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AP CALCULUS AB and BC
Final Notes
Trigonometric Formulas
1. sin2θ +cos2θ =1 sinθ 1
2. 1+tan2θ =sec2θ 13. tanθ = cosθ = cotθ
3. 1+cot2θ =csc2θ cosθ 1
4. sin(−θ) = −sinθ 14. cotθ = sinθ = tanθ
5. cos(−θ) = cosθ 15. secθ = 1
6. tan(−θ) = −tanθ cosθ
7. sin(A+ B) = sin AcosB+sinBcosA 16. cscθ = 1
8. sin(A−B) =sin AcosB−sinBcosA sinθ
9. cos(A+B)=cosAcosB−sinAsinB 17. 2 1
cos θθ=1 cos+2
2( )
10. cos(A− B) = cos AcosB+sin AsinB
18. 2 1
sin θθ=1 cos−2
11. sin 2θ = 2sinθ cosθ 2( )
12. 222 2
cos2θ=cos θ−sin θ=2cos θ−=−1 1 2sin θ
Differentiation Formulas
1. d (xn) = nxn−1 11. d (ex) = ex
dx dx
2. d (fg) = fg′+ gf ′Product rule 12. d (ax) = ax lna
dx dx
3. d ( f ) = gf ′− fg′ Quotient rule 13. d (lnx) = 1
dx g g2 dx x
d ′ ′ 14. d (Arcsinx) = 1
4. dx f (g(x)) = f (g(x))g (x)Chain rule dx 1−x2
5. d (sin x) = cosx 15. d (Arctanx) = 1
dx dx 1+x2
6. d (cosx) = −sin x 16. d (Arcsecx) = 1
dx dx | x | x2 −1
7. d (tan x) = sec2 x d
dx 17. dx [c] = 0
8. d (cot x) = −csc2 x
dx 18. d cf x =cf ' x
( ) ( )
9. d (secx) = secxtanx dx
dx
10. d (cscx) = −cscxcotx
dx
Integration Formulas
1. ∫a dx = ax+C
n 1
x +
2. xn dx C, n
∫ = n+1+ ≠ −1
3. ∫ 1 dx = ln x + C
x
4. ∫ex dx = ex +C
ax
5. ∫axdx = +C
lna
6. ∫lnx dx = xlnx− x+C
7. ∫sin x dx = −cosx+C
8. ∫cosx dx = sin x +C
9. ∫tanx dx = lnsecx +C or −lncosx +C
10. ∫cot x dx = lnsin x +C
11. ∫secx dx = lnsecx + tan x +C
12. ∫cscx− dx = ln++cscx cotx C
13. ∫sec2 x dx = tan x +C
14. ∫secxtan x dx = secx +C
15. ∫csc2 x dx = −cot x +C
16. ∫cscxcot x dx = −cscx +C
17. ∫ tan2 x dx = tan x − x + C
18. dx = 1 Arctan x+C
∫ a2 + x2 a a
19. dx = Arcsin x+C
∫ 2 2 a
a −x
20. ∫ dx = 1 Arcsec x +C = 1 Arccos a +C
x x2 −a2 a a a x
Formulas and Theorems
1. Limits and Continuity:
A function y = f (x) is continuous at x = a if
i). f(a) exists
ii). lim fx
( ) exists
→
xa
iii). lim fx= fa( )
( )
x→a
Otherwise, f is discontinuous at x = a.
The limit lim f (x) exists if and only if both corresponding one-sided limits exist and are equal –
xa→
that is,
lim f→=x =L =lim ffx L lim x
( ) ( ) ( )
xa→ +−
xa→→xa
2. Even and Odd Functions
1. A function y = f (x) is even if f (−x) = f (x) for every x in the function’s domain.
Every even function is symmetric about the y-axis.
2. A function y = f (x) is odd if f (−x) = − f (x) for every x in the function’s domain.
Every odd function is symmetric about the origin.
3. Periodicity
A function f (x) is periodic with period p (p > 0) if f (x + p) = f (x) for every value of x
.
Note: The period of the function y = Asin(Bx + C) or y = Acos(Bx +C) is 2π .
B
The amplitude is A . The period of y = tan x is π .
4. Intermediate-Value Theorem [ ]
A function y = f (x) that is continuous on a closed interval a,b takes on every value
between f (a) and f (b) .
Note: If is continuous on [ ] and and differ in sign, then the equation
f a,b f (a) f (b)
f (x) = 0 has at least one solution in the open interval (a,b).
5. Limits of Rational Functions as x → ±∞
i). fx() if the degree of
lim =0 f (x) < the degree of g(x)
x→±∞ gx()
2
xx−2
Example: lim =0
x→∞ x3+3
ii). lim f (x) is infinite if the degrees of f (x) > the degree of g(x)
x→±∞ g(x)
3
xx+2
lim =∞
Example:
x→∞ x2 −8
iii). lim f (x) is finite if the degree of f (x) = the degree of g(x)
x→±∞ g(x)
2
2xx−+32 2
Example: lim =−
x→∞ 2 5
10x−5x
6. Horizontal and Vertical Asymptotes
1. A line y = b is a horizontal asymptote of the graph y = f (x) if either
.(Compare degrees of functions in fraction)
lim fx( ) =b or lim fx( ) =b
x→∞ x→−∞
2. A line x = a is a vertical asymptote of the graph y = f (x) if either
lim fx( ) = ±∞ or lim fx= ±∞
( ) (Values that make the denominator 0 but not
+−
x→→a xa
numerator)
7. Average and Instantaneous Rate of Change
i). Average Rate of Change: If x, y and xy, are points on the graph of
( ) ( )
00 11
y = f (x), then the average rate of change of y with respect to x over the interval
f (x ) − f (x ) y − y ∆y
[ ] is 1 0 = 1 0 = .
x ,x
0 1 x −x x −x ∆x
1 0 1 0
ii). Instantaneous Rate of Change: If (x0, y0 ) is a point on the graph of y = f (x), then
the instantaneous rate of change of with respect to x at is ′ .
y x0 f (x0)
8. Definition of Derivative
ff
fx(+−h) fx() xa−
′ or f ' a =lim ( ) ( )
fx( ) = lim ( )
h→0 h xa→ xa−
The latter definition of the derivative is the instantaneous rate of change of f (x) with respect to
x at x = a.
Geometrically, the derivative of a function at a point is the slope of the tangent line to the graph of
the function at that point.
9. The Number e as a limit
1 n
i). lim 1+=e
n→∞
n
ii). 1/n
lim 1+=ne
( )
n→0
10. Rolle’s Theorem (this is a weak version of the MVT)
If is continuous on [ ] and differentiable on ( ) such that , then there
f a,b a,b f (a) = f (b)
is at least one number cin the open interval ( ) such that ′ .
a,b f (c) = 0
11. Mean Value Theorem ( )
If is continuous on [ ] and differentiable on , then there is at least one number c
f a,b a,b
( ) such that f (b) − f (a) ′ .
in a,b b−a = f (c)
12. Extreme-Value Theorem [ ]
If f is continuous on a closed interval a,b , then f (x) has both a maximum and minimum
on [ ].
a,b
13. Absolute Mins and Maxs: To find the maximum and minimum values of a function y = f (x),
locate
1. the points where ′ is zero or where ′ fails to exist.
f (x) f (x)
2. the end points, if any, on the domain of f (x) .
3. Plug those values into f (x) to see which gives you the max and which gives you this
min values (the x-value is where that value occurs)
Note: These are the only candidates for the value of x where f (x) may have a maximum or a
minimum.
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