of the function f and the set of its maximisers over the set X = [−3, b] where b ∈ [−3, 3]. Your answer should depend on the parameter b. Check that it coincides with your results for the problem D1 for b ∈ {3, 2, −2.5, −3}.
A small hint: suppose that f (xo) = yo and we want to find another x s xo such that f (x) = yo. Consider a polynomial g(x) = f (x) − yo; that is, g(x) = 2x3 + 3x2 − 12x − yo. We are interested in finding its roots (values of variable x at which g(x) = 0) other than xo.
At the first glance, we need to solve a cubic equation, which is much more difficult than solving a quadratic equation. However, we have an additional piece of informationabout g(x); namely, xo is its root. It follows that g(x) = (x − xo) · h(x), where h(x) is some quadratic polynomial, and the roots of g(x) that differ from xo should be the roots of the polynomial h(x) as well. Therefore, we can divide polynomial g(x) by the polynomial x − xo to obtain h(x).
Problem D3. Consider the function f from problems D1-D2. Consider function
M : [−3, 3] → R given by
Is it true that the function M is continuous at all points of the interval (−3, 3)? Try to make a formal argument.
Problem E1. Calculate the derivatives of the following functions at an arbitrary point x of their domains (try to simplify your answer when possible):
a) f (x) = 7.
b) g(x) = 7x −9.
c) h(x) = 5x2 − x + 1. d) u(x) = (x − 5)3.
e) v(x) = (x − 5)3 (x + 2)4.
Problem E2. For functions f , g, h, u, v, w, q, considered in the Problem E1, what are the values of their derivatives at point x = 5? At point x = −2?
Problem F. Consider function f : R → R given by 数学exam代考
F1. Is the function f continuous (i.e. continuous at any point x of its domain R)? Does your answer depend on the parameter b?
F2. At which points is the function f differentiable? Does your answer depend on the parameter b?
F3. Find the derivative of the function f at an arbitrary point where it is differen- tiable. Your answer may depend on the parameter b. You can express your answer in terms of formulas that depend on whether x < 1, x > 1, etc. (for instance, we describe the function f itself in this way).
Hint. To answer questions F1,F2,F3, it is useful to consider auxiliary functions g, h, u :
R → R given by
You can first answer questions F1,F2,F3 for functions g, h, u and use your results to analyse function f . In particularly, recall that whether function is continuous (differ- entiable), or not, and the value of its derivative at point x depends ONLY on the values of the function on an (arbitrary) small open interval around point x.
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