Braincast File - AB (1) Flashcards
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 discontinous.
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 take 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.
Derivative Applications-30Q:What are the different words for saying the derivative equals zero?
Derivative Applications-30A:Horiztonal Tangent Line
Critical Number
Derivative Applications-31Q:How do you find if a graph is increasing or decreasing?
Derivative Applications-31A:Do a sign chart:
1. Find Critical Numbers and setup intervals around them
2. Test values in each interval to determine + or -
3. Positive: increasing. Negative: Decreasing
4. Write answer and provide justifications with f’(x)>0 or f’(x)<0
Derivative Applications-32Q:What is the average rate of change and the instantaeous rate of change on a graph?
Derivative Applications-32A: The average rate of change is the same as the slope of the secant line which gives the slope from one point to another on the graph.
The instantaneous rate of change is the same as the slope of the tangent line which gives the slope of one point on the graph.
Derivative Applications-33Q:What is the other word for an instantaneous rate of change?
Derivative Applications-33A:The other word for an instantaneous rate of change is the derivative.
Derivative Applications-34Q:What does the word extrema mean?
Derivative Applications-34A: The word extrema is plural for maximums and minimums.
Derivative Applications-35Q:How do you find relative extrema?
Derivative Applications-35A:To find relative extrema for f(x):
1. find critical numbers
2. sign chart testing intervals around critical numbers
3. if f’(x) changes from (-) to (+) → min, or if f’(x) changes from (+) to (-) → max
4. See if asking what or when
5. answer
6. justifications: relative max b/c f’(x) changes from (+) to (-) OR relative min b/c f’(x) changes from (-) to (+) at that value
Derivative Applications-36Q:What is the second derivative test? How does it work?
Derivative Applications-36A:The second derivative test is a shortcut to find relative extrema.
f’‘(x) < 0 means concave down and relative max
f’‘(x) > 0 means concave up and relative min
Derivative Applications-37Q:How do you find absolute extrema?
Derivative Applications-37A:1. Find critical #’s
2. Plug crit #’s and ENDPOINTS into original equation
3. Largest answer is max, smallest is min
Derivative Applications-38Q:How do you find intervals of concavity for a function?
Derivative Applications-38A:Do a sign chart for f’‘(x)
1. Find critical #’s for f’‘(x)=0 or DNE
2. Setup intervals and test values inbetween
3. f’‘(x) > 0 concave up, f’‘(x) < 0 concave down
4. Answer and use justifications from step 3.
Derivative Applications-39Q:How do you find points of inflection?
Derivative Applications-39A:Do a sign chart or look at given information to find any of the following
- f’‘(x) change in sign
- f’(x) change in slope
Derivative Applications-40Q:What is a secant line?
What is a tangent line?
How do you find the slope of secant line?
How do you find the slope of a tangent line?
Derivative Applications-40A:Secant line is a line connecting two points on a graph. A tangent line is a line that has the same slope and height of a graph at one point on a graph.
Slope of secant: (y₂-y₁)/(x₂-x₁)
Slope of tangent: derivative at that x value
Derivative Applications-41Q:How do you find the equation of a tangent line?
How is this different than when a problem says to find the
slope of the tangent line?
Derivative Applications-41A:Use y-[y₁]=m
or y=m+y₁.
x₁ is given, find y1 by substituting x₁ into original equation, m is slope of the line (derivative at x₁).
This is different than asking for the slope of the tangent line because the slope is just the derivative value at x₁ and the equation is the full y-[y₁]=m .
Derivative Applications-42Q:How do you find the equation of a secant line?
Derivative Applications-43Q:How do you approximate f(#)? How do you know if this approximation is an over or an under approximation?
Derivative Applications-43A:To approximate f(#), just plug # into the equation of the tangent line, substitute for the x value. Remember eqn of tan. line: y=m+y₁
ex) Approximation of f(2.1) = f’(2)([2.1]-2)+f(2)
Derivative Applications-44Q:How do you find the equation of a secant line?
