Formula sheet
Math 105
Midterm 2
1 − sin2 x = cos2 x, 1 + tan2 x = sec2 x, sec2 x − 1 = tan2 x
Half-angle formulas:
cos2 x =
1 + cos2x
1 − cos 2x
, sin 2 x =
2
2
Double-angle formulas:
sin 2x = 2sin x cos x,
cos 2x = cos2 x − sin2 x = 2cos2 x − 1 = 1 − 2sin2 x
Indefinite integrals:
Z
Z
sec x dx = ln|sec x + tan x| + C,
csc x dx = −ln|csc x + cot x| + C
Summation identities:
n
X
k=1
n
n
n(n + 1) 2
n(n + 1) X 2 n(n + 1)(2n + 1) X 3 n2 (n + 1)2
k =
k =
,
,
=
.
k=
2
6
4
2
k=1
k=1
Approximation’s Rules and their error bound: Let y = f (x) be a continuous function defined on the interval [a, b] such that its derivatives f 00 and f (4) are
continuous. The midpoint Rule approximation M (n), the Trapeziod Rule approxRb
imation T (n) and the Simpson’s Rule approximation S(n) to a f (x)dx using n
equally spaced subintervals on [a, b] are given by the following formulas:
n
X
1
M (n) =
f a + (k − )∆x ∆x,
2
k=1
!
n−1
X
1
1
T (n) =
f (x0 ) +
f (a + k∆x) + f (xn ) ∆x,
2
2
k=1
S(n) = (f (x0 ) + 4f (x1 ) + 2f (x2 ) + 4f (x3 ) + ... + 4f (xn−1 ) + f (xn ))
where ∆x = b−a
n . The absolute errors in approximating the integral
the Midpoint Rule, Trapezoid Rule satisfy the inequalities
EM ≤
k(b − a)
k(b − a)
(∆x)2 and ET ≤
(∆x)2 ,
24
12
where k is a real number such that |f 00 (x)| ≤ k for all x in [a, b].
Rb
a
∆x
,
3
f (x)dx by
Formula sheet
Math 105
Midterm 2
Rb
The absolute errors in approximating the integral a f (x)dx by the Simpson’s Rule
satisfies the inequalities
K(b − a)
ES ≤
(∆x)4
180
where K is a real number such that |f (4) (x)| ≤ K for all x in [a, b].
Equivalently, the error bounds can also be written as:
k(b − a)3
,
24n2
k(b − a)3
,
ET ≤
12n2
K(b − a)5
ES ≤
.
180n4
EM ≤
(you can use any of these two versions for error bounds, as they give the same result.)