AP Calculus AB Review Week 1 (Basics, Limits/Asmyptotes, Derivatives)

The Basics-1Q:How do you multiply fractions, divide fractions, and add fractions?

The Basics-1A:You multiply fractions straight across, divide fractions by flipping the second fraction and multiplying straight across, and add fractions by using the butterfly method.

The Basics-2Q:How do you find a fraction power [such as 9^(3/2)]
How do you find a negative power like 2⁻³

The Basics-2A:You do a fraction power by taking the root of the bottom number and then the power of the top number (therefore you would square root the nine and then third power it to get 27).
You do a negative power by putting it on the bottom of the fraction and then powering it (therefore 2⁻³ = 1/(2³) = 1/8

The Basics-3Q:What can xᵃ⁺ᵇ be rewritten as?
What can xᵃ⁻ᵇ be rewritten as?
How can you rewrite xᵃ*ᵇ?

The Basics-3A:xᵃ⁺ᵇ=xᵃ•xᵇ
xᵃ⁻ᵇ=xᵃ/xᵇ
xᵃ*ᵇ=(xᵃ)ᵇ

The Basics-4Q:Formula for area of a circle? Quarter circle?
Formula for area of a rectangle? Triangle? Square?
Formula for circumference of a circle?

The Basics-4A:Circle = 𝝅r²
Quarter Circle = 𝝅r²/4
Rectangle = base* height
Triangle = base * height / 2
Square = base²
Circumference = 2𝝅r

The Basics-5Q:Formula for Surface area of a cube and sphere?
Volume of a cube and sphere?

The Basics-5A:SA cube: 6* side², SA sphere: 𝝅r²
Vol cube: side³, Vol sphere: 4/3 * 𝝅r³

The Basics-6Q:What is e to the zero? When is eˣ =0? What will eˣ always be equal to?

The Basics-6A:e⁰=1. eˣ is always positive, never equal to zero.

The Basics-7Q:What is ln(0)? What is ln(1)? What is ln(- #)?

How can you rewrite ln(a*b), ln(a/b) and ln(a)ʳ?

The Basics-7A:ln(0) DNE. ln(1) = 0. ln(-#) DNE.

ln(a)+ln(b)=ln(a+b).
ln(a)-ln(b)=ln(a/b).
r*ln(a) = ln(a)ʳ

The Basics-8Q:What is the formula for finding the slope between two points?

The Basics-8A:m=(y₂-y₁)/(x₂-x₁)

The Basics-9Q:What can sec(x) be algebraically simplified to? What about csc(x) , tan(x) , and cot(x)?

The Basics-9A:sec(x)=1/cos(x)
csc(x)=1/sin(x)
tan(x)=sin(x)/cos(x)
cot(x)=cos(x)/sin(x)

The Basics-10Q:What is the difference between sec(x) and arccos(x)?
What is the difference between csc(x) and arcsin(x)?

The Basics-10A:sec x is 1 over cosine and arccos is the opposite of cosine (it undoes a cosine in solving a problem).
Csc x is 1 over sine and arcsine is the opposite of sine (it undoes a sine in solving problems). They are not related to each other at all.

The Basics-11Q:What is SOHCAHTOA and the pythagorean theorem?

The Basics-11A:sin( ) = opp/hyp,
cos( ) = adj/hyp
tan( ) = opp/adj
Py Thm: (adj)²+(opp)²=(hyp)²

The Basics-12Q:When solving an equation how do you know if you have to use SADMEP or factoring? Explain what SADMEP
means?

The Basics-12A:When solving equations, you will use SADMEP when the variable in the problem is only written once.
And, you will use factoring when the variable is written more than one time.

The Basics-13Q:What is the opposite of adding, the opposite of multiplying, the opposite of squared, the opposite of cubed,
the opposite of sin, the opposite of cos, the opposite of tan, the opposite of ln?

The Basics-13A:The opposite of adding is subtracting. The opposite of multiplying is dividing. The opposite of squared is plus and minus square root. The opposite of cubed is cube root. The opposite of sine is arcsine (sin⁻¹ x in your calculator). The opposite of cosine is arccosine (cos⁻¹ x in your calculator). The opposite of tangent is arctangent (tan⁻¹ x in your calculator). The opposite of ln is e to the power.

The Basics-14Q:Calculator:
*What do you do first before doing a calculator with an equation?
a) How do you find f(#)?
b) How do you find when an equation equals zero?
c) How do you find the intersection of two graphs?
d) How do you find the derivative?
e) How do you find the integral?

The Basics-14A:FIRST, input equation into [y=].
a) Then graph, use trace calc menu, select [1:value]
b) Then graph, use trace calc menu, select [2:zero], set left bound, right bound, and guess
c) Then graph, use trace calc menu, select [3:intersection], set left bound, right bound, and guess
d) graph, use trace calc menu, select [6:dy/dx], input x value
e) Then graph, use trace calc menu, select [7:integral], input x value lower bound, and then upper bound

The Basics-15Q:How do you know if numbers with parenthesis like (2,7) are an interval or a point? What is the difference between an interval and a point?

The Basics-15A:Words before the parenthesis:
on, in: interval
at: point
An interval is a distance from one x to another, a point is one spot on the graph.

Continuity and Asymptotes-16Q:How do you know if a graph is continuous?

How do you know if a graph is differentiable?

Which is true?: if a graph is continuous it is also differentiable OR if a graph is differentiable it is also continuous.

Continuity and Asymptotes-16A:A graph is continuous if you can graph the graph without lifting your pencil. The left limit = the middle = the right limit.

