AP CALCULUS AB and BC Final Notes - Folsom Cordova Unified ...

AP CALCULUS AB and BC

Final Notes

Trigonometric Formulas

1. sin 2 + cos2 = 1 2. 1 + tan 2 = sec2

13. tan = sin = 1 cos cot

3. 1 + cot 2 = csc2 4. sin(- ) = - sin

14. cot = cos = 1 sin tan

5. cos(- ) = cos 6. tan(- ) = - tan

15. sec = 1 cos

7. sin( A + B) = sin Acos B + sin B cos A 8. sin( A - B) = sin Acos B - sin B cos A

16. csc = 1 sin

9. cos( A + B) = cos Acos B - sin Asin B

17. cos= 2 1 (1+ cos 2 )

2

10. cos( A - B) = cos Acos B + sin Asin B

11. sin 2 = 2sin cos

18. sin= 2 1 (1- cos 2 )

2

12. cos 2 = cos2 - sin2 = 2 cos2 -1 = 1- 2 sin2

Differentiation Formulas

1. d (x n ) = nx n-1 dx

2. d ( fg) = fg + gf Product rule dx

3.

d

(

f

)=

gf -

fg

Quotient rule

dx g

g2

4. d f (g(x)) = f (g(x))g (x) Chain rule dx

5. d (sin x) = cos x dx

6. d (cos x) = - sin x dx

7. d (tan x) = sec2 x dx

8. d (cot x) = - csc2 x dx

9. d (sec x) = sec x tan x dx

10. d (csc x) = - csc x cot x dx

11. d (e x ) = e x dx

12. d (a x ) = a x ln a dx

13. d (ln x) = 1

dx

x

14. d ( Arc sin x) = 1

dx

1- x2

15. d ( Arc tan x) = 1

dx

1+ x2

16. d ( Arc sec x) =

1

dx

| x | x2 -1

17. d [c] = 0

dx

18.

d dx

cf

(

x )

=

cf

'(

x)

Integration Formulas

1. a dx = ax + C

2. x n dx = x n+1 + C, n -1

n +1

3.

1 x

dx

=

ln

x

+

C

4. e x dx = e x + C

5. a x dx = a x + C

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

13. sec2 x dx = tan x + C

14. sec x tan x dx = sec x + C

15. csc2 x dx = - cot x + C

16. csc x cot x dx = - csc x + C

17. tan 2 x dx = tan x - x + C

18.

dx a2 + x2

=

1 a

Arc tan x + C a

19.

dx = Arc sin x + C

a2 - x2

a

20.

dx = 1 Arc sec x + C = 1 Arc cos 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 f ( x) exists xa

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

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 f (x) = L = lim f (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

.

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

B

The amplitude is A . The period of y = tan x is .

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 ?

i).

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

x? g(x)

Example:

lim

x

x2 - 2x x3 + 3

=

0

ii).

lim

f (x)

is infinite if the degrees of

f (x) > the degree of g(x)

x? g(x)

Example:

lim x3 + 2x = x x2 - 8

f (x)

iii). lim

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

x? g(x)

Example:

lim 2x2 - 3x + 2 = x 10x - 5x2

-2 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

xa+

xa-

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 -

y0

=

y

.

x1 - x0

x1 - x0 x

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 y with respect to x at x0 is f (x0 ) .

8.

Definition of Derivative

f (x) = lim f (x + h) - f (x) or f '(a) = lim f (x) - f (a)

h0

h

xa 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

i).

lim

n

1

+

1 n

n

= e

ii).

lim (1+ )n 1/n = e

n0

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

in (a,b) such that f (b) - f (a) = 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 xn .

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

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

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

We say, that as x :

1.

f

(x)

grows faster than

g(x)

if

lim

x

f (x) g(x)

=

or if

lim

x

g(x) f (x)

= 0.

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

x.

2.

f (x) and g (x) grow at the same rate as x if

f (x)

lim

x

g

( x )=

L 0 (L is finite

and nonzero).

For example,

1.

e x

grows faster than

x 3

as

x

since

e x

lim

x

x3

=

2. x4 grows faster than ln x as x since lim x4 = x ln x

3.

x2 + 2x

grows at the same rate as

x 2

as

x

since

x2 + 2x

lim

x

x2

= 1

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'H?pital'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

xa g(x)

0

xa g(x)

lim f (x) = lim f (x) . xa g(x) xa 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

g = f -1 (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

g '(b) =

f

1

'(a) .

20.

Properties of y = e x

1.

The exponential function y = e x 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 (e x ) = e x and d e f (x) = f '( x) e f (x)

dx

dx

5.

e x1 e x2 = e x1 + x2

6.

