1
Formula Sheet
Newtons Law of Cooling/Heating:
dT
= k(T − T1 )
dt
Useful Trig Identities:
sin2 (x) + cos2 (x) = 1
tan2 (x) + 1 = sec2 (x)
1 − cos(2x)
sin2 (x) =
2
1
+
cos(2x)
cos2 (x) =
2
sin(2x) = 2 sin(x) cos(x)
cos(2x) = cos2 (x) − sin2 (x)
√
1 − x2
sin(cos−1 (x)) =
√
cos(sin−1 (x)) =
1 − x2
√
sec(tan−1 (x)) =
1 + x2
√
x2 − 1
x≥1
√
tan(sec−1 (x)) =
2
− x − 1 x ≤ −1
1
sin(mx) cos(nx) =
[sin(m + n)x + sin(m − n)x]
2
1
sin(mx) sin(nx) = − [cos(m + n)x − cos(m − n)x]
2
1
cos(mx) cos(nx) =
[cos(m + n)x + cos(m − n)x]
2
Useful Derivatives:
1
−1<x<1
Dx sin−1 (x) = √
1 − x2
1
Dx cos−1 (x) = − √
−1<x<1
1 − x2
1
Dx tan−1 (x) =
1 + x2
1
√
Dx sec−1 (x) =
|x| > 1
|x| x2 − 1
1
Some Useful Integral Forms:
Z
tan udu = − ln | cos(u)| + C
Z
cot(u)du = ln | sin(u)| + C
Z
du
u
√
= sin−1 ( ) + C
2
2
a
a −u
Z
du
u
1
= tan−1 ( ) + C
2
2
a +u
a
a
Z
du
1
u
√
= sec−1 ( ) + C
2
2
a
a
u u −a
Note: The practice midterm may be a little longer than the actual midterm,
but it reflects problems similar in difficulty to what you may encounter in
the actual exam.
2
2
True/False
1. L’Hospital’s Rule can be used to solve for the limit of any sequence
{an }.
2. Any infinite series of all positive terms will diverge.
√
2
3. lim ln( x2 ex )e−x diverges
x→∞
4. The limit of an infinite sequence {an } = f (n) is the same as the limit
of the generating function f (x) as x approaches infinity.
5. We can solve analytically
(write down a formula/answer for) any inR
m
tegral of the form tan (x) secn (x)dx where m,n are both any real
numbers.
3
3
Free Response
Evaluate the following integrals:
R 2 −2x
1. x√x−2
dx
4
x+4
dx
x2 −9
2.
R
3.
R∞
1
1
dx
(x−1)2
5
Evaluate the Following Limits:
4. lim+
x→1
Rx
1
sin tdt
x−1
6
5. lim xx
x→0
7
6. lim an where {ai } = 1, 13 , 15 , 71 , ...
n→∞
8
7. Find the limit of the corresponding series,
9
P∞
i=1
ai