![CALCULUS
Greatest integer function
A collection of exercises
The function [ ∙ ] ∶ ℝ ⟶ ℝ defined by [𝑥] = greatest integer not larger than 𝑥 is usually called the greatest integer
function.
1. Specific properties
Let 𝑥, 𝑦 ∈ ℝ. Show that:
1A. [𝑥 + 𝑦] = [𝑥] + [𝑦] ∨ [𝑥 + 𝑦] = [𝑥] + [𝑦] + 1.
1B. [2𝑥] = [𝑥] + [𝑥 +
2
1
].1
(Hint: use the previous result).
2. Image
2A. Find the image of [ ∙ ].
3. Graph
3A. Draw the graph of the greatest integer function over the interval [−4, 4].
4. Injectivity and Surjectivity
4A. Verify that the greatest integer function is neither injective nor onto.
5. Limits
5A. Consider 𝑎 ∈ ℤ.
What is [𝑎+] − [𝑎−]? Discuss its meaning.
(Remark: [𝑎+] = lim
𝑥⟶𝑎+
[𝑥] and [𝑎−] = lim
𝑥⟶𝑎−
[𝑥]).
5B. Use the definition of limit to show that lim
𝑥⟶+∞
[𝑥] = +∞.
Conclude that lim
𝑥⟶−∞
[𝑥] = −∞ (make 𝑦 = −𝑥 in 1A.) and therefore [ ∙ ] is not bounded.
6. Continuity
6A. Study the continuity of [ ∙ ].
7. Derivative and Antiderivative
7A. For what values is the derivative of [ ∙ ] defined?
Write the expression of that derivative.
7B. Can we use the derivative to study the monotonicity of the function [ ∙ ] on ℝ? Justify.
7C. Is it possible to find an antiderivative for [ ∙ ]? Justify.
8. Integrability
8A. Justify that the greatest integer function is integrable on any interval [0, 𝑛], 𝑛 ∈ ℕ.
8B. Prove that
n nn
t
0 2
)1(
.
1
More generally, [𝑛𝑥] =
1
0
n
k n
k
xx (Hermite’s identity)](https://image.slidesharecdn.com/gifunction-181129015640/85/Greatest-Integer-Function-A-collection-of-Calculus-problems-1-320.jpg)
![More problems…
1. Let 𝑛 ∈ ℕ and 𝑥 ∈ ℝ.
Show that:
n
x
n
x
2. Let 𝑛 ∈ ℕ.
Prove that:
[√𝑛 + 1] − [√ 𝑛] = {
1, if 𝑛 is a perfect square
0, otherwise
3. Determine whether the following improper integral converges or diverges.
∫ (−1)[𝑒 𝑥]
+∞
0
𝑑𝑥
(Remark: [ ∙ ] denotes the greatest integer function).
4. Show that [√𝑛2 + 2𝑛] = 𝑛, for 𝑛 ∈ ℕ.
5. Determine whether the following series converges or diverges.
1
2
)1(
n
n
n
6. Find all values of 𝑥 that solve the equation below:
2
1
1
2
2
xx
7. The function { ∙ } ∶ ℝ ⟶ ℝ defined by {𝑥} = 𝑥 − [𝑥] is called the fractional part function.
What are the possible values of {𝑥} + {−𝑥}?
8. Evaluate the following integral:
∫ {𝑥}2[𝑥]
5
3
9. Show that:
1
!
!
1
n n
en
(Hint: start by using Taylor’s formula and then apply Abel’s summation by parts to evaluate the series).
Miguel Fernandes](https://image.slidesharecdn.com/gifunction-181129015640/85/Greatest-Integer-Function-A-collection-of-Calculus-problems-2-320.jpg)
Greatest Integer Function - A collection of Calculus problems
- 1. CALCULUS Greatest integer function A collection of exercises The function [ ∙ ] ∶ ℝ ⟶ ℝ defined by [𝑥] = greatest integer not larger than 𝑥 is usually called the greatest integer function. 1. Specific properties Let 𝑥, 𝑦 ∈ ℝ. Show that: 1A. [𝑥 + 𝑦] = [𝑥] + [𝑦] ∨ [𝑥 + 𝑦] = [𝑥] + [𝑦] + 1. 1B. [2𝑥] = [𝑥] + [𝑥 + 2 1 ].1 (Hint: use the previous result). 2. Image 2A. Find the image of [ ∙ ]. 3. Graph 3A. Draw the graph of the greatest integer function over the interval [−4, 4]. 4. Injectivity and Surjectivity 4A. Verify that the greatest integer function is neither injective nor onto. 5. Limits 5A. Consider 𝑎 ∈ ℤ. What is [𝑎+] − [𝑎−]? Discuss its meaning. (Remark: [𝑎+] = lim 𝑥⟶𝑎+ [𝑥] and [𝑎−] = lim 𝑥⟶𝑎− [𝑥]). 5B. Use the definition of limit to show that lim 𝑥⟶+∞ [𝑥] = +∞. Conclude that lim 𝑥⟶−∞ [𝑥] = −∞ (make 𝑦 = −𝑥 in 1A.) and therefore [ ∙ ] is not bounded. 6. Continuity 6A. Study the continuity of [ ∙ ]. 7. Derivative and Antiderivative 7A. For what values is the derivative of [ ∙ ] defined? Write the expression of that derivative. 7B. Can we use the derivative to study the monotonicity of the function [ ∙ ] on ℝ? Justify. 7C. Is it possible to find an antiderivative for [ ∙ ]? Justify. 8. Integrability 8A. Justify that the greatest integer function is integrable on any interval [0, 𝑛], 𝑛 ∈ ℕ. 8B. Prove that n nn t 0 2 )1( . 1 More generally, [𝑛𝑥] = 1 0 n k n k xx (Hermite’s identity)
- 2. More problems… 1. Let 𝑛 ∈ ℕ and 𝑥 ∈ ℝ. Show that: n x n x 2. Let 𝑛 ∈ ℕ. Prove that: [√𝑛 + 1] − [√ 𝑛] = { 1, if 𝑛 is a perfect square 0, otherwise 3. Determine whether the following improper integral converges or diverges. ∫ (−1)[𝑒 𝑥] +∞ 0 𝑑𝑥 (Remark: [ ∙ ] denotes the greatest integer function). 4. Show that [√𝑛2 + 2𝑛] = 𝑛, for 𝑛 ∈ ℕ. 5. Determine whether the following series converges or diverges. 1 2 )1( n n n 6. Find all values of 𝑥 that solve the equation below: 2 1 1 2 2 xx 7. The function { ∙ } ∶ ℝ ⟶ ℝ defined by {𝑥} = 𝑥 − [𝑥] is called the fractional part function. What are the possible values of {𝑥} + {−𝑥}? 8. Evaluate the following integral: ∫ {𝑥}2[𝑥] 5 3 9. Show that: 1 ! ! 1 n n en (Hint: start by using Taylor’s formula and then apply Abel’s summation by parts to evaluate the series). Miguel Fernandes