Problem 1. [1 point each] Compute:

(a) ∫_{1}^{3}(3x^{2} − 5x)dx

(b) ∫_{1}^{5} [\frac{1}{2}x + 1] dx

Problem 1. Continued.

(c) ∫_{0}^{4} |(x − 1)(x − 3)|dx

Midterm : November 14, 2013 3

Mathematical induction

Problem 2. Do either part (a) or part (b)

(a) Let c be any fixed natural number. Prove that 1^{c} + 2^{c} + ⋯ + (n − 1)^{c} < \frac{n^{c+1}}{c + 1} for every n ∈ ℕ.

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Problem 2. Continued.

(b) Let F_{k} denote the k-th Fibonacci number, and let c be any fixed natural number. Prove that F_{n+c} = F_{c}F_{n+1} + F_{c−1}F_{n} for every n ∈ ℕ.

Midterm : November 14, 2013 5

Integration

Problem 3.

(a) Let s : [a, b] → ℝ be a step function. Define ∫_{a}^{b} s.

(b) Let f : [a, b] → ℝ be any bounded function. Define the statement “f is integrable”, and the expression ∫_{a}^{b} f.

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Short Answer: 2 points each

Problem 4.

(a) Find a rational number and an irrational number in the interval (0, 1).

(b) Compute ∫_{0}^{4} |(x − 1)(x − 3)|dx

(c) Compute \underline{I}(f) and \overline{I}(f) for f : [0, 1] → ℝ given by f(x) = \begin{cases}1 + x & if x ∈ ℚ \\ 2 − x & if x ∉ ℚ \end{cases}.

(d) Compute ∫_{1}^{5} [\frac{1}{2}x + 1] dx

(b) ∫_{1}^{5} [\frac{1}{2}x + 1] dx

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Problem 6. True or false:

(a) For all functions f, g, h : ℝ → ℝ, f ∘ (g + h) = f ∘ g + f ∘ h.

(b) The function given by

f(x) = \frac{2x + 1}{x^{2} + 2x + 3}

is neither even nor odd, but it can be written as the sum of an even and an odd function:

\frac{2x + 1}{x^{2} + 2x + 3} = \frac{3 − 3x^{2}}{x^{4} + 2x^{2} + 9} + \frac{2x (x^{2} + 2)}{x^{4} + 2x^{2} + 9}.

Here’s the statement that you are to decide is either true or false: Every function f : ℝ → ℝ can be written as the sum of an even and an odd function.

(c) If s : [a, b] → ℝ is a step function, then s([a, b]) is a finite set.

(d) Let π_{1} : ℝ × ℝ → ℝ and π_{2} : ℝ × ℝ → ℝ be the canonical projections. Let A be the

ordinate set of a nonegative function f : [a, b] → ℝ. Then

π_{1}(A) = the domain of f and π_{2}(A) = the range of f.

(e) If s : [a, b] → ℝ is a step function, then for any y ∈ ℝ, the sets s^{−1}({y}) is either empty or an interval.

(f) For any a, b, c, d ∈ ℝ, (ac + bd)^{2} ≤ (a^{2} + c^{2})(b^{2} + d^{2}).

(g) If A and B satisfy

\frac{17 − 6n}{3} ≤ A ≤ B ≤ \frac{17 + 6n}{3}

for every n ∈ ℕ, then A = B

Midterm : November 14, 2013 10

Problem 7. True or false:

(a) For all functions f, g, h : ℝ → ℝ, f ∘ (g + h) = f ∘ g + f ∘ h.

(b) If x is any rational number and y is any irrational number, then x + y is irrational.

(c) If f : A → ℝ and g : B → ℝ then the domain of \frac{f}{g} is A ∩ B ∩ (ℝ \ g^{−1}(0)).

(d) The function given by

f(x) = \frac{2x + 1}{x^{2} + 2x + 3}

is neither even nor odd, but it can be written as the sum of an even and an odd function:

\frac{2x + 1}{x^{2} + 2x + 3} = \frac{3 − 3x^{2}}{x^{4} + 2x^{2} + 9} + \frac{2x (x^{2} + 2)}{x^{4} + 2x^{2} + 9}.

Here’s the statement that you are to decide is either true or false: Every function f : ℝ → ℝ can be written as the sum of an even and an odd function.

