Calculus AB Topics

Limits

◦ Continuity, Asymptotes

Consider 2 1 3

13

3

x x

f xx

x

Is there a vertical asymptote at x = 3?

Do not give a Precalculus answer on a Calculus exam.

Consider 2 1 3

13

3

x x

f xx

x

Is there a horizontal asymptote at y = 0?

Definition of Continuity

• Continuity at an interior point –

• Continuity at an endpoint -

limx c

f x f c

lim limx a x b

f x f a or f x f b

Derivative Highlights

Limit Definitions of Derivative

Do you REALLY know the definitions of a derivative???

• What are they???

( ) ( )limx a

f x f a

x a

0

( ) ( )limh

f x h f x

h

Example

3 3

0( ) lim

h

x h xg x

h

(5) ?g

’

’

• What does it mean for a function to be differentiable?

– The derivative exists at all x values or at a specified x value

– For a derivative to exist…the LEFT hand derivative, must equal the RIGHT hand derivative.

• What is the example of a function that is continuous at a specific x value but not differentiable at that x-value?

product rule: d dv du

uv u vdx dx dx

Notice that this is not just the product of two derivatives.

quotient rule:

2

du dvv u

d u dx dx

dx v v

change in positionVelocity =

change in time

Average velocity = slope between two positions

Instantaneous velocity = velocity at instant in time

= derivative at a point

Speed = velocity

change in velocityAcceleration =

change in time

derivative of velocity

= 2nd derivative of position

Summary of Trig Derivatives

d

dxsin x cos x

d

dxcsc x csc x cot x

d

dxcos x sin x

d

dxsec x sec x tan x

d

dxtan x sec2 x

d

dxcot x csc2 x

Remember: sec2 x sec x2

GOLDEN TICKET!!!

Definition of the Chain Rule

d

dxf g x f ' g x g ' x

Derivative of “outside” function

Derivative of “inside” function

Implicit Differentiation

1. x2 y2 25 Find dy

dx

2x 2ydy

dx0

Power rule Chain rule

Derivative of Constant is 0

Inverse Trig Rules

dy

dxsin 1u

1

1 u2du

dy

dxtan 1u

1

1 u2du

dy

dxsec 1u

1

u u2 1du

dy

dxcos 1u

1

1 u2du

dy

dxcot 1u

1

1 u2du

dy

dxcsc 1u

1

u u2 1du

GOLDEN TICKET!!!

Exponentials and Logs

lnu u u ud du d due e a a a

dx dx dx dx

1 1ln log ln

lna

dy du dy duu u

dx u dx dx u a dx

GOLDEN TICKET!!!

Applications of Derivatives

Extrema and Concavity • Critical points are where the derivative is zero or

undefined.

• Extrema may occur at critical points and endpoints

• Finding extrema on a closed interval—TEST the ENDPOINTS (Extreme Value Theorem)

• 1st derivative tells you where the function is increasing/decreasing and possible extrema

• 2nd derivative tells you where the function is concave up/down and possible points of inflection.

Sign charts • Sign charts are valuable tools and are allowed, BUT

THEY ARE NEVER NEVER NEVER SUFFICIENT TO EARN A POINT

• To earn the test points you must interpret the sign chart using words

• Your words should demonstrate that you understand the connection between the positive/negative behavior of a graph and the increasing/decreasing/extrema behavior of the parent function and how the increasing/decreasing behavior determines the extrema behavior

Intermediate Value Theorem If a function is continuous on [a,b] then the function takes on all values between f(a) and f(b).

Mean Value Theorem If f is continuous on [a,b] and differentiable on (a,b) Then there exists a number, c, in (a,b) such that

f ' cf b f a

b a

Optimization

• Write one or two equations that model the situation described.

• You may need to write two equations and use substitution.

• Take the derivative to find the maximum or minimum.

An open top box is to be made by cutting congruent squares from the corners of a 20-by-25 inch sheet of tin and bending up the sides. What dimensions give the box

the LARGEST volume? Steps:

1. Set up an equation to maximize or minimize

1a. You may have 2 equations and substitute into one.

2. Take its derivative.

3. Set derivative = 0 and check slopes to show it is a max/min.

4. Answer the question asked.

Linearization

• A linearization is just another term for tangent line!

• You can use a linearization to estimate the value of a function at a given x-value.

• The closer the x-value is to the point of tangency the

better the estimation.

Related Rates

• Draw a picture to model the problem

• Write an equation that reflects the model

Pythagorean Theorem

Trig

Similar Figure (such as cones)

• Plug in values that never change

• Implicit differentiation

• Plug in values that do change and solve

Integration

Riemann Sums LRAM RRAM Midpoint - midpoint of x-interval, not the geometric midpoint Trapezoidal

A1

2h b0 2b1 2b2 ... 2bn 1 bn

h width of partition

b value of function at the specific x value

If intervals are not uniform width, rectangles and trapezoids must be calculated individually

Fundamental Theorem of Calculus

d

dxF x

d

dxf t

a

x

dt f x

d

dxf t

a

u

dt f udu

dx

533

0

3 1 5 3 5 1 5

xd

Example t t dt x xdx

FTC – Evaluative Part

f x dxa

b

F b F a

Average Value Theorem: If a function is integrable on [a,b], then its average y-value is:

Average1

b af x

a

b

dx

x2 5dx

x2 5dx2

5

Is called a definite integral. We can evaluate it and get a numerical answer.

Is called an indefinite integral. Its solution is the set of all possible antiderivatives.

Finding “c”

Solve with the initial condition F(1) = 4 3x2 dx

1. Find the antiderivative family

2. Substitute the given initial condition values

3. Solve for c

4. Substitute back into antiderivative

U - Substitution:

2 3( 5) (2 ) x x dx

Inverse Trig Integrals

1.4

4x2 1 dx

What inverse trig function looks like the best fit? Get the 1 in the bottom Factor constants outside the integral Write as (something)2

Use u-Substitution with inverse trig formula

1

2

1

2

1

2

1sin

1

1tan

1

1sec

1

x cx

x cx

x cx x

Applications of Integrals

Velocity is the derivative of position. Position is the integral of velocity.

Displacement is the distance from an arbitrary starting point at the end of some interval. It is the difference between ending position and starting position. Total distance is the how far an object travelled regardless of direction.

Displacement v ta

b

dt

Total Distance v ta

b

dt

Position

Velocity

Acceleration

Deriv

ativ

es

Inte

gra

ls

Summary

To find You

Displacement Total Distance Position at a specific time

v ta

b

dt

v ta

b

dt

v t dt P t c

Solve the indefinite integral and use given initial condition to find “c”

Other Applications

• Velocities (distance and displacement) • Rates of flow (e.g. oil leaking from a tanker or water

flowing into a container) • How many cars flow through an intersection • How much money has accumulated in a bank account

• Anytime you know the rate of something and what to

know how something has accumulated.

What if we want area between two curves?

S top bottoma

b

S f x g xa

b

dxS

dx

x value

x valuefunction in x dx

a b

a

b

dy

y value

y valuefunction in y dy

Why does x always have all the fun?

Let’s integrate with respect to y

Disks and Washers

Disk method Washer method

2 2

b

a

R r dx2

b

a

r dx

The axis of rotation is important

But different axis of rotation = different solid

Same equations = same area being rotated.