## 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.