9.2 Second Derivatives of Parametric Equations
Practice
Calculus
Given the following parametric equations, find
1. 𝑥 𝑡
𝑒
and 𝑦 𝑡
3. 𝑥 𝑡
𝑎𝑡 and 𝑦 𝑡
positive constants.
5. 𝑥
𝑒 and 𝑦
𝑡𝑒 .
𝒅𝟐 𝒚
𝒅𝒙𝟐
in terms of 𝒕.
𝑒 .
2. 𝑥 𝑡
𝑏𝑡, where 𝑎 and 𝑏 are
4.
6. 𝑥
𝑡 and 𝑦 𝑡
4 and
𝑡
𝑡
sin 𝑡 .
1 and 𝑦
2𝑡 .
1 for 𝑡
0.
7. Given a curve defined by the parametric equations
𝑡
𝑡 . Determine the
𝑥 𝑡
2 𝑡 and 𝑦 𝑡
open 𝑡-intervals on which the curve is concave up
or down.
8. If 𝑥 𝜃
2 sec 𝜃 and 𝑦 𝜃
1 2 tan 𝜃, Find
the slope and the concavity at 𝜃
.
9. If 𝑥 cos 𝜃 and 𝑦 3 sin 𝜃, find the slope and
concavity at 𝜃 0.
10. If 𝑥 𝑡
𝑡 ln 𝑡 and 𝑦 𝑡
𝑡 ln 𝑡, determine
values of 𝑡 where the graph is concave up.
Test Prep
9.2 Second Derivatives of Parametric Equations
11. If 𝑥
A.
3𝑡
B.
𝑡
12. If 𝑥
A. Slope:
1 and 𝑦
𝜃
cos 𝜃 and 𝑦
ln 𝑡, what is
𝑡
1
1, Concave down
D. Slope: 1, Concave up
in terms of 𝑡?
C.
𝑡
D.
sin 𝜃, find the slope and concavity at 𝜃
B. Slope: 𝜋, Concave up
E. Slope: , Concave up
𝑡
E. 6𝑡
𝜋.
C. Slope: 1, Concave down