Textbook Question
Even and odd at the origin
a. If ƒ(0) is defined and ƒ is an even function, is it necessarily true that ƒ(0) = 0? Explain.
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Ch. 1 - Functions
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 1, Problem 89c
{Use of Tech} Sum of squared integers Let T (n) = 1² + 2² + ... + n², where n is a positive integer. It can be shown that T (n) = n (n + 1) (2n + 1) / 8
c. What is the least value of n for which T(n) > 1000?
Verified step by step guidance1
Step 1: Understand the problem. We need to find the smallest positive integer n such that the sum of the squares of the first n integers, T(n), is greater than 1000. The formula for T(n) is given as T(n) = \( \frac{n(n + 1)(2n + 1)}{6} \).
Step 2: Set up the inequality. We want T(n) > 1000, so substitute the formula into the inequality: \( \frac{n(n + 1)(2n + 1)}{6} > 1000 \).
Step 3: Clear the fraction by multiplying both sides of the inequality by 6 to get: n(n + 1)(2n + 1) > 6000.
Step 4: Solve the inequality. This involves finding the smallest integer n that satisfies the inequality. You can start by testing integer values of n to see when the inequality holds true.
Step 5: Verify your solution. Once you find a candidate for n, substitute it back into the original formula for T(n) to ensure that T(n) > 1000.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Sum of Squares Formula
The sum of the squares of the first n positive integers is given by the formula T(n) = n(n + 1)(2n + 1) / 6. This formula allows for the efficient calculation of the sum without needing to manually add each squared integer, which is particularly useful for large values of n.
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Introduction to Riemann Sums
Inequalities
An inequality is a mathematical statement that compares two expressions, indicating that one is greater than, less than, or not equal to the other. In this context, solving T(n) > 1000 requires finding the smallest integer n such that the sum of squares exceeds 1000, which involves manipulating the inequality to isolate n.
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Integer Solutions
In many mathematical problems, especially those involving sums or sequences, we often seek integer solutions. For this question, we need to find the least positive integer n that satisfies the inequality, which may involve testing successive integer values or using numerical methods to find the threshold where T(n) first exceeds 1000.
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Related Practice
Textbook Question
{Use of Tech} Polynomial calculations
Find a polynomial ƒ that satisfies the following properties. (Hint: Determine the degree of ƒ; then substitute a polynomial of that degree and solve for its coefficients.)
ƒ(ƒ(x)) = x⁴ - 12x² + 30
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Textbook Question
Composition of even and odd functions from tables Assume ƒ is an even function, g is an odd function, and both are defined at 0. Use the (incomplete) table to evaluate the given compositions. <IMAGE>
g. ƒ (g(g(-2)))
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Textbook Question
Composition of even and odd functions from tables Assume ƒ is an even function, g is an odd function, and both are defined at 0. Use the (incomplete) table to evaluate the given compositions. <IMAGE>
i. g(g(g(-1)))
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Textbook Question
{Use of Tech} Polynomial calculations
Find a polynomial ƒ that satisfies the following properties. (Hint: Determine the degree of ƒ; then substitute a polynomial of that degree and solve for its coefficients.)
ƒ(ƒ(x)) = 9x - 8
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Textbook Question
Express in terms of using the inverse sine, inverse tangent, and inverse secant functions. <IMAGE>
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