Solutions to midterm 1
By H˚
akan Nordgren
Question 1:
Find the most general antiderivative of
1.
f
(
t
) = 3 cos
t
+
t
2
and
2.
g
(
t
) = 2
√
t
+
e
t
.
Answer:
1. We need to find a function with we can differentiate to obtain 3 cos
t
and one
which differentiates to
t
2
. The functions 3 sin
t
and
1
3
t
3
work. Thus the most
general antiderivative of
f
is
F
(
t
) = 3 sin
t
+
1
3
t
3
+ constant. Make sure that
you always check your answers.
2. This time, we need to find a function with we can differentiate to obtain 2
√
t
and one which differentiates to
e
t
. The functions 2
2
3
t
3
2
and
e
t
work. Thus the
most general antiderivative of
g
is
G
(
t
) =
4
3
t
3
2
+
e
t
+ constant.
Question 2:
A particle is moving along a straight line with velocity
v
(
t
) = sin
t

cos
t
meters per second, and has initial displacement
s
(0) = 0 meters.
Find its
position at time
t
.
Answer:
So we know that
ds
(
t
)
dt
=
v
(
t
) = sin
t

cos
t
. Thus we can integrate with
respect to
t
to obtain
s
(
t
) =

cos
t

sin
t
+
C
, where
C
is a constant. Since we
know that 0 =
s
(0) =

1+
C
we must have
C
= 1. Thus the position of the particle
at time
t
is
s
(
t
) =

cos
t

sin
t
+ 1.
Question 3:
With the aid of a sketch, evaluate the following integral by interpreting
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 Spring '08
 Justin
 Calculus, Derivative, Fundamental Theorem Of Calculus, Cos, akan Nordgren Question

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