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The Greatest and Least Integer Functions
Does ⌊x⌋ = ⌈x⌉ for all real x? Give reasons for your answer.
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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 1.3.42
In Exercises 39–42, express the given quantity in terms of sin x and cos x.
cos (3π/2 + x)
Guida verificata passo dopo passo1
Recognize that the expression cos(3π/2 + x) involves a trigonometric identity for cosine of a sum, which is cos(a + b) = cos(a)cos(b) - sin(a)sin(b).
Identify the values of a and b in the expression: here, a = 3π/2 and b = x.
Substitute these values into the identity: cos(3π/2 + x) = cos(3π/2)cos(x) - sin(3π/2)sin(x).
Recall the exact trigonometric values: cos(3π/2) = 0 and sin(3π/2) = -1.
Substitute these values back into the expression: cos(3π/2 + x) = 0 * cos(x) - (-1) * sin(x), which simplifies to sin(x).
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Trigonometric Identities
Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. They are essential for simplifying expressions and solving equations in trigonometry. Key identities include the Pythagorean identity, angle sum and difference identities, and double angle formulas, which help express trigonometric functions in various forms.
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Verifying Trig Equations as Identities
Angle Addition Formulas
Angle addition formulas are specific trigonometric identities that express the sine and cosine of the sum or difference of two angles. For example, the cosine of the sum of two angles is given by cos(a + b) = cos(a)cos(b) - sin(a)sin(b). These formulas are crucial for transforming expressions involving sums of angles into simpler forms using basic trigonometric functions.
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Percorso guidato
05:56Additional Rules for Indefinite Integrals
Unit Circle
The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It is a fundamental tool in trigonometry, as it provides a geometric interpretation of the sine and cosine functions. The coordinates of points on the unit circle correspond to the values of cosine and sine for various angles, making it easier to evaluate trigonometric functions for any angle.
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5:10Evaluate Composite Functions - Values on Unit Circle
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Even and Odd Functions
In Exercises 47–62, say whether the function is even, odd, or neither. Give reasons for your answer.
g(x) = x⁴ + 3x² − 1
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In Exercises 3 and 4, find the domains of f, g, f/g and g/f.
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In Exercises 9–16, determine whether the function is even, odd, or neither.
𝔂 = x cos x
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Finding Formulas for Functions
Express the side length of a square as a function of the length d of the square’s diagonal. Then express the area as a function of the diagonal length.
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