1. Parametric Equations and Calculus
Section 10.3

2. Maximum and Minimum Values
Rectangular
Find the coordinates of the maximum of the function 𝑦=−2 𝑥 2 +𝑥−1.
Parametric
Find the highest and lowest points on the graph of the function x=3 cos 𝑡 +1, 𝑦=5 sin 𝑡 −2, 0≤𝑡≤2𝜋 .
Find the rightmost point and the leftmost point of the same function.

3. More with Max and Mins
Find all horizontal and vertical tangent lines to the graph of 𝑥= sec 𝑡 , 𝑦= tan 𝑡 , 0≤𝑡≤2𝜋.

4. Arc Lengths Rectangular Parametric
Find the length of the curve 𝑦=2 𝑥 2 −4 from −2≤𝑥≤2.
Parametric
Find the length of the curve parameterized by 𝑥= 4 cos 𝑡 , 𝑦=3 sin 𝑡 , 0≤𝑡≤2𝜋.

5. Separation of Variables
Rectangular
𝑑𝑦 𝑑𝑥 =2𝑥−3. Find 𝑦 in terms of 𝑥 given that 𝑦 −2 =4
Parametric
A particle moves along a curve defined by 𝑦= 𝑥 2 + 5𝑥−3. The x-coordinate of the particle, 𝑥(𝑡), satisfies the equation 𝑑𝑥 𝑑𝑡 =2𝑡−1 with initial condition 𝑥 2 = 0.
Find 𝑥(𝑡) in terms of 𝑡.
Find 𝑑𝑦 𝑑𝑡 .

6. Putting it Together…
A bug travels along a curve parameterized by x= 6 sin 𝑡 −3 sin 7𝑡 , 𝑦=6 cos 𝑡 −3 cos (7𝑡) for 2𝜋 seconds.
What is the position of the bug after 𝜋 seconds?
How far did the bug travel?
What was the speed of the bug when 𝑡= 𝜋 2 ?