AP CALCULUS AB and BC

Final Notes

Trigonometric Formulas

1. sin θ + cos θ = 1

2 2

sin θ 1

13. tan θ = =

2. 1 + tan 2 θ = sec 2 θ cosθ cot θ

1 + cot 2 θ = csc 2 θ cosθ 1

14. cot θ =

3.

=

4. sin( −θ ) = − sin θ sin θ tan θ

5. cos(−θ ) = cosθ 1

15. secθ =

6. tan(−θ ) = − tan θ cosθ

7. sin( A + B ) = sin A cos B + sin B cos A 16. cscθ =

1

8. sin( A − B ) = sin A cos B − sin B cos A sin θ

cos( A + B ) = cos A cos B − sin A sin B 1

9.

θ

17. cos =

2

(1 + cos 2θ )

2

10. cos( A − B ) = cos A cos B + sin A sin B

1

11. sin 2θ = 2 sin θ cos θ

θ

18. sin =

2

(1 − cos 2θ )

2

12. cos 2θ = cos 2 θ − sin 2 θ = 2 cos 2 θ − 1 = 1 − 2sin 2 θ

Differentiation Formulas

d n d x

1. ( x ) = nx n −1 11. (e ) = e x

dx dx

d d x

2. ( fg ) = fg ′ + gf ′ Product rule 12. (a ) = a x ln a

dx dx

d f gf ′ − fg ′ d 1

3. ( )= Quotient rule 13. (ln x) =

dx g g2 dx x

d d 1

4. f ( g ( x)) = f ′( g ( x)) g ′( x) Chain rule 14. ( Arc sin x) =

dx dx 1− x2

d d 1

5. (sin x) = cos x 15. ( Arc tan x) =

dx dx 1+ x2

d d 1

6. (cos x) = − sin x 16. ( Arc sec x) =

dx dx | x | x2 −1

d

(tan x) = sec 2 x d

7.

dx 17. [c] = 0

dx

d

8. (cot x) = − csc 2 x

dx d

cf ( x ) = cf ' ( x )

dx

18.

d

9. (sec x) = sec x tan x

dx

d

10. (csc x) = − csc x cot x

dx

Integration Formulas

1. ∫ a dx = ax + C

x n +1

∫ x dx = + C , n ≠ −1

n

2.

n +1

1

3. ∫ x dx = ln x + C

∫ e dx = e + C

x x

4.

ax

∫ a dx = +C

x

5.

ln a

6. ∫ ln x dx = x ln x − x + C

7. ∫ sin x dx = − cos x + C

8. ∫ cos x dx = sin x + C

9. ∫ tan x dx = ln sec x + C or − ln cos x + C

10. ∫ cot x dx = ln sin x + C

11. ∫ sec x dx = ln sec x + tan x + C

12. ∫ csc x dx = − ln csc x + cot x + C

∫ sec x d x = tan x + C

2

13.

14. ∫ sec x tan x dx = sec x + C

∫ csc x dx = − cot x + C

2

15.

16. ∫ csc x cot x dx = − csc x + C

∫ tan x dx = tan x − x + C

2

17.

dx 1 x

18. ∫a 2

+x 2

= Arc tan + C

a a

dx x

19. ∫ a2 − x2

= Arc sin + C

a

dx 1 x 1 a

20. ∫x x2 − a2

=

a

Arc sec + C = Arc cos + C

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 f ( x ) exists

x→a

iii). lim f ( x ) = f (a )

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 –

x →a

that is,

lim f ( x ) =

L → lim+ f ( x ) =lim− f ( x )

L=

x →a x →a x →a

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

.

2π

Note: The period of the function y = A sin( Bx + C ) or y = A cos( Bx + C ) is .

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 f is continuous on [a, b] and f (a ) and f (b) differ in sign, then the equation

f ( x) = 0 has at least one solution in the open interval (a, b) .

5. Limits of Rational Functions as x → ±∞

f ( x)

i). lim = 0 if the degree of f ( x) < the degree of g ( x)

x →±∞ g ( x )

x2 − 2 x

Example: lim =0

x3 + 3

x →∞

f ( x)

ii). lim is infinite if the degrees of f ( x) > the degree of g ( x)

x → ±∞ g ( x )

x3 + 2 x

Example: lim = ∞

x2 − 8

x →∞

f ( x)

iii). lim is finite if the degree of f ( x) = the degree of g ( x)

x → ±∞ g ( x )

2 x2 − 3x + 2 2

Example: lim = −

10 x − 5 x 2

x →∞ 5

6. Horizontal and Vertical Asymptotes

1. A line y = b is a horizontal asymptote of the graph y = f (x) if either

=

lim f ( x) b=

or lim f ( x) b .(Compare degrees of functions in fraction)

x →∞ x →−∞

2. A line x = a is a vertical asymptote of the graph y = f (x) if either

lim+ f ( x) = ±∞ or lim− f ( x ) = ±∞ (Values that make the denominator 0 but not

x→a x→a

numerator)

7. Average and Instantaneous Rate of Change

i). Average Rate of Change: If ( x0 , y0 ) and ( x1 , y1 ) are points on the graph of

y = f (x) , then the average rate of change of y with respect to x over the interval

[x0 , x1 ] is f ( x1 ) − f ( x0 ) = y1 − y 0 = ∆y .

x1 − x0 x1 − x0 ∆x

ii). Instantaneous Rate of Change: If (x0 , y 0 ) is a point on the graph of y = f (x) , then

the instantaneous rate of change of y with respect to x at x0 is f ′( x0 ) .

