9.2 2nd Derivatives of Parametric Equations Notes

Calculus 9.2 2nd Derivatives of Parametric Equations

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Second Derivative of a Parametric Equation

The second derivative of a parametric given by and

is

Notes

Given

the

following

parametric

equations,

find

in

terms

of

.

1. and

2 for 0. 2. 3 cos and

4 sin .

3. At 1, find the concavity of the graph defined parametrically by 1 and .

9.2 Second Derivatives of Parametric Equations

Calculus

Practice

Given

the

following

parametric

equations,

find

in

terms

of

.

1. and .

2. and 1 for 0.

3. and , where and are positive constants.

4.

4 and

sin .

5. and .

6. 1 and 2 .

7. Given a curve defined by the parametric equations 2 and . Determine the open -intervals on which the curve is concave up

or down.

8. If 2 sec and 1 2 tan , Find the slope and the concavity at .

9. If cos and 3 sin , find the slope and concavity at 0.

10. If ln and ln , determine values of where the graph is concave up.

9.2 Second Derivatives of Parametric Equations

11. If 3 1 and ln , what is in terms of ?

Test Prep

A.

B.

C.

D.

12. If cos and 1 sin , find the slope and concavity at .

E. 6

A. Slope: 1, Concave down D. Slope: 1, Concave up

B. Slope: , Concave up E. Slope: , Concave up

C. Slope: 1, Concave down

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