﻿ ﻿POLAR FUNCTIONS - Functions - Calculus AB and Calculus BC

## CHAPTER 1 Functions

### G. POLAR FUNCTIONS

Polar coordinates of the form (r, ) identify the location of a point by specifying , an angle of rotation from the positive x-axis, and r, a distance from the origin, as shown in Figure N1–14. FIGURE N1–14

A polar function defines a curve with an equation of the form r = f ( ). Some common polar functions include: EXAMPLE 17

Consider the polar function r = 2 + 4 sin .

(a) For what values of in the interval [0,2π] does the curve pass through the origin?

(b) For what value of in the interval [0,π/2] does the curve intersect the circle r = 3?

SOLUTION:

(a) At the origin r = 0, so we want 2 + 4 sin = 0. Solving for yields which occurs at (b) The curves r = 2 + 4 sin and r = 3 intersect when 2 + 4 sin = 3, or From the calculator we find = arcsin  FIGURE N1–15

A polar function may also be expressed parametrically:

x = r cos , y = sin In this form, the curve r = 2 + 4 sin from Example 17 would be defined by:

x( ) = (2 + 4 sin ) cos , y( ) = (2 + 4 sin ) sin EXAMPLE 18

Find the (x, y) coordinates of the point on r = 1 + cos where  Chapter Summary

This chapter has reviewed some important precalculus topics. These topics are not directly tested on the AP exam; rather, they represent basic principles important in calculus. These include finding the domain, range and inverse of a function; and understanding the properties of polynomial and rational functions, trigonometric and inverse trig functions, and exponential and logarithmic functions.

For BC students, this chapter also reviewed parametrically defined functions.

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