How do you know if a secant line approximation is an over or under approximation?
Derivative Applications-44A:To approximate f(#) using a secant line, just plug # into the equation of the tangent line. To see if it is an over or under approximation you will need to draw it and know that it involves concavity.
If the interval is concave up then the approximation is an over approximation (⊻)
If the interval is concave down then the approximation is an under approximation (⊼).
Derivative Applications-45Q:What are the justifications for increasing, decreasing, relative maximum, relative minimum, relative maximum, point of inflection, concave up, concave down?
Derivative Applications-45A:increasing = 𝑓’(𝑥) > 0
Decreasing = 𝑓’(𝑥) < 0
Relative maximum = 𝑓’(𝑥) changes from positive to negative there Relative minimum = 𝑓’(𝑥) changes from negative to positive there Point of inflection = 𝑓”(𝑥) changes signs there
Concave up = 𝑓”(𝑥) > 0
Concave down = 𝑓”(𝑥) < 0
Derivative Applications-46Q:How do you know which theorem to use?
Derivative Applications-46A: How you know which theorem to use:
1. If asks for max or mins then it is the EVT
2. If given f and asks for f or given 𝑓’ and asks for 𝑓’ then IVT
3. If given f and asks for 𝑓’ or given 𝑓′ and asks for 𝑓’’ then MVT
Derivative Applications-47Q:What are the constraints for the EVT to be true and what does it tell you?
Derivative Applications-47A:IF: f is continuous on [a, b]
THEN: f has an absolute maximum and minimum value in [a, b]
Derivative Applications-48Q:What are the constraints for the IVT to be true and what does it tell you?
Derivative Applications-48A:IF: f is continuous on [a, b] and f(a) > k and f(b) <k or f(a) < k and f(b) > k (one endpoint height is above and one is below the height of k)
THEN: f(c) = k (the graph will cross the height of k somewhere in the interval)
Derivative Applications-49Q:What are the constraints for the MVT to be true and what does it tell you?
Derivative Applications-49A:IF: f is differentiable on [a, b]
THEN: f’(x) = (y₂-y₁)/(x₂-x₁)
(the dervative somewhere in (a,b) equals the slope from one endpoint to another)
Related Rates-50Q:How do you know that a problem is a related rate? What are the other words for rates? What are the steps to related rates?
Related Rates-50A:It is a good chance a problem is a related rate when the word “when” or “at the instant” is in the problem. You know for sure when there is more than one rate. Any word that means something is changing is a rate (like moving, or shrinking, or increasing or such). The steps to related rates are SREDWU:
1. Sketch (only if it doesn’t give a shape in the problem)
2. Find the Rates
3. Make an Equation (only involving the letters from the rates)
4. Take the Derivative of that equation
5. Plug in the When
6. Find the Units
Related Rates-51Q:When doing a related rate cone or cylinder problem, how do you know if you have to do the product rule or not? If you are doing the kind that doesn’t involve the product rule, what are going to do?
Related Rates-51A:For a related rate cylinder or cone problem you will need to do the product rule if there are three rates in the problem. You do not do the product rule if there are two rates given in the problem. If the product rule is not required (only two rates given) you will need to do the following:
Cylinder: replace r with the given radius
Cone: Use similar triangles to replace either r or the h
Integrals and Differential Equations-52Q:What are the steps you should follow when trying to find integrals?
Integrals and Differential Equations-52A:The steps to integration are:
1. u-sub
2. arcs?
3. simplify
Integrals and Differential Equations-53Q:What are the steps to finding a particular solution of a differential equation?
Integrals and Differential Equations-53A:The steps to find a particular solution:
1. get y’s with dy’s and x’s with dx’s
2. integrate
3. put the +C with x’s
4. solve for y (if you e the plus C, it becomes times C) (stop here for general solutions)
5. plug in the initial condition to solve for the C
6. find the domain
Integrals and Differential Equations-54Q:How do you know that a problem is asking you to find the particular solution?
