AP Calculus Review PDF

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A review of many topics ...

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MEGA REVIEW FOR AB CALCULUS Format for the Exam This year the AP exam will be 2 Free response questions. The first will be 25 minutes and the second will be 15 minutes. You will have 5 minutes to upload. This year’s test is open note, so please familiarize yourself with tis document thoroughly. Items that are not on this year’s test will be highlighted with ** - **

Topics I.

Limits, Continuity, Differentiability

A.

Limits L'Hopital's Rule

If lim

𝑓(𝑥)

𝑛→∞ 𝑔(𝑥)

, 𝑎𝑛𝑑 lim 𝑓(𝑥) = 0 = lim 𝑔(𝑥) 𝑜𝑟 lim 𝑓(𝑥) = ∞ = lim 𝑔(𝑥) , 𝑛→∞

𝑛→∞

𝑛→∞

𝑛→∞

f (x ) f ( x) . (This can be repeated as needed)  lim x a g ( x ) x a g (x )

then lim

Limits at Infinity

lim x 

f (x ) g ( x)

Case 1:

Higher degree on top, limit DNE (infinity or negative infinity)

Case 2:

Equal degree, limit is ratio of leading coefficients

Case 3:

Higher degree on bottom, limit is zero

Squeeze Theorem:

B.

Continuity Definition

For a function, (x), to be continuous at some point, x  a , the following three conditions must be met: I. II.

f (a ) must be defined lim f ( x ) must exist

III.

f (a ) must be equal to lim f ( x)

x a

x a

Types of Discontinuities Jump

Point (or Removable)

Infinite

1

Intermediate Value Theorem: If 𝑓(𝑥) is continuous on [a,b] and 𝑓(𝑎) < 𝑢 < 𝑓(𝑏) (or 𝑓(𝑎) > 𝑢 > 𝑓(𝑏)), then there is a value c (maybe more than one), with 𝑎 < 𝑐 < 𝑏 (c is between the x-values a and b) such that 𝑢 = 𝑓(𝑐). (can also prove the existence of a zero if f(a) and f(b) have a sign change C.

Differentiability For a function to be differentiable at a point (i.e. in order for the derivative to exist at a point), the slope must exist and be the same both from the right and the left. Examples of a function failing to be differentiable at x  a :

II.

Not Continuous The Derivative at a Point

A.

Rates of Change f (b )  f (a ) b a

Sharp Corner (Cusp)

Vertical Tangent

Average Rate of Change of  over [a, b] Slope of Secant Line to  over [a, b] Difference Quotient Instantaneous Rate of Change of  at x  a

f ( a  h)  f ( a) f (a )  lim h 0 h

Slope of the tangent to  at x  a Limit of the Difference Quotient Derivative of  at x  a

B.

Numerical Derivative of  at x  a

C.

Derivative from a graph

D.

Tangent to curve  at x  a

f (a  x )  f (a ) (the smaller x , the better the approximation) x Write equations of lines and curves whenever possible to find EXACT values. Estimate only as a last resort.

y  f (a )x  a   f (a )

Use the tangent line equation to approximate values of  close to x  a .

Tangent Line Approximation Local Linearization

If  is concave up at x  a , the tangent line approximation is an underestimate. If  is concave down at x  a , the tangent line approximation is an overestimate.

2

III.

Derivative as a Function

A.

The limit definition of the derivative

B.

Horizontal Tangents, 1st and 2nd Derivative Tests, Critical Points

f ( x )  lim h 0

f ( x  h)  f ( x) h

If f ( a)  0 , then the point ( a, f (a)) is called a critical point, and we know that  has a horizontal tangent at x  a . The following three possibilities exist:

local maximum

local minimum

plateau

To determine which of these conditions exist, we must use one of the tests below:

First Derivative Test

(We generally use a sign chart to keep organized, but must explain in words) Let  be a differentiable function with f ( a)  0 1.

If𝑓 ′ (𝑥) changes from positive to negative at 𝑥 = 𝑎, then  has a relative (or local) maximum at x  a .

2.

If 𝑓 ′ (𝑥) changes from negative to positive at 𝑥 = 𝑎, then  has a relative (or local) minimum at x  a .

3.

If 𝑓 ′ (𝑥) does not change sign at 𝑥 = 𝑎, then  has a plateau at x  a .

Second Derivative Test (Works great for implicitly defined derivatives and series at the center) Let  be a differentiable function with f ( a)  0 1.