8. Definition of Derivative

f ( x + h) − f ( x ) f ( x ) − f (a )

f ′( x) = lim or f ' ( a ) = lim

h →0 h x →a x −a

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

n

1

i). lim 1 + =

e

n →∞

n

lim (1 + n ) =

1/ n

ii). e

n →0

10. Rolle’s Theorem (this is a weak version of the MVT)

[ ]

If f is continuous on a, b and differentiable on (a, b ) such that

f (a ) = f (b) , then there

is at least one number c in the open interval (a, b ) such that f ′(c) = 0 .

11. Mean Value Theorem

If f is continuous on [a, b] and differentiable on (a, b ) , then there is at least one number c

f (b) − f (a )

in (a, b ) such that = f ′(c) .

b−a

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 f ′(x) is zero or where f ′(x) fails to exist.

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.

14. Increasing and Decreasing: Let f be differentiable for a < x < b and continuous for a

a ≤ x ≤ b,

1. If f ′( x) > 0 for every x in (a, b ) , then f is increasing on [a, b] .

2. If f ′( x) < 0 for every x in (a, b ) , then f is decreasing on [a, b ] .

15. Concavity: Suppose that f ′′(x) exists on the interval (a, b )

1. If f ′′( x) > 0 in (a, b ) , then f is concave upward in (a, b ) .

2. If f ′′( x) < 0 in (a, b ) , then f is concave downward in (a, b ) .

To locate the points of inflection of y = f (x) , find the points where f ′′( x ) = 0 or where

f ′′(x) fails to exist. These are the only candidates where f (x) may have a point of inflection.

Then test these points to make sure that f ′′( x) < 0 on one side and f ′′( x) > 0 on the other.

16a. If a function is differentiable at point x = a , it is continuous at that point. The converse is false,

in other words, continuity does not imply differentiability.

16b. Local Linearity and Linear Approximations

The linear approximation to f (x ) near x = x0 is given by y = f ( x0 ) + f ′( x0 )( x − x0 ) for

x sufficiently close to x0 . In other words, find the equation of the tangent line at ( x0 , f ( x0 ) )

and use that equation to approximate the value at the value you need an estimate for.

17. ***Dominance and Comparison of Rates of Change (BC topic only)

Logarithm functions grow slower than any power function ( x n ) .

Among power functions, those with higher powers grow faster than those with lower powers.

All power functions grow slower than any exponential function ( a x , a > 1) .

Among exponential functions, those with larger bases grow faster than those with smaller bases.

We say, that as x → ∞ :

f (x) g(x)

1. f ( x ) grows faster than g ( x ) if lim = ∞ or if lim = 0.

x →∞ g(x) x →∞ f (x)

If f ( x ) grows faster than g ( x ) as x → ∞ , then g ( x ) grows slower than f ( x ) as

x→∞.

f (x)

2. f ( x ) and g ( x ) grow at the same rate as x → ∞ if lim = L ≠ 0 (L is finite

x →∞ g(x)

and nonzero).

For example,

ex

1. e x grows faster than x 3 as x → ∞ since lim = ∞

x →∞ x 3

x4

2. x 4 grows faster than ln x as x → ∞ since lim = ∞

x →∞ ln x

x 2 + 2x

3. x 2 + 2x grows at the same rate as x 2 as x → ∞ since lim =1

x →∞ x2

To find some of these limits as x → ∞ , you may use the graphing calculator. Make sure that an

appropriate viewing window is used.

18. ***L’Hôpital’s Rule (BC topic, but useful for AB)

f ( x) 0 ∞ f ′( x)

If lim is of the form or , and if lim exists, then

x→a g ( x) 0 ∞ x → a g ′( x )

f ( x) f ′( x)

lim = lim .

x→a g ( x) x → a g ′( x )

19. Inverse function

1. If f and g are two functions such that f ( g ( x)) = x for every x in the domain of

g and g ( f ( x)) = x for every x in the domain of f , then f and g are inverse

functions of each other.

2. A function f has an inverse if and only if no horizontal line intersects its graph more

than once.

3. If f is strictly either increasing or decreasing in an interval, then f has an inverse.

4. If f is differentiable at every point on an interval I , and f ′( x) ≠ 0 on I , then

−1

g= f ( x) is differentiable at every point of the interior of the interval f (I ) and if

the point ( a, b ) is on f ( x ) , then the point ( b, a ) is on g= f −1

( x) ; furthermore

1

g '(b) = .

f '(a)

20. Properties of y = e

x

1.

x

The exponential function y = e is the inverse function of y = ln x .

2. The domain is the set of all real numbers, − ∞ < x < ∞ .

3. The range is the set of all positive numbers, y > 0 .

4.

d x

dx

(e ) = e x and

d f ( x)

dx

e ( )

= f ' ( x ) e f ( x)

x x x + x2

5. e 1 ⋅e 2 = e 1

6. y = e x is continuous, increasing, and concave up for all x .

7. lim e x = +∞ and lim e x = 0 .

x →∞ x →−∞

8. e ln x = x , for x > 0; ln(e x ) = x for all x .

21. Properties of y = ln x

1. The domain of y = ln x is the set of all positive numbers, x > 0.

2. The range of y = ln x is the set of all real numbers, − ∞ < y < ∞ .

3. y = ln x is continuous and increasing everywhere on its domain.

4. ln (ab ) = ln a + ln b .

a

5. ln = ln a − ln b .

b

6. ln a r = r ln a .

7. y = ln x < 0 if 0 < x < 1 .

8. lim ln x = +∞ and lim ln x = −∞ .

x → +∞ x → 0+

ln x

9. log a x =

ln a

f '( x)

10.

d

dx

( ln f ( x ) ) =

f ( x)

and

d

dx

( ln ( x ) ) =