Integrals and Differential Equations-54A:A problem is asking you to find the particular solution if a differential equation is given (dy/dx=equation) and it asks for what the original equation or solution was.
Calculus and Physics-55Q:What are all the relationships between velocity, acceleration, and position (for both derivatives and integrals)?
Calculus and Physics-55A:p ‘’ = v ‘ = a
p ‘ = v
∫∫a = ∫v = p
∫a = v
Calculus and Physics-56Q:What does the word initially, originally, or at the beginning mean? What does at rest or dropped mean? What
does on the ground or at the origin mean?
Calculus and Physics-56A:The word initially, originally, or at the beginning means time = 0. At rest or dropped means velocity = 0.
On the ground or at the origin means position = 0.
Calculus and Physics-57Q:How do you know if a particle changes directions? How do you know if a particle is moving to the left or moving to the right?
Calculus and Physics-57A:A particle changes direction if you do a sign chart for velocity and the velocity changes signs. You know a particle is moving left if the velocity is negative and moving right if the velocity is positive.
Calculus and Physics-58Q:How do you know if a particle changes which side it is on? How do you know if a particle is on the right or on the left?
Calculus and Physics-58A:A particle changes which side it is on if you do a sign chart for position and the position changes signs. You know a particle is on the right if position is positive and you know it is on the left if the position is negative.
Calculus and Physics-59Q:How do you find where a particle is farthest to the right or left?
Calculus and Physics-59A:You find where a particle is farthest right or left by finding the absolute max/min of the x position.
Calculus and Physics-60Q:How do you find where a particle is moving fastest or slowest?
Calculus and Physics-60A:To find when a particle is moving fastest or slowest you find the absolute max/min of the velocity.
Calculus and Physics-61Q:Conceptually, how should you memorize position, velocity, and acceleration?
Calculus and Physics-61A:For position, velocity, and acceleration you need to think of it as a person standing in roller skates on a flat surface and…
Position = where you at
Velocity = where you going
Acceleration = where the wind pushing you
Calculus and Physics-62Q:How do you find speed and total distance?
Calculus and Physics-62A:Speed = |velocity| =
Total Distance = ₐ∫ᵇ speed dt = ₐ∫ᵇ |velocity| dt
Calculus and Physics-63Q:How do you find if a particle moving towards the origin or away from the origin?
Calculus and Physics-63A:To find if the particle is moving towards the origin or away do a double sign chart for position and velocity and if they match then the particle is moving away from the origin and if they are opposite then they are moving towards the origin. (You really need to understand why these work and not just memorize it)
Calculus and Physics-64Q:How do you know if an objects speed is increasing or decreasing?
Calculus and Physics-64A:To find if speed is increasing or decreasing you need to do a double sign chart with velocity and acceleration and if they match signs then speed is increasing and if they are opposite signs then the speed is decreasing. (You really need to understand why these work and not just memorize it)
Integrals and Word Problems-65Q:How do you take the derivative of an integral? How do you take an integral of a derivative?
Integrals and Word Problems-65A:When you take the derivative of a definite integral you must use the 2nd fundamental theorem of calculus.
d/dx[ₐ∫ᵇ f(x)dx]= b’•f(b) - a’•f(a)
When you take the definite integral of a derivative you must use the 1st fundamental theorem of calculus.
ₐ∫ᵇ f ‘(x)dx]= f(b) - f(a)
Integrals and Word Problems-66Q:What is the blanket statement for the definite integral?
Integrals and Word Problems-66A:The blanket statement is: “[Answer] [units] is the amount of [possessive noun] that the [proper noun] has [gained/lost] from t = a to t = b.”
Integrals and Word Problems-67Q:How do you find the units of an integral? How do find the units of the derivative?
Integrals and Word Problems-67A:To find the units of an integral you must multiply the units of the equation times the units of the x. To find the units of the derivative you must divide the units of the equation by the units of the x.