If f (a )  0 , then  has a relative (or local) maximum at x  a .

2.

If f ( a)  0 , then  has a relative (or local) minimum at x  a .

3.

If f ( a)  0 , then the test fails. Use another test.

3

C.

Intervals of Increasing and Decreasing

1.

If f ( x)  0 over the interval (a, b), then  is decreasing over the interval (a, b).

2.

If f ( x)  0 over the interval (a, b), then  is increasing over the interval (a, b).

D.

Concavity

1.

If f ( x)  0 over the interval (a, b), then f  is decreasing over the interval (a, b) and therefore,  is concave down over the interval (a, b).

2.

If f ( x)  0 over the interval (a, b), then f  is increasing over the interval (a, b) and therefore,  is concave up over the interval (a, b).

E.

Points of Inflection Inflection points are where a function changes concavity. To determine whether a function, , has an inflection point at x  a , ALL of the following conditions must be met: 1.

f (a )  0 or f (a ) does not exist

2.

f (x) must change from positive to negative or negative to positive at x  a .

3.

f (x) must change from increasing to decreasing or decreasing to increasing at x  a .

Table of justifications – the above statements can be summarized by this table Positive Negative

Increasing Decreasing

Concave up Concave Down

𝑓(𝑥)

𝑓’(𝑥)

𝑓′′(𝑥)

The three double arrows imply equivalence. Example: If 𝑓′(𝑥) is increasing, then 𝑓(𝑥) is concave up and 𝑓′′(𝑥) is positive. 4

IV.

Derivative Rules

A.

Properties of Derivatives

B.

d c   0 where c is a constant dx

d c f ( x)  c d f ( x) where c is a constant dx dx

d  f (x ) g (x )  d f (x )  d g ( x) dx dx dx

d  f ( x)  g( x)   d f ( x)  d g (x ) dx dx dx

The shortcuts

d sin x  cos x dx

d cos x    sin x dx

d ln x  1 dx x

d x e  ex dx

d sec x   sec x tan x dx

d arctan x  1 2 dx 1 x

d arcsin x  1 2 dx 1x

 

 

d x 2  ln 2  2 x dx

d tan x  sec2 x dx

d arccos x    1 2 dx 1 x

C.

Special Rules

 

Power Rule:

d n x  n x n 1 dx

Product Rule:

d  f (x ) g (x )  f (x ) g (x ) g '(x )  f (x ) dx

Quotient Rule:

d  f (x )  f '(x )  g (x )  g '(x ) f (x )  dx  g (x )  g 2 (x )

V.

Applications of Derivatives

A.

Mean Value Theorem

Chain Rule:

d  f (g (x ))  f g ( x)  g  ( x) dx

"Lo dHi minus Hi dLo, square the bottom and away we go"

If  is continuous on [a,b] and differentiable over the interval (a, b), then there must exist at least one value, x  c, such that:

f (c ) 

f (b)  f ( a) b a

Rolle's Theorem is the special case where f (b)  f (a ) , so f (c ) must equal 0. 5

B.

C.

Optimization Step 1:

Determine what quantity you want to optimize (maximize or minimize)

Step 2:

Write an equation for that quantity

Step 3:

Make sure the equation is in terms of one variable only. Use a secondary equation to remove a variable if necessary

Step 4:

Take the derivative and set it equal to zero

Step 5:

Solve the equation. The answer may maximize or minimize the quantity. Test the value along with possible endpoints for the answer.

Step 6:

Re-read the question to make sure you have found what is being asked for.

Position, Velocity, Acceleration Velocity: x(t ) or v (t )

Position: x (t )

Acceleration: x(t ) , v(t ) , or a (t )

b

Distance Traveled over (a, b):

 v (t )dt a

b

Displacement (Change in position) over (a, b):

x (b )  x (a )

or

 v(t )dt a

Average Velocity over (a, b):

x (b)  x( a ) ba

or

1 b v( t)dt b  a a

Average Acceleration over (a, b):

v( b)  v( a) ba

or

1 b a( t)dt b  a a

VI.

Implicit Differentiation

A.

Take first and second derivatives of implicit functions with respect to x and be able to evaluate them at a point.

B.

Find horizontal and vertical tangents

C.

