Step by step solution to "Higher-order derivatives Compute r"(t) and r""(t) for the..."

khi1la2f1qv

Answered question

2021-11-28

Higher-order derivatives Compute r"(t) and r""(t) for the following functions
$r (t) = ⟨ \cos 3 t, \sin 4 t, \cos 6 t ⟩$

Ida Perry

Beginner2021-11-29Added 20 answers

Step 1
Given parametric curve:

r (t) = ⟨ \cos 3 t, \sin 4 t, \cos 6 t ⟩

Step 2
Now,
Differentiate the above curve using standard derivate:

\frac{d (\sin (a x))}{d x} = a \cos (a x)

\frac{d (\cos (a x))}{d x} = - a \sin (a x)

Step 3
Therefore,

r (t) = ⟨ \cos 3 t, \sin 4 t, \cos 6 t ⟩

First derivate is:

r ‘ (t) = ⟨ - 3 \sin 3 t, 4 \cos 4 t, - 6 \sin 6 t ⟩

Second derivate is:

r^{″} (t) = ⟨ - 9 \cos 3 t, - 16 \sin 4 t, - 36 \cos 6 t ⟩

Third derivate is:

r^{‴} (t) = ⟨ 27 \sin 3 t, - 64 \cos 4 t, 216 \sin 6 t ⟩

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