Domanda del libro di testo
Piecewise-Defined Functions
Find a formula for each function graphed in Exercises 29–32.
b. <IMAGE>
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Ch. 1 - Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 1, Problema 1.1.33b
The Greatest and Least Integer Functions
For what values of x is
b. ⌈x⌉ = 0
Guida verificata passo dopo passo1
Understand the ceiling function ⌈x⌉, which rounds a number up to the nearest integer. For example, ⌈2.3⌉ = 3 and ⌈-1.7⌉ = -1.
To find the values of x for which ⌈x⌉ = 0, consider the definition: ⌈x⌉ is the smallest integer greater than or equal to x.
Since ⌈x⌉ = 0, x must be less than or equal to 0 but greater than -1, because ⌈x⌉ rounds up to the nearest integer.
Therefore, the values of x that satisfy ⌈x⌉ = 0 are those in the interval (-1, 0].
Express the solution in interval notation: x ∈ (-1, 0].
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Greatest Integer Function (Ceiling Function)
The greatest integer function, denoted as ⌈x⌉, returns the smallest integer that is greater than or equal to x. For example, ⌈2.3⌉ equals 3, while ⌈-1.5⌉ equals -1. This function is crucial for understanding how to manipulate and solve equations involving integer constraints.
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6:04
Intro to Rational Functions
Understanding Zero in the Context of the Ceiling Function
To solve the equation ⌈x⌉ = 0, we need to determine the range of x values that yield a ceiling of zero. Since the ceiling function rounds up to the nearest integer, this means x must be in the interval [-1, 0). Thus, any value of x within this range will satisfy the equation.
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Percorso guidato
06:37Average Value of a Function
Intervals and Inequalities
Understanding intervals and inequalities is essential for solving equations involving functions like the ceiling function. In this case, the solution involves identifying the interval where the function holds true, which is critical for determining valid x values that meet the specified condition.
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02:59Finding Area Between Curves that Cross on the Interval
Pratica correlata
Domanda del libro di testo
Combining Functions
Assume that f is an even function, g is an odd function, and both f and g are defined on the entire real line (−∞,∞). Which of the following (where defined) are even? odd?
b. f/g
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Domanda del libro di testo
Composition of Functions
Copy and complete the following table.
b. <IMAGE>
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Domanda del libro di testo
Piecewise-Defined Functions
Find a formula for each function graphed in Exercises 29–32.
b. <IMAGE>
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Domanda del libro di testo
Functions
In Exercises 7 and 8, which of the graphs are graphs of functions of x, and which are not? Give reasons for your answers.
b. <IMAGE>
369
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Domanda del libro di testo
Piecewise-Defined Functions
Find a formula for each function graphed in Exercises 29–32.
b. <IMAGE>
302
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