Final Review 1: Definition and Calculation
•
Limit:
lim f (x) = A if f (x) is close to A when x is close to c and x 6= c
x→c
lim f (x) = A if f (x) is close to A when x is close to c and x > c
x→c+
lim f (x) = A if f (x) is close to A when x is close to c and x < c
x→c−
Continuity:
Derivative:
Indefinite integral:
f (x) is continuous at c if lim f (x) = f (c)
x→c
f (x + h) − f (x)
df
= f 0 (x) = lim
Dx (f ) =
h→0
dx
h
Z
f (x) dx = F (x) + C if F 0 (x) = f (x)
Z
Definite integral:
b
f (x) = lim
n→∞
a
n
X
f (x̄i ) ·
i=1
b−a
n
•
– Limits of polynomials, trigonometric functions are plugging in.
0
– Limits of rational functions of the type : Eliminating the common factor.
0
1 − cos x
sin x
= 1, lim
= 0. Method of substitution.
– lim
x→0
x→0 x
x
1
1
1
1
– lim+ = ∞, lim− = −∞, lim
= 0, lim
= 0.
x→∞ x
x→−∞ x
x→0 x
x→0 x
6= 0
– Limits of the type
: Must be ∞ or −∞, check the sign.
0
1
– Limits at infinity: convert all x’s into ’s.
x
•
– Dx (xr ) = rxr−1 (r can be any real number), Dx (sin x) = cos x, Dx (cos x) = − sin x
– Dx4 (sin x) = sin x,
–
Dx4 (cos x) = cos x
(cf )0 = c · f 0 if c is a constant
(f + g)0 = f 0 + g 0 , (f − g)0 = f 0 − g 0
µ ¶0
f
f 0 g − g0 f
(f g)0 = f 0 g + g 0 f ,
=
g
g2
Constant multiple:
Sum and difference rule:
Product and quotient rule:
(f ◦ g)0 = f 0 (g(x)) · g 0 (x)
Chain Rule:
•
– Implicit derivative: Take derivative on both sides of the equation then solve for f 0 or y 0 .
Z u(x)
d
– First fundamental theorem of calculus:
f (t) dt = f (u(x)) · u0 (x) − f (l(x)) · l0 (x)
dx l(x)
Z b
Z
– Second fundamental theorem of calculus:
f (x) dx = F (b) − F (a) if f (x) dx = F (x) + C
a
Z
xr+1
xr dx =
–
+ C (r can be any real number but −1)
r+1
Z
Z
sin x dx = − cos x + C,
cos x dx = sin x + C
–
Z
Z
– Constant multiple:
cf (x) dx = c f (x) dx
Z
Z
Z
Sum and difference rule:
(f + g) dx = f dx + g dx,
Z
Generalized power rule:
f 0 (x) · f r (x) dx =
f r+1 (x)
+C
r+1
Z
Z
(f − g) dx =
Z
f dx −
g dx
– No product, quotient or chain rule.
– Method of substitution: Let the part that you don’t like be u, find du and then take the
integral for the variable u. Don’t forget to change the upper and lower limits when you are
doing definite integral.
– Symmetry and periodicity: Definite integrals of even, odd and periodic functions, especially
for
is an absolute value.
Z the case that there
Z
a
a
f (x) dx = 2
−a
Z a
f (x) dx
when f (x) is even, i.e. f (−x) = f (x)
0
f (x) dx = 0
when f (x) is odd, i.e. f (−x) = −f (x)
−a
Find the number of bumps for sin and cos by sketching the graph.