Ch. 1 - Functions

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

Hass 15th Edition

Ch. 1 - Functions

Problema 1

Capitolo 1, Problema 1

Functions and Graphs

Express the area and circumference of a circle as functions of the circle’s radius. Then express the area as a function of the circumference.

Guida verificata passo dopo passo

Start by recalling the formulas for the area and circumference of a circle. The area \( A \) of a circle is given by \( A = \pi r^2 \), where \( r \) is the radius. The circumference \( C \) is given by \( C = 2\pi r \).

Express the area \( A \) as a function of the radius \( r \). This is already given by the formula \( A(r) = \pi r^2 \).

Express the circumference \( C \) as a function of the radius \( r \). This is given by the formula \( C(r) = 2\pi r \).

To express the area as a function of the circumference, first solve the circumference formula for \( r \). From \( C = 2\pi r \), we get \( r = \frac{C}{2\pi} \).

Substitute \( r = \frac{C}{2\pi} \) into the area formula \( A = \pi r^2 \) to express the area as a function of the circumference: \( A(C) = \pi \left(\frac{C}{2\pi}\right)^2 \). Simplify this expression to complete the function.

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Area of a Circle

The area of a circle is calculated using the formula A = πr², where A represents the area and r is the radius. This formula shows that the area is directly proportional to the square of the radius, meaning that as the radius increases, the area increases quadratically.

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Percorso guidato

05:59

Estimating the Area Under a Curve with Right Endpoints & Midpoint

Circumference of a Circle

The circumference of a circle is given by the formula C = 2πr, where C is the circumference and r is the radius. This formula indicates that the circumference is directly proportional to the radius, meaning that if the radius doubles, the circumference also doubles.

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Evaluate Composite Functions - Values on Unit Circle

Function Relationships

In this context, expressing the area as a function of the circumference involves manipulating the formulas for area and circumference. By substituting the expression for radius from the circumference formula into the area formula, we can derive a new function A(C) that relates area directly to circumference.

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Percorso guidato

6:22

Graphs of Secant and Cosecant Functions

Pratica correlata

Domanda del libro di testo

Functions

In Exercises 1–6, find the domain and range of each function.

f(x) = 1 + x²

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Domanda del libro di testo

[Technology Exercise]

You want to make an 80° angle by marking an arc on the perimeter of a 12-in.-diameter disk and drawing lines from the ends of the arc to the disk’s center. To the nearest tenth of an inch, how long should the arc be?

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Domanda del libro di testo

Increasing and Decreasing Functions

Graph the functions in Exercises 37–46. What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing.

y = x³/8

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Domanda del libro di testo

In Exercises 9–16, determine whether the function is even, odd, or neither.

𝔂 = x² + 1

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Domanda del libro di testo

In Exercises 39–42, express the given quantity in terms of sin x and cos x.

sin (2π − x)

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