Derivatives of Parametric Equations

Lesson 10.2

2

Studying Graphs

• Recall that a function, y = f(x) is intersected no more than once by a vertical line

• Other graphs exist that are not functions We seek to study characteristics

of such graphs How do we determine the slope

at a point on the graph (for aparticular value of t)?

We will use parametric equations

We will use parametric equations

3

Derivative of Parametric Equations

• Consider the graph ofx = 2 sin t, y = cos t

• We seek the slope, thatis

• For parametric equations

For our example

dy

dx

/

/

dy dy dt

dx dx dt

sin tan

2cos 2

dy t t

dx t

4

Try It Out

• Find dy/dx for the given parametric equations x = t + 3

y = t2 + 1

• What is the slope of the line tangent to the graph when t = 2?

• What is the slope of the line tangent to the graph when x = 2?

5

Second Derivatives

• The second derivative is the derivative of the first derivative

• But the first derivative is a function of t We seek the derivative with respect to x We must use the chain rule

2

2

d dyd y dt dx

dxdxdt

6

Second Derivatives

• Find the second derivative of the parametric equations x = 3 + 4cos t

y = 1 – sin t

• First derivative

• Second derivative

cos 1cot

4sin 4

dy tt

dx t

2 2

2 3

1 csc 1

4 4sin 16sin

d y t

dx t t

7

Try This!

• Where does the curve described by the parametric equations have a horizontal tangent? x = t – 4

y = (t 2 + t)2

• Find the derivative

• For what value of t does dy/dx = 0?

8

Assignment

• Lesson 10.2A

• Page 412

• Exercises 1 – 21

• Lesson 10.2B

• Page 413

• Exercises 22 – 26 all