MATH1130: Calculus II
Exercise Sheet 5: Limits, derivatives and integrals of vector functions
Please hand solutions in at the lecture on Tuesday 2nd March.
1.) Let r : R → R3 be defined by
r(t) =
sin(t)
2
, cos(t), 3 t .
t
Evaluate the following limits.
(a) limt→π r(t)
(b) limt→1 r(t)
(c) limt→0 r(t)
2.) Discuss the continuity of each of the following functions
(a) r(t) = (t2 + 1, cos(2 t), sin(3 t))
√
(b) r(t) =
t + 1, tan(t)
√
(c) r(t) = t21−1 , 1 − t2 , 1t
(d) r(t) = cos(4 t), 1 −
√
3 t + 1, sin(5 t), sec(t)
3.) Find the derivative of each of the following functions.
(a) r(t) = (t3 , t, 2 t + 4)
(b) r(t) = (3 t cos(2 t), 4 t sin(2 t))
(c) r(t) = (4 t3 − 3, sin(t), e−2 t )
(d) r(t) = (e−t sin(3 t), e−t cos(3 t), t e−t )
d
For (b) compare dt
kr(t)k and kr 0 (t)k.
For (d) let f (t) = e−t and p(t) = (sin(3 t), cos(3 t), t) and verify that
df
(t) · p(t) + f (t) · dp
(t).
dt
dt
d
dt
(f (t) · p(t)) =
Please turn over!
4.) Evaluate the following integrals.
R1
(a) −1 r(t) dt where r(t) = (t, −t e−t ).
(b)
R2
(c)
R2
1
1
r(t) dt where r(t) =
r(t) dt where r(t) =
√
t .
√1 , −
t3
1
, −t
t
√
t2 + 1, (t − 1) sin(t/2) .