Ch. 1 - Functions

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 1 - Functions

Problema 1.3.65

Capitolo 1, Problema 1.3.65

Refer to the given figure. Write the radius r of the circle in terms of α and θ.

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Step 1: Begin by understanding the relationship between the angle α, angle θ, and the radius r of the circle. Typically, these angles are part of a geometric configuration involving circles, such as sectors or segments.

Step 2: Consider the geometric properties of the circle. If α and θ are angles subtended by arcs or sectors, they might relate to the radius through trigonometric identities or geometric formulas.

Step 3: Use trigonometric identities or geometric relationships to express r in terms of α and θ. For example, if α and θ are angles in a right triangle formed by the radius, you might use sine, cosine, or tangent functions.

Step 4: If the figure involves a sector of the circle, recall that the arc length or area of the sector can be used to relate the radius to the angles. The formula for arc length is L = rθ, where θ is in radians.

Step 5: Combine the relationships and formulas derived from the figure to express the radius r solely in terms of α and θ. Ensure that any assumptions made about the figure are consistent with the given angles and their geometric context.

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Circle Geometry

Understanding the properties of circles is essential for solving problems involving their dimensions. The radius is a key feature, defined as the distance from the center of the circle to any point on its circumference. In terms of angles, the relationship between the radius and angles like α and θ can often be explored using trigonometric functions.

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Changing Geometries

Trigonometric Functions

Trigonometric functions such as sine, cosine, and tangent relate the angles of a triangle to the ratios of its sides. In the context of a circle, these functions can help express the radius in terms of angles. For example, if α and θ represent angles in a right triangle inscribed in the circle, the radius can be derived using these functions.

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Introduction to Trigonometric Functions

Coordinate Systems

Coordinate systems, particularly polar coordinates, are often used to describe points in relation to a circle. In polar coordinates, a point is defined by its distance from the origin (the radius) and the angle from the positive x-axis. Understanding how to convert between polar and Cartesian coordinates is crucial for expressing the radius in terms of angles like α and θ.

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Critical Points Example 2

Pratica correlata

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[Technology Exercise]

a. Graph the functions f(x) = x/2 and g(x) = 1 + (4/x) together to identify the values of x for which

x/2 > 1 + 4/x

b. Confirm your findings in part (a) algebraically.

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Finding a Viewing Window

In Exercises 5–30, find an appropriate graphing software viewing window for the given function and use it to display that function’s graph. The window should give a picture of the overall behavior of the function. There is more than one choice, but incorrect choices can miss important aspects of the function.

y = x + (1/10) sin 30x

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Shifting Graphs

The accompanying figure shows the graph of y = −x² shifted to two new positions. Write equations for the new graphs.

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Using the Half-Angle Formulas

Find the function values in Exercises 47–50.