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

7.

lim ex = + and lim ex = 0 .

x

x-

8.

eln 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 .

5.

ln a = ln a - ln b .

b

6.

ln ar = r ln a .

7.

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

8.

lim ln x = + and lim ln x = - .

x +

x 0+

9.

loga

x

=

ln ln

x a

10.

d (ln

dx

f

(x)) =

f '(x) f (x)

and

d (ln ( x)) =

dx

1 x

22. Trapezoidal Rule

[ ] [ ] If a function f is continuous on the closed interval a, b where a, b has been equally

[ ] [ ] [ ] partitioned into n subintervals

x0 , x1 ,

x1, x2 , ... xn -1, xn

, each length

b-a

, then

n

[ ] b f (x) dx b - a

a

2 n

f (x0 ) + 2 f (x1) + 2 f (x2 ) + ... + 2 f (xn -1) + f (xn ) , which is

equivalent to 1 ( Leftsum + Rightsum)

2

23a. Definition of Definite Integral as the Limit of a Sum

[ ] Suppose that a function f (x) is continuous on the closed interval a, b . Divide the interval into

n equal subintervals, of length x = b - a . Choose one number in each subinterval, in other n

words, x1 in the first, x2 in the second, ..., xk in the k th ,..., and xn in the n th . Then

n

b

( ) lim f

n k =1

xk

=x

f ( x) d=x

a

F (b) - F (a) .

23b. Properties of the Definite Integral

[ ] Let f (x) and g(x) be continuous on a, b .

b

b

i). c f (x) dx = c f (x) dx for any constant c .

a

a

a

ii). f (x) dx = 0 a

b

a

iii). f (x) dx = - f (x) dx

a

b

b

c

b

iv). f (x) dx = f (x) dx + f (x) dx , where f is continuous on an interval

a

a

c

containing the numbers a, b, and c . a

v). If f (x) is an odd function, then f (x) dx = 0

-a

a

a

vi). If f (x) is an even function, then f (x) dx = 2 f (x) dx

-a

0

b

vii). If f (x) 0 on [a,b], then f (x) dx 0 a

b

b

viii). If g(x) f (x) on [a,b], then g(x) dx f (x) dx

a

a

24. Fundamental Theorem of Calculus:

b

a

f (x) dx =

F (b) - F (a), where F (x) =

f (x), or

d dx

b

a

f (x) dx

=

f (x) .

25. Second Fundamental Theorem of Calculus (Steve's Theorem):

d dx

x

a

f (t)

dt

=

f

(x)

or

= d g(x) f (t) dt

dx h(x)

g '(x) f (g (x)) - h '(x)f (h (x))

26. Velocity, Speed, and Acceleration 1. The velocity of an object tells how fast it is going and in which direction. Velocity is an instantaneous rate of change. If velocity is positive (graphically above the "x"-axis), then the object is moving away from its point of origin. If velocity is negative (graphically below the "x"-axis), then the object is moving back towards its point of origin. If velocity is 0 (graphically the point(s) where it hits the "x"-axis), then the object is not moving at that time.

2. The speed of an object is the absolute value of the velocity, v (t ) . It tells how fast it is going

disregarding its direction. The speed of a particle increases (speeds up) when the velocity and acceleration have the same signs. The speed decreases (slows down) when the velocity and acceleration have opposite signs. 3. The acceleration is the instantaneous rate of change of velocity ? it is the derivative of the

velocity ? that is, a (t ) = v '(t ) . Negative acceleration (deceleration) means that the velocity is

decreasing (i.e. the velocity graph would be going down at that time), and vice-versa for acceleration increasing. The acceleration gives the rate at which the velocity is changing.

Therefore, if x is the displacement of a moving object and t is time, then:

i) velocity = = v (t) x= '(t ) dx

dt

ii) acceleration = a (=t)

x ''(=t)

v '(=t)

d=v dt

d2x dt 2

iii) v (t) = a (t)dt

iv) x (t) = v (t)dt

Note: The average velocity of a particle over the time interval from t0 to another time t, is

Average Velocity = Change in position = s (t ) - s (t0 ) , where s (t ) is the position of the particle

Length of time

t - t0

( ) 1 b

at time t or

v t dt if given the velocity function.

b-a a

27.

[ ] The average value of f (x) on a, b is

1

b f (x) dx .

b-a a

28. Area Between Curves

[ ] If f and g are continuous functions such that f (x) g(x) on a, b , then area between the

b

b

d

curves is [ f (x) - g(x)]dx or [top - bottom] dx or [right - left]dy .

a

a

c

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