5 Steps to a 5 AP Calculus BC 2019

128 STEP 4. Review the Knowledge You Need to Score High

7.5 Parametric, Polar, and Vector Representations

Main Concepts:Parametric Curves, Polar Equations, Types of Polar Graphs,

Symmetry of Polar Graphs, Vectors, Vector Arithmetic

Parametric Curves

Parametric curves are relations (x(t),y(t)) for which bothxandyare defined as functions

of a third variable,t, that is,x=f(t) andy=g(t).

Example 1

A particle is moving in the coordinate plane in such a way thatx(t)= 2 t−5 andy(t)=

4 sin

(

π

t+ 1

)

for 0≤t≤5. Sketch the path of the particle and indicate the direction of

motion.

Step 1: Create a table of values.

t 0 1 2 3 4 5

x(t) − 5 − 3 − 1 1 3 5

y(t) 0 4 3.464 2.828 2.351 2

Step 2: Plot the points and sketch the path of a particle as a smooth curve. Place arrows to

indicate the direction of motion.

Example 2

A parametric curve is defined byx= 2 +etandy=e^3 t. Find the Cartesian equation of the

curve.

Step 1: Solvex= 2 +etfort.x− 2 =etsot=ln(x−2).

Step 2: Substitutet=ln(x−2) intoy=e^3 t.y=e3ln(x−2)=(x−2)^3.

Step 3: Note thatt=ln(x−2) is defined only whenx>2. The equation of the curve is

y=(x−2)^3 with domain (2,∞).