A graph is differentiable if it has no gaps in the graph, sharp turns, or vertical points.

IF a graph is differentiable, it is also continuous.

Continuity and Asymptotes-17Q:There are 4 reasons a function is discontinuous.

Name them

Continuity and Asymptotes-17A:1 - divide by zero
2 - piecewise functions
3 - ln of zero or a negative
4 - square root of a negative

Continuity and Asymptotes-18Q:How do you find the intervals for which a function is continuous if there is a ln or square root involved?

Continuity and Asymptotes-18A:To find continuity algebraically for ln or square root make a discontinuity sign chart:
1. Find the discontinuous boundary point(s) (the inside equation equaling zero)
2. Start at negative infinity in top row of chart and end at infinity, stopping at each discontinuity point
3. Pick values in between stopping points and check if positive or negative value for the inside equation
4. Negative DNE for both ln and √, zero DNE for ln only
5. answer in interval form

Continuity and Asymptotes-19Q:How do you know if piecewise function is continuous or not?

Continuity and Asymptotes-19A:To find continuity of a piecewise function check to see if the heights match at the boundary. If they do not, then the boundary is a discontinuity point.

Continuity and Asymptotes-20Q:Of the 4 reasons for discontinuity (divide by zero, piecewise functions, ln of 0 or negative, square root of negative) WHICH ONE IS A REMOVEABLE DISCONTINUITY?

Continuity and Asymptotes-20A:Divide by zero is the only removable discontinuity.
If you plug the zero from the bottom into the top and get zero then you have a hole/removable discontinuity. If you plug the zero on the bottom into the top and get anything other than zero then you have a vertical asymptote/non-removable discontinuity.

Continuity and Asymptotes-21Q:Graphically and Mathematically:
What is a jump discontinuity?
What is an infinite discontinuity?
What is an essential discontinuity?

Continuity and Asymptotes-21A:Jump: heights do not match from left and right. Happens with absolute value center or piecewise
Infinite: Line goes vertical . Equation = #/0.
Essential: sin(#/0) cos(#/0), aka sin(∞) or cos(∞) which would be oscillating

Continuity and Asymptotes-22Q:What is the blanket statement for theorems? What gives you a hint that a problem is probably one of these?

Continuity and Asymptotes-22A:Due to the name of the theorem, since constraints necessary, there is a c in (a, b) such that what the theorem says. If the problem asks if there “must be” or if it is a yes or no question it is a good chance it’s a theorem problem.

Continuity and Asymptotes-23Q:How do you find the height of a hole?

How do you find one-sided limits?

Continuity and Asymptotes-23A:You find the height of a hole by taking the limit there. You find a one sided limit with the same steps as limits

Continuity and Asymptotes-24Q:What does a limit tell you about a graph?
What are the steps to finding limits?
How do you know if a limit is undefined?

Continuity and Asymptotes-24A:A limit tells you what height (y-value) that a graph is approaching as the graph approaches the x value you are taking the limit at, or you can talk about it is the height that two roller coasters collide at or not.

Steps to limits: 1-plug in the number, 2 – l’hopitals rule 3-table (using your calculator). A limit is undefined if the y-values approach different numbers in your table.

A limit is undefined if the y-values approach different numbers in your table.

Continuity and Asymptotes-25Q:How do you find horizontal asymptotes?

How do you find limits as x approaches infinity?

What is the order of functions from slowest to fastest?

Continuity and Asymptotes-25A:You find a horizontal asymptote by finding the limit as x approaches infinity and negative infinity.
You find the limit as x approaches infinity by comparing the top and bottom of an equation (if the top is faster then you get infinity or undefined, if the bottom is faster then you get zero, and if they match then you get their coefficients).
The order of functions from slowest to fastest: trig, constant, lnx, squareroot, linear, polynomial, exponential

Derivatives-26Q:When taking the derivative or integral of an equation like 1/(x²), what would you need to do first? What about for ∜(x^5)

Derivatives-26A:Change terms with powers so that it is x^( ). A power in the denominator will become negative. Roots will turn into fractional powers.

Derivatives-27Q:How do you take the derivative of two equations being multiplied together?

How do you take the derivative of two equations being divided?

How do you take the derivative of an equation that is inside of another equation?

Derivatives-27A:To take the derivative of two equations being multiplied use the Product Rule:
d/dx[f(x)g(x)]=f'(x)g(x)+f(x)g'(x)
To take the derivative of two equations being divided use the Quotient Rule:
d/dx[f(x)/g(x)]=[f'(x)g(x)-f(x)g'(x)]/[(g(x))^2]
To take the derivative of an equation that is inside of another equation use the Chain Rule:
d/dx[f(u)]=f'(u)•u'

Derivatives-28Q:What are the two limit definitions of the derivative?

What is each one used for?

Derivatives-28A:Limit of the difference quotient
lim h→0 [(f(x+h)-f(x))/h]

and the alternative form
lim x→# [(f(x)-f(#))/(x-#)]

The limit of the difference quotient will be on multiple choice and will need to be recognized. The alternative form is used to define differentiability.

Derivatives-29Q:How do you find the derivative of an equation that has x and y on the same side?

Derivatives-29A:Use implicit differentiation. To implicit differentiate:
1. Take the derivative of each piece
2. Put dy/dx when taking the derivative of a y.
3. Get dy/dx terms on one side and the other stuff on the other side.
4. Factor out dy/dx (if more than one)
5. Divide the dy/dx stuff to the other side.