(e) If possible, find a left inverse and a right inverse of the function f : ℕ → ℕ be defined by f(n) = n + 2.

(f) Find a rational number and an irrational number in the interval (0, \frac{1}{2}).

(g) Let f : ℚ → ℚ be given by f(a) = 5a + 2. Then g : ℚ → ℚ given by g(b) = \frac{1}{5}(b − 2) satisfies f ∘ g = g ∘ f = id_{ℚ}.

(h) Find a rational number and an irrational number in the interval (0.99, 1).

(i) If s : [a, b] → ℝ is a step function, then s([a, b]) is a finite set.

(j) Let π_{1} : ℝ × ℝ → ℝ and π_{2} : ℝ × ℝ → ℝ be the canonical projections. Let A be the

ordinate set of a nonegative function f : [a, b] → ℝ. Then

π_{1}(A) = the domain of f and π_{2}(A) = the range of f.

(k) If s : [a, b] → ℝ is a step function, then for any y ∈ ℝ, the sets s^{−1}({y}) is an interval.

(l) ∫_{0}^{4} |(x − 1)(x − 3)|dx = 4.

(m) For any a, b, c, d ∈ ℝ, (ac + bd)^{2} ≤ (a^{2} + c^{2})(b^{2} + d^{2}).

(n) If A and B satisfy

\frac{17 − 6n}{3} ≤ A ≤ B ≤ \frac{17 + 6n}{3}

for every n ∈ ℕ, then A = B

(o) If A is any nonempty subset of real numbers bounded below, then there is an element z ∈ A with z ≤ a for all a ∈ A.

(p) If A is any nonempty subset of natural numbers, then there is an element z ∈ A with z ≤ a for all a ∈ A.

(q) Let g : [0, 1] → ℝ defined by

g(x) = \begin{cases}x & if x = \frac{1}{n} for some n ∈ ℕ, \\ 1 & otherwise. \end{cases}

Then g is integrable and ∫_{0}^{1} g = 1.

(r) Let g : [0, 1] → ℝ defined by as above. There are step function s and t with ∫_{0}^{1} s = \underline{I}(g) and ∫_{0}^{1} t = \overline{I}(g).

(s) Let h(x) = [\sqrt{x}]. Then h is a step function and ∫_{1}^{16} h = 34.

(t) If f is decreasing on [a, b] then f is integrable and (b − a)f(b) ≤ ∫_{a}^{b} f ≤ (b − a)f(a).

Problem 8. Consider the function f : [0, 1] → ℝ whose graph consists of straight lines connecting

(1, 0) and (\frac{1}{2}, 1)
(\frac{1}{2}, 1) and (\frac{1}{3}, 0)
(\frac{1}{3}, 0) and (\frac{1}{4}, 1)
(\frac{1}{4}, 1) and (\frac{1}{5}, 0)
⋮

The value of f at zero is immaterial, say f(0) = 0 if you like. One could give a formula for f on each interval (\frac{1}{n}, \frac{1}{n+1}) but a picture is worth a thousand words:

(a) [1 point] Is f is piecewise monotonic on [0, 1]?

(b) [1 point] Is f integrable on [0, 1]?

(c) [bonus point] Determine \underline{I}(f) and \overline{I}(f).

Problem 9. Bonus. Suppose that instead of the usual definition, we had defined the integral of a step function s by the formula

∫_{a}^{b} s = ∑_{i=1}^{n} (s_{i}^{2}) (x_{i} − x_{i−1})

where {x_{i}} is a partition of [a, b] for which s takes the constant value s_{i} on the open subinterval (x_{i−1}, x_{i}). Then, a different theory of integration would result, with possibly different properties. Of the following two familiar properties of the integral, only one is true under the modified definition. For two bonus points, prove the claim that is true and disprove the claim that is false:

Claim. For all step functions s and all constants c ∈ ℝ, ∫_{a}^{b} cs = c ∫_{a}^{b} s.

Claim. For all step functions s and all a, b, c ∈ ℝ with a < b < c, ∫_{a}^{c} s = ∫_{a}^{b} s + ∫_{b}^{c} s.

EXAM

Midterm Math 157 November 14, 2013

• Show your work, but not your scratchwork. Neatness counts.

• Problems 1 and 2 are worth five points each, and each part of

problem 3 is worth one point for a total of 20 possible points. Success!