all right here's my 10 minute ap calculus a b review here we go so first start with limits and continuity all right limits from the left and the right okay and then a limit only exists if it approaches the same value from the left and the right some tricks for evaluating limits you're always going to plug in the value first when you get a number you're done if you get 0 over 0 or infinity over infinity you're going to use l'hopital's rule and if you get any some number divided by 0 then it's some kind of infinity so if you get like 1 over 0 or 10 over 0 and you plug it in some kind of infinity you have to decide whether the sign of that 0 is positive or negative because you're thinking like a 1 divided by a tiny number it's a huge number it could be positive or negative horizontal asymptotes are limits at infinity so either left or right direction they can be two different horizontal asymptotes by the way so make sure you check the both positive infinity and negative infinity a function is continuous this is the continuity limit here right here on this guy that is if the limit approaches the value of the function at that point it's considered continuous continuous basically means no breaks and no gaps so a function is continuous when the limit exists from the left and the right and the value of the function matches what the limit approaches there okay derivatives remember the limit definition of the derivative not a big topic on the ap exam but just make sure you understand what those limit definitions are as well as all your basic derivative rules know all of know how to symbolically do differentiation for both product rule quotient rule and chain rule because they love to do derivatives where it's like you don't have the function exactly but it's like a function and you get a table and you have to extract the information okay understand differentiability when is a function not differentiable one if it's not there's no derivative and that could be because one it's not continuous if the slopes from the left and the right mismatch and that we call like a corner so it's continuous if it touches but then it like you know it like sort of bends into a corner or the function or the tangent line is vertical at that point we call that a cusp or if the function is just simply undefined at that point there's no possible way it can have a slope of a tangent line there or derivative implicit differentiation is when we take derivatives of equations because you can't solve for y directly so you take an equation and you just take the derivative of both sides note that every time you take the derivative with a y in there you're going to treat it you're going to get a d y d x pop out and your goal is to isolate the d y d x and solve for it now you're going to get an answer in terms of x and y that's the downside of implicit differentiation the derivative is not just in terms of x you'll have x's and y's in there but that's okay logarithmic differentiation is a kind of derived differentiation where if the func if you're doing a function to an exponent function you actually take the natural log of both sides and then do implicit differentiation instead okay um big things to remember on applications of derivatives this is super important know how to do a tangent line how do i ever find when i teach my students if you're asked to find the equation of a tangent line you are always needs two pieces of information you need a slope you need a point the point comes from either it's given to you or you're given the x value you plug it into the function to get the y value the slope comes from taking the derivative at that point and then you do slope point form no need to simplify slope point form you can just leave it like that in the free response questions okay cool um no curve behavior for differentiation know that when they're asking you if a function is increasing that means the derivative is greater than zero if it's decreasing the derivative is less than zero if it's concave up the second derivative is positive is concave down the second derivative is negative you find relative mins and max's by doing the first derivative test or the second derivative test okay points of inflection are where the concavity changes so the second derivative goes from concave or negative to positive or positive or negative or you just going from concave up to concave down and we do absolute min and max's by using the candidates test that is the end points plus any critical points optimization and related rates are not heavily tested on ap calculus so i don't really focus too much on them but just as a quick overview optimization is where you're trying to maximize something that means you want to create a function that you're trying to maximize or minimize identify that function make it in terms of one variable take the derivative set it equal to zero just like you would maximize or minimize any other function related rates is you need to identify the rates that you're given and the rate that you're trying to find in the setup of the problem you want an equation that relates those two variables and then you're going to take the derivative with respect to time when you're doing those related rates like i said optimization related rates not heavily tested so i don't really spend a huge amount of time on them i haven't seen an frq that focuses on optimization or related rates although if they want to throw curveball that could happen one year integration okay make sure you know how to do riemann sums and make sure you know how to do them from a table okay and know that right or left sums are under or over estimates depending on whether the function is increasing or decreasing which again remembers the derivative is positive or negative trapezoid sums are over under estimates depending on the concavity of the curve that is if it's concave up or concave down and the way i like to do some of those to remember them is just think about if i have a concave up and i do trapezoid rule and i connect the points like that and i'm making these trapezoids that's going to be an overestimate because that area is bigger whereas if it's concave down and i connect the points and make the trapezoids like that okay then that's an underestimate same with the increasing and decreasing i think it was increasing and it's a left sum then your rectangles will look like this and that is an underestimate so think about the increasing or decreasing just make a little quick sketch like that and help remind yourself whether or not you think it's uh overall underestimate any area below the x-axis is negative area and any when you're integrating from left to right and then understand fundamental theorem of calculus the two forms they're basically telling you the same thing is that when you integrate you take the antiderivative you plug in the endpoints and you subtract if you take the derivative of something with an x in it you just take that x and you plug it in this is the more complicated version if these functions are not of x then what you do is you're going to plug in h of x you're going to plug in g of x like if the bounds are weird functions of