Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
Related questions
Question
a. Use the sum formulas to find the numerical value.
| 50 | (4k2 + 3) |
| k = 1 |
b.
Use the sum formulas to express the following without the summation symbol.
| n |
|
||||||||
| i = 1 |
![**Problem Statement**
Use the sum formulas to find the numerical value.
\[
\sum_{k=1}^{50} (4k^2 + 3)
\]
**Explanation**
To solve the problem, follow these general steps:
1. **Understand the Expression:**
- This is a summation expression, where \(k\) takes on integer values from 1 to 50.
- For each value of \(k\), compute \(4k^2 + 3\).
2. **Apply Sum Formulas:**
- The expression can be separated into two sums: \(\sum_{k=1}^{50} 4k^2\) and \(\sum_{k=1}^{50} 3\).
- Use the sum formula for squares: \(\sum_{k=1}^{n} k^2 = \frac{n(n + 1)(2n + 1)}{6}\).
- Use the sum formula for constants: \(\sum_{k=1}^{n} c = cn\), where \(c\) is a constant.
3. **Calculation:**
- Calculate \(\sum_{k=1}^{50} 4k^2 = 4 \sum_{k=1}^{50} k^2\).
- Calculate \(\sum_{k=1}^{50} 3 = 3 \times 50\).
4. **Combine the Results:**
- Add the results of the two sums to get the final numerical value.
This mathematical approach leads to the solution of the given summation problem.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff3df1ae4-b332-4fdc-9f25-7646d53876c7%2Fed714f94-aaba-408b-9913-8d52dc94935b%2Fufvr65r_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement**
Use the sum formulas to find the numerical value.
\[
\sum_{k=1}^{50} (4k^2 + 3)
\]
**Explanation**
To solve the problem, follow these general steps:
1. **Understand the Expression:**
- This is a summation expression, where \(k\) takes on integer values from 1 to 50.
- For each value of \(k\), compute \(4k^2 + 3\).
2. **Apply Sum Formulas:**
- The expression can be separated into two sums: \(\sum_{k=1}^{50} 4k^2\) and \(\sum_{k=1}^{50} 3\).
- Use the sum formula for squares: \(\sum_{k=1}^{n} k^2 = \frac{n(n + 1)(2n + 1)}{6}\).
- Use the sum formula for constants: \(\sum_{k=1}^{n} c = cn\), where \(c\) is a constant.
3. **Calculation:**
- Calculate \(\sum_{k=1}^{50} 4k^2 = 4 \sum_{k=1}^{50} k^2\).
- Calculate \(\sum_{k=1}^{50} 3 = 3 \times 50\).
4. **Combine the Results:**
- Add the results of the two sums to get the final numerical value.
This mathematical approach leads to the solution of the given summation problem.
![**Expression Simplification Using Sum Formulas**
**Objective:**
Simplify the given expression without using the summation symbol.
**Expression:**
\[
\sum_{i=1}^{n} \left(1 - \frac{i^2}{n^2}\right) \left(\frac{3}{n}\right)
\]
**Approach:**
1. **Expand the Expression:**
Expand the terms within the summation.
2. **Apply Sum Formulas:**
Use relevant sum formulas to simplify.
3. **Final Expression:**
Simplify to remove the summation symbol.
A detailed explanation involves multiplying and collecting like terms, then using known sum formulas such as the sum of integers or sum of squares to simplify fully.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff3df1ae4-b332-4fdc-9f25-7646d53876c7%2Fed714f94-aaba-408b-9913-8d52dc94935b%2Feu02qcp_processed.png&w=3840&q=75)
Transcribed Image Text:**Expression Simplification Using Sum Formulas**
**Objective:**
Simplify the given expression without using the summation symbol.
**Expression:**
\[
\sum_{i=1}^{n} \left(1 - \frac{i^2}{n^2}\right) \left(\frac{3}{n}\right)
\]
**Approach:**
1. **Expand the Expression:**
Expand the terms within the summation.
2. **Apply Sum Formulas:**
Use relevant sum formulas to simplify.
3. **Final Expression:**
Simplify to remove the summation symbol.
A detailed explanation involves multiplying and collecting like terms, then using known sum formulas such as the sum of integers or sum of squares to simplify fully.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 4 steps with 4 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning