Solved: "Working with vector-valued functions For each vector-valued function..."

meplasemamiuk

Answered question

2021-12-05

Working with vector-valued functions For each vector-valued function r, carry out the following steps.
a.

E v a l u a t e lim_{t \Rightarrow 0} r (t) and lim_{t \Rightarrow \infty} r (t)

, if each exists.
b. Find r'(t) and evaluate r'(0).
c. Find r"(t).
d. Evaluate

\int r (t) d t .

r (t) = ⟨ \sin 2 t, 3 \cos 4 t, t ⟩

Zachary Pickett

Beginner2021-12-06Added 17 answers

Step1
Part (a)
Let the vector valued function

r (t) = \sin 2 t i + 3 \cos 4 t j + t k

Find the limit

L t_{t \Rightarrow 0} r (t)

L t_{t \Rightarrow 0} r (t) = L t_{t \Rightarrow 0} \sin 2 t i + 3 \cos 4 t j + t k = 0 i + 3 j + 0 k

Therefore, the value is 3j.
Find the limit

L t_{t \Rightarrow \infty} r (t)

L t_{t \Rightarrow \infty} r (t) = L t_{t \Rightarrow \infty} \sin 2 t i + 3 \cos 4 t j + t k = \infty

Therefore, the value is not defined.
Step2
Part (b)
Differentiate the expression

r (t) = \sin 2 t i + 3 \cos 4 t j + t k

with respect to t to find r'(t).

r^{'} (t) = (\cos 2 t x 2) i + (- 12 \sin 4 t) j + k

= (2 \cos 2 t) i - (12 \sin 4 t) j + k

Therefore,

r^{'} (t) i s (2 \cos 2 t) i - (12 \sin 4 t) j + k

.
Substitute 0 for t to find r'(O).

r^{'} (0) = (2 \cos O) i - (12 \sin 0) j + k

= 2 i + k

Therefore, the value of r'(O) is 2i+k.
Step3
Part (c)
Differentiate the expression

(2 \cos 2 t) i - (12 \sin 4 t) j + k

to find the value of r'(t).

r^{'} (t) = (- 4 \sin 2 t) i - (48 \cos 4 t) j

Therefore, the value of

r^{'} (t) i s (- 4 \sin 2 t) i - (48 \cos 4 t) j

.
Step4
Part (d)
Integrate the expression

r (t) = \sin 2 t i + 3 \cos 4 t j + t k

with respect to t to find the value of

\int r (t)

\int r (t) d t = (\begin{matrix} - \frac{\cos 2 t}{2} \end{matrix}) i + (\begin{matrix} \frac{3}{4} \sin 4 t \end{matrix}) j + (\begin{matrix} \frac{t^{2}}{2} \end{matrix}) k

Therefore,

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