x's you plug in h of x and g of x you subtract them but you multiply this by chain rule these two pop out because of chain rule okay so make sure you know how to take derivatives especially if the bounds of that integral are functions of x's instead okay some applications of integrations you need to know make sure you know how to draw area between curves make sure you know how to do volumes in general i know volumes are tricky you spend a lot of time it's conceptually difficult there may be one part of an frq and a few multiple choice questions that are related to volumes so it's useful to know for volumes in general your goal is to find the area of the shape times its thickness okay so you're going to take that rectangle you're either going to make a cross-sectional shape or you're going to revolve it around axes and you want to um you want to you want to work on that now the ap exam only covers disk and washer method some of you might have learned shell method shell method is technically not on the ap exam i think it used to be they took it off at some point so you can only you only need to know for volumes of revolution additional disk and washer method you want to make sure you draw the rectangle are you drawing a vertical rectangle or a horizontal rectangle and the thickness of your rectangle is either d y if you're a horizontal rectangle because the thickness is cut in the y direction or in the vertical direction if you slice it up the thickness is dx and so that's the thickness of your rectangle if you're doing the rectangles horizontally okay um and then you want to for volumes of revolution you want to draw the appropriate shape it's going to either make a solid disc because there's no hole in it or there's going to be a hole in it and so then you want to find the area as either just the area of the whole thing if it's a disc or the area of the bigger circle minus the little circle if you're doing the area of a washer right and so that's pi big r squared minus pi little r squared area of the big circle minus area of the little circle okay and that's what you want to set up when you do cross-sectional areas you take that rectangle it might make a square so you just take the height of that rectangle and square it and that's your area it could be a semi-circle so you take the height of that rectangle is a diameter and it draws a semicircle and you find the area of the semicircle which is half of a circle that's basically it um for that net change okay so there's almost always one frq on this topic so you always think of a rate as something the net rate of something is the rate coming in minus the rate that's leaving rate in minus rate out that's the rate of change of something and that the integral of that rate is the is the net change it's the change in that um amount so for example if i'm integrating the rate at which water is flowing in minus water flowing out the integral of that is how much water has the change in the amount of water in the tank or something like that average value you just do the integral over the region divide by the the integral width it's basically taking sums and dividing it by the interval width but instead of a sum you do an integral motion so velocity's derivative of position acceleration's derivative of velocity when you're moving left and right you want to decide uh if you if you're trying to decide which direction it's going you look at the sign of the velocity curve positive velocity means it's moving right or up negative velocity means moving left or down speeding up and slowing down speeding up is when the velocity and acceleration have the same sign slowing down is when velocity acceleration have opposite signs and then just a couple of uh things to interpret integral of velocity is the displacement or change in position integral of the absolute value of the velocity which is the speed is the total distance traveled differential equations know how to solve separation of variables no slope fields general solutions are solutions to differential equations where you haven't solved for the c because remember you always integrate you get a plus c in there if you haven't solved for c that's a general solution particular solution adds one more step is you find the value of c because you know a value that the graph goes through okay no and then um so that's that's called the particular solution when you find the value of c and know how to draw slope fields and sketch solutions to them or sketch themselves some theorems okay so there's three theorems you're supposed to know the big ones are intermediate value theorem and mean value theorem to be honest okay but there's also extreme value theorem all of the when you're doing theorems you always need to invoke the conditions the conditions all of them require continuity so all three theorems extreme value intermediate value mean value all require the function to be continuous just that mean value requires that um the function also be differentiable over those regions so make sure you state those conditions and somehow why you think they're true continuity is usually given or at least if it's like a polynomial or a sine curve or something like that then you know it's continuous sometimes they'll tell you differentiable and differentiable functions are also continuous so that's how you know it's continuous that's it what does the extreme value theorem tell you is that basically f x has an absolute max absolute min there's a largest value and a smallest value as long as it's continuous intermediate value theorem basically says if f basically if it pat if it if there's a value above zero and below zero then it passes through zero but in the general sense if the function is above like a value and below another value then if it's continuous it must have passed through all the values in between and mean value theorem just tells you that the derivative there's a some point where the slope of the tangent line matches the slope of the secant line that means the slope like this f of b minus f of a over b minus a is the slope of a secant line there's there's a tangent line that has the same slope as the slope of that secant line okay and then just a general quick review of the general frq types we talked about the raid in versus raid out frqs you're going to get data table ones so that's making sure you know how to estimate integrals with riemann sums make sure you know how to estimate a derivative and interpret these in context and make sure you write units on these so know how the derivative unit's going to work how the riemann sum area is going to work the units wise you usually get a graph where they give you some graphs of f prime and you have to relate it to f of x to the main function f of x now it's that's a combination of making sure you know some of the net change fundamental theorem of calculus and some of the curve features we talked about okay differential equations we talked about there's usually one on there and then areas and volumes of revolution all right so i hope that helped for those of you taking bc calculus i'm going to make a separate video where i go through bc calculus topics too