Math 1401 Spring 2000 Cheat Sheet
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1MATH 1401 SPRING 2000 CHEAT SHEET
FINAL
JAN MANDEL
1. Important formulas from algebra.
sin(a + b) = sin(a) cos(b) + cos(a) sin(b),
2
2
b+c
b
c
m/n
b
(log a)b
2
m
sin
x + cos
x = 1, a
= a
a
, a
=
a
, a
= e
. Solution of ax
+ bx + c = 0 is
b
b
4ac
x
=
1,2
2a
2. Limits and continuity.
lim
f (x) = f (c)
f is continuous at c
x
c
sin(x)
1 cos(x)
1/x
lim
= 1, lim
= 0, lim
(1 + x)
= e
x
0
x
0
x
0
x
x
lim
f (x) = L
lim
f (x) = lim
f (x) = L
x
c
x
c
x
c
Intermediate value theorem: If f is continuous on [a,b] and k is between f (a) and f (b), then
there exists c
[a, b] such that f (c) = k.
Infinite limits: The formulas for the limit of sum, product, and quotient apply unless they
lead to undefined expressions of the form
,
.0, L/0,
/ .
If lim
f (x) = 0 and lim
g(x) = 0, with g(x) = 0 on a neighborhood of c, then the
x
c
x
c
graph of f /g has vertical asymptote x = c.
3. Differentiation.
The equation of the line passing through (x
, y
) with slope s is
0
0
y
y
= s(x
x
). The equation of the tangent to the graph of f at (x
, y
), y
= f (x
), is
0
0
0
0
0
0
y
y
= f (x
)(x
x
).
0
0
0
f (c) = lim
(f (x)
f (c))/(x
c). If f (c) exists, f is continuous at c.
x
c
n
n 1
x
x
(x
) = nx
, (sin x) = cos x, (cos x) =
sin x, (ln x) = 1/x, (e
) = e
2
1
sin x = cos x, cos x =
sin x, (arctan x) = 1/(1 + x
), arcsin x =
, arcsec x =
1 x
2
1
, (uv) = u v + uv , (u/v) = (u v
uv )/v
, f (g(x)) = f (g(x))g (x)
x
1 x
1
If g = f
and y = g(x), f (y) = 0, then g (x) = 1/f (y).
4. Applications and extrema.
If f is continuous on [a, b], then f attains maxi-
mum and minimum on [a, b]. f can attain extremum on [a, b] only at endpoints or critical
numbers (where f does not exist or f = 0). f can attain relative extremum in (a, b) only
at a critical number.
Mean value theorem: If f is continuous on [a, b] and differentiable on (a, b) then there exists
c
(a, b) such that f (c) = (f (b)
f (a))/(b
a). (The case when f (a) = f (b) is Rolle’s
theorem.)
If f > 0 in (a, b) and f is continuous on [a, b], then f is increasing on [a, b].
If f is continuous at c, f (x) < 0 for x < c and f (x) > 0 for x > c, then f has relative
minimum (c, f (c)). (Or, relative minimum f (c) at x = c.)
If f in increasing in interval I, then f is concave upward in I.
If f > 0 in (a, b), then f is concave upward in (a, b).
If f (c) = 0 and f (c) > 0, then f has relative minimum at c.
e
e
e +e
2
2
5. Hyperbolic functions.
sinh x =
, cosh x =
, cosh
x sinh
x = 1,
2
2
1
2
cosh x = sinh x, (tanh
) = 1/(1
x
)
6. Integration.
f (x) dx = F (x) + C, F = f .
n
n+1
x
dx = x
/(n + 1) + C, n =
1,
f (g(x))g (x) dx =
f (u) du, u = g(x)
1
x
1
1
x
= arcsin
+ C,
dx =
arctan
+ C
a
a +x
a
a
a
x
1
1 x
1
1
1 x
= sinh
+ C,
dx =
tanh
+ C
a
a
x
a
a
a +x
b
f (x) dx = F (b)
F (a), F = f .
a
x
(d/dx)
f (t) dt = f (x)
a
1
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