calculus ab topics limits continuity,...
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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.