Integrals and Word Problems-68Q:What does the average value stand for on a graph? What does average value stand for in a word problem? How
do you know that a question is asking you to find the average value? What is the formula for the average value?
Integrals and Word Problems-68A:Average value gives you the average height of the graph over that interval.
Average value gives you the average amount of the words of the equation for a word problem.
Find the average value if it asks for the average words of the equation.
1/(b-a) ⋅ₐ∫ᵇ f(x)dx
Integrals and Word Problems-69Q:What does the average rate of change stand for on a graph? What does average rate of change stand for in a
word problem? How do you know that a question is asking you to find the average rate of change? What is the
formula for average rate of change?
Integrals and Word Problems-69A:The average rate of change is the average slope from one point on a graph to another.
The average rate of change is the average change in the possessive noun in a word problem.
Find the average rate of change if it asks for the average rate of the words of the equation.
avg rate = (y₂-y₁)/(x₂-x₁)
Integrals and Word Problems-70Q:Mathematically, what is the difference between how much you have of something and how much you gain of
something?
Integrals and Word Problems-70A:To find how much of something you have you must find the initial amount plus the integral of gain minus the integral of the loss. To find how much you gain or loss of something you just take the integral of the rate of gain or rate of loss.
Integrals and Word Problems-71Q:What does the definite integral (ₐ∫ᵇ f(x) dx) stand for on a graph? What does it stand for in a word problem?
Integrals and Word Problems-71A:The definite integral stands for the area under the curve for a graph. The definite integral stand for how much was gained or lost in a word problem.
Integrals and Word Problems-72Q:In calculus word problems, if a problem asks what happens to the amount at a specific time (example: “at time t = 2 is the particle on the right or left”), what do you do?
Integrals and Word Problems-72A:If a problem asks for what happens at a specific time, simply plug that number into the necessary equation.
Integrals and Word Problems-73Q:In calculus word problems, if a problem asks “at what time” (as in a single time) something happens (example: “at what time is the particle at rest”), what do you do?
Integrals and Word Problems-73A:If a problem asks for what time (single time) something happens, then you must set the necessary equation equal to the necessary number (example: velocity equals zero) and then solve the equation for the time.
Integrals and Word Problems-74Q:In calculus word problems, if a problem asks “at what times” (as in a multiple times) something happens (example: “at what times is the particle moving to the right”), what do you do?
Integrals and Word Problems-74A:If a problem asks what times (multiple times) something happens you must do a sign chart for the necessary equation.
Area and Volume-75Q:In area and volume problems, when you see the word x-axis cross it out and rewrite what? When you see the word y-axis cross it out and rewrite what?
Area and Volume-75A:When you see x-axis cross it out with y = 0.
When you see y-axis cross it out and write x = 0.
Area and Volume-76Q:What are the steps to find the area between two graphs?
Area and Volume-76A:To find the area between two graphs…
1. Sketch the graphs and shade the region R (graph the graph if not graphed for you)
2. find a and b (when the graphs intersect) and label the a and b on the graph
3. label top equation and bottom equation on the graph
4. take the integral ₐ∫ᵇ top − bottom dx
Area and Volume-77Q:What method do you use to find the volume of a figure rotated around y = #? What are the steps?
Area and Volume-77A:This is called Washers method. To find the volume of a figure rotated around y = #:
1. Sketch the graph (if not already done for you), shade the area, and label the revolution line.
2. Find a and b and label on the graph.
3. Label the graph that is farthest from the revolution line and the graph that is closest to the revolution line.
4. find R(x) and r(x)
R(x) : using only the graph that is farthest from the revolution line and the revolution line itself: R(x) = the higher – lower one
r(x) : using only the graph that is closest to the revolution line and the revolution line itself: r(x) = the higher – lower one
5. Find the integral: π ∫ (R(x))² − ((r(x))²
Area and Volume-78Q:What method do you use to find the volume of a figure rotated around x = #? What are the steps?