Related Rates

(horiz set top = 0, vertical set bottom = 0)

Step 1:

Determine what rates you have and what rate you need to find

Step 2:

Write an equation that has the variables of the rates you have and the rates you want. Only use constants in your equation, not temporary values

Step 3:

Differentiate implicitly 6

Step 4:

Substitute rates and values into derivative

Step 5:

Re-read the question to make sure you have found what is being asked for.

VII.

Approximate Integration

A.

Riemann Sums

b a n

n = number of partitions,  x 

LRAM(n) =x  f (a )  f (a  x )  f (a  2x )  ...  f (b   x )  RRAM(n) = x  f (a  x )  f (a  2x )  ...  f (b  x)  f (b) 

x  f (a )  2 f (a  x )  2 f (a  2 x )  ...  2 f (b  x )  f (b )  2

TRAP(n) =

MID(n) =  x( f ( a 

x x 3 )  f ( a  )  ...  f ( b )) 2 2 2

 increases overestimates underestimates -----------

RHS LHS TRAP MID B.

RHS( n)  LHS( n) 2

or

 decreases underestimates overestimates -----------

 concave up ----------overestimates underestimates

 concave down ----------underestimates overestimates

Writing Definite Integrals as Riemann Sums b

 f (x )dx



a

n

lim n 

 f  a  x i  x 

where

x 

i1

ba n

VIII. Integration Rules A.

Simple Integrals

 cos x dx  sin x  C 

dx  ln x  C x

x  2 dx 

x

2 C ln 2

 sin x dx  cos x  C

 sec

 e dx  e

 sec x tan xdx  sec x  C

x

1

x

2

C 1

𝑥

∫ 𝑎2 +𝑥2 𝑑𝑥 = 𝑎 arctan 𝑎 + 𝐶

∫

x dx  tan x  C

1 √𝑎2 +𝑥2

𝑥

𝑑𝑥 = arcsin 𝑎 + 𝐶

DON’T FORGET THE + C!! 7

B.

Properties of Integrals

b

b

b

b

a

a

a

a

 f ( x)  g ( x) dx   f ( x)dx   g ( x)dx

 cdx  c  b  a where c is any constant

b

b

b

b

b

a

a

a

a

a

 f ( x)  g ( x) dx   f ( x)dx   g ( x) dx

 c f ( x) dx  c f ( x) dx where c is any constant b

a

a

b

 f (x )dx    f (x )dx C.

c

b

b

a

c

a

 f (x )dx   f (x )dx   f (x )dx

Advanced Techniques of Integration

Long Division and Completing the Square U-Substitution:

If u  g ( x) , is a differentiable function, then

 f  g ( x) g ( x) dx   f ( u) du

The idea behind u-substitution is to replace a relatively complicated integral by a simpler integral by changing the original variable x to a new variable u that is a function of x. The main challenge is thinking of an appropriate substitution. Try to let u equal the inside function and then look for du. The derivative should differ by at most a constant. For definite integrals, be sure to change the limits of integration! IX.

The Fundamental Theorem of Calculus (FTC)

A.

Total Change in F over [a, b]

B.

FTC

F (b)  F (a) b

If F   f , then

 f (t )dt  F (b )  F (a ) a

C.

𝑏

If you know Some value of 𝐹(𝑥) (𝐹(𝑎)) and want a new value (𝐹(𝑏)): 𝐹(𝑏) = 𝐹(𝑎) + ∫𝑎 𝑓(𝑡)𝑑𝑡

“The definite integral of  over the interval [a, b] is equal to the total change in F over [a, b].” “The area under the curve  over the interval [a, b] is equal to the change in y-values in F over [a, b].” x

C.

Second FTC

If g ( x)   f (t )dt , then g ( x )  f ( x ) a

a

Note: g (a )  0 because g (a )   f (t )dt  0 a

8

D.

Finding Absolute Maximums and Absolute Minimums –

Candidates test – Abs Max and min can only occur at critical points (where f  ( x)  0 ) or endpoints. Find the critical points and endpoints, plug into original function to determine largest/smallest values. (make a table) First derivative test for Abs. Extrema -If 𝑓(𝑥) is continuous on any interval and 𝑓 ′ (𝑥)changes sign only once, 𝑓(𝑥) will have a max or min at that critical point. XI.

Differential Equations and Slope Fields

A.

Slope Fields: What variables are in the DE? Only x: slopes change horizontally only y: slopes change vertically Both x and y: slopes change both horizontally and vertically Set D.E = 0. Look for horizontal tangents here. Set D.E.> 0 Look for positive slopes here. Set DE...