9/19/12
Worksheet: Implicit Diff., Higher Derivatives
Exercise 1. Use implicit differentiation to find y 0 =
dy
dx
GSI: Ralph Morrison
for the following equations.
• x3 + 2y 2 = 4xy 3 .
• sin(x/y) = 3x − 2 arctan(y 2 ).
• ex+y − log(x) = y 2 .
Exercise 2. As we saw in lecture, the first, second, and third derivatives of position with respect
to time are called velocity, acceleration, and jerk. The fourth, fifth and sixth derivatives of position with respect to time are called snap, crackle, and pop. (Like, for real. Wikipedia the word
“jounce.”)
• What are the snap, crackle, and pop if position is given by f (x) = sin(x)? What about
f (x) = sin(2x)?
• Suppose position is given by g(x) = xn for some positive n. If n ≥ 6, what are the snap,
crackle, and pop? What about if n < 4? (What about for n = 4 and n = 5?)
2
2
dy
, and evaluate
Exercise 3. The equation for a particular ellipse is given by x9 + y4 = 1. Find dx
√
√
it at the points (2, −2 5/3), (2, 2 5/3), and (3, 0). What’s the geometric interpretation of what
you’ve calculated?
Math 10A: Methods of Mathematics
1
Fall 2012