Area and Volume-78A:This is called Shells Method. To find the volume of a figure rotated around x = #:
1. Sketch the graphs (if not already done for you), shade the area, label the revolution line
2. find a and b and label them on the graph.
3. find p(x) and h(x)
p(x) =
x – rev line (if rev line is on the left of the graph) OR
rev. line – x (if rev line is on the right of the graph
h(x) = top – bottom
4. Find the integral:
2π ₐ∫ᵇ p(x)h(x)dx
Area and Volume-79Q:What are the steps to find the volume of a figure with known cross sections?
Area and Volume-79A:To find the volume of a figure with known cross
1. Sketch the graphs and shade the region R (graph the graph if not graphed for you)
2. find a and b (when the graphs intersect) and label the a and b on the graph
3. label top equation and bottom equation on the graph
4. plug (top – bottom) into s for A(s)
5. Find the integral:
ₐ∫ᵇA(x)dx =
Area and Volume-80Q:What are the area formulas for cross sectional volume?
Area and Volume-80A:square: (s²)
semi-circle:(π/8)(s²)
equil tri:(√3)/4
isos. tri: (½)(s²)
rect: (#)(s²)
Derivative and Integral Approximations-81Q:How do you approximate a derivative? How do you approximate an integral?
Derivative and Integral Approximations-81A:You approximate a derivative by finding the slope of two points right next to the derivative. Approximate an integral by doing the sums from trapezoid, left, right, etc.
Derivative and Integral Approximations-82Q:What are the steps to doing right, left and midpoint approximations?
Derivative and Integral Approximations-82A:To find right, left, or midpoint approximations of a definite integral…
1. Find the width of the rectangles (either they will be given or by using (b-a)/n, where n is the number of rectangles).
2. Draw the base of the rectangles.
3. Find the x values you will use (on the right, left, or midpoint)
4. Find the height (plugging into the equation if not given a table) at each x value from step 3 and write in the rectangle you drew.
5. Multiply each used heights by the widths
6. Add them all up
Derivative and Integral Approximations-83Q:How do you know if a right or left hand Riemann sum is an over or under approximation?
Derivative and Integral Approximations-83A:If the function is increasing then the right hand is an over approximation and the left hand is an under approximation.
If the function is decreasing then the right hand approximation is an underapproximation and the left hand is an overapproximation.
(Don’t just memorize this, know that it involves increasing and decreasing and be able to draw it).
Derivative and Integral Approximations-84Q:What are the steps to doing upper, lower, circumscribed, and inscribed Riemann sum approximations?
Derivative and Integral Approximations-84A:To find an upper, lower, circumscribed, or inscribed sum approximations of a definite integral…
1. Find the width of the rectangles (either they will be given or by using (b-a)/n, where n is the number of rectangles).
2. Draw the base of the rectangles.
3. Find the height (plugging into the equation if not given a table) at all x values.
4. Use the height of the larger one if it asks for upper or circumscribed and the smaller one if it asks for lower or inscribed.
5. Multiply each used heights by the widths
6. Add them all up
Derivative and Integral Approximations-85Q:What are the steps to doing a trapezoidal approximation?
Derivative and Integral Approximations-85A:To find a trapezoidal approximation of a definite integral…
1. Find the width of the rectangles (either they will be given or by using (b-a)/n, where n is the number of rectangles).
2. Find the height at the endpoints of each individual trapezoid (the first x and last x should be listed only
once and all other should be listed twice)
3. Multiply each trapezoid height by their widths.
4. Add them all up
5. Divide by 2.
Derivative and Integral Approximations-86Q:How do you know if a trapezoidal approximation is an over or under approximation?
Derivative and Integral Approximations-86A:If the graph is concave up then the trapezoidal approximation is an overapproximation. If the graph is concave down then the trapezoidal approximation is an underapproximation. (Don’t just memorize this, just know that it involves concavity and draw the scenario).
