Ap Calculus AB Multiple Choice 012 number 1 5 practice differentiation limits continuity
Business Contact: mathgotserved@gmail.com 2012 Ap Calculus Multiple Choice Part I Good day student welcome to mathgotserved.com and this clip we're going to be going over problems 1 to 5 of the multiple choice AP calc exam 01 2 All right let's take a look at question number 1 E38 if Y equals X sinx then the derivative dy/dx equals what so that if you want to keep in mind for this problem is that you do not distribute the derivative across a product or quotient of functions okay if you have a product or a quotient of functions you use the product or quotient rule but you can distribute the derivative across sums and differences okay so let's all take a look at the formula that will be guiding our problem solving process for today See the formulas are first one is the product rule which you should have mastered before the AP exam the product rule is if you're looking for the derivative UV Prime of two functions you NV the derivative is Vu prime plus UV Prime okay and then you need to remember your trigonometric differentiation rule the tree differentiation rule that we are going to be using for this examples the derivative of sine the derivative of sinx is cosine X and lastly the power of row we're going to be applying that's to the derivative of x x Prime equals 1 okay now it's go ahead and I'll solve the problem we have Y equals X sinx so we have a product of two functions delenia function X and the trigonometric function sinx so we're going to do a label them you and be okay so the first function is you and the second function sinx is V okay so what is dy/dx DUI DX is equal to X sign X sinx Prime remember Prime end dy/dx are the same thing okay so let's see what we have apply in the product rule will have Vu Prime is going to be Sinex Furby X you prime which is X Prime plus you which is x times the prime sign X Prime okay now is differentiate and simplify sine X times the derivative of x is just one plus x times of derivative of sine is cosine X okay it would simplify will have sinx + x cosine X as our final answer and that is option letter b now let's take a look at? To let F be the function given by f of x equals 300 x minus x to the Third on which of the following intervals is a function is increasing one penny one to keep in mind is the connection between the sign of f Prime and the behavior of f if a firm is positive on an interval then it will be increasing on that interval okay so let's keep that in mind now what's right on the procedure for solving this problem there are three parts to this process part one involves finding the critical values okay so you find the critical values what are the critical values of the X values where the derivative is equal to 0 or does not exist okay after that we're going to set up a number line sign chart okay so set up line sign chart that enable us to see how the sign of f Prime changes across the critical values and then lastly we will find the intervals find intervals where F Prime is positive okay let's do it Part 1 we want to find the critical values so find the critical values physically where is f Prime is equal to zero or does not exist start with the function f of x equals 300 x minus x to the third we're going to find the first derivative of prime of X applying the power rule the derivative of 300 X 300 in the derivative of x to the third is -3x Square now to find the critical values we're going to see wherever Prime of x equals 0 and wherever Prime of X does not exist okay since we do not have a denominator here there isn't an ex value where this function is going to be on the find this is a polynomial function it is smooth and continuous is differentiable everywhere okay so are we just have to do is find where the derivative is a pool to 0 this situation is not applicable so we'll take the derivative and set it equal to zero and let's solve this we can subtract 3x square from both sides and flipped equation will have 3 x squared equals 300 divide both sides by 3 that you check square equals 100 and then we are going to take the square root of both sides not one thing you want to keep in mind is that anytime you take the square root of a square you have to introduce the positive and negative roots Plus or minus 10 because when you square root when you squirt end or -10 you end up with positive 100 okayNow we have our critical values left list them are critical values are x equals negative 10 and positive 10 we are done with part 1 of the procedure now it's Advanced 2 part 2 which involves creating a number line sign chart let me partition my workspace here part two and three are basically combined in one okay so we have our number line sign chart we're going to grab the critical values on a number line this will be an open interval because we do not have any cut strings on the domain okay so we're going to grab a -10 and positive 10 now let's pick up some partitions so we can have a visual distinction as to what's going on where because we have two
Business Contact: mathgotserved@gmail.com 2012 Ap Calculus Multiple Choice Part I Good day student welcome to mathgotserved.com and this clip we're going to be going over problems 1 to 5 of the multiple choice AP calc exam 01 2 All right let's take a look at question number 1 E38 if Y equals X sinx then the derivative dy/dx equals what so that if you want to keep in mind for this problem is that you do not distribute the derivative across a product or quotient of functions okay if you have a product or a quotient of functions you use the product or quotient rule but you can distribute the derivative across sums and differences okay so let's all take a look at the formula that will be guiding our problem solving process for today See the formulas are first one is the product rule which you should have mastered before the AP exam the product rule is if you're looking for the derivative UV Prime of two functions you NV the derivative is Vu prime plus UV Prime okay and then you need to remember your trigonometric differentiation rule the tree differentiation rule that we are going to be using for this examples the derivative of sine the derivative of sinx is cosine X and lastly the power of row we're going to be applying that's to the derivative of x x Prime equals 1 okay now it's go ahead and I'll solve the problem we have Y equals X sinx so we have a product of two functions delenia function X and the trigonometric function sinx so we're going to do a label them you and be okay so the first function is you and the second function sinx is V okay so what is dy/dx DUI DX is equal to X sign X sinx Prime remember Prime end dy/dx are the same thing okay so let's see what we have apply in the product rule will have Vu Prime is going to be Sinex Furby X you prime which is X Prime plus you which is x times the prime sign X Prime okay now is differentiate and simplify sine X times the derivative of x is just one plus x times of derivative of sine is cosine X okay it would simplify will have sinx + x cosine X as our final answer and that is option letter b now let's take a look at? To let F be the function given by f of x equals 300 x minus x to the Third on which of the following intervals is a function is increasing one penny one to keep in mind is the connection between the sign of f Prime and the behavior of f if a firm is positive on an interval then it will be increasing on that interval okay so let's keep that in mind now what's right on the procedure for solving this problem there are three parts to this process part one involves finding the critical values okay so you find the critical values what are the critical values of the X values where the derivative is equal to 0 or does not exist okay after that we're going to set up a number line sign chart okay so set up line sign chart that enable us to see how the sign of f Prime changes across the critical values and then lastly we will find the intervals find intervals where F Prime is positive okay let's do it Part 1 we want to find the critical values so find the critical values physically where is f Prime is equal to zero or does not exist start with the function f of x equals 300 x minus x to the third we're going to find the first derivative of prime of X applying the power rule the derivative of 300 X 300 in the derivative of x to the third is -3x Square now to find the critical values we're going to see wherever Prime of x equals 0 and wherever Prime of X does not exist okay since we do not have a denominator here there isn't an ex value where this function is going to be on the find this is a polynomial function it is smooth and continuous is differentiable everywhere okay so are we just have to do is find where the derivative is a pool to 0 this situation is not applicable so we'll take the derivative and set it equal to zero and let's solve this we can subtract 3x square from both sides and flipped equation will have 3 x squared equals 300 divide both sides by 3 that you check square equals 100 and then we are going to take the square root of both sides not one thing you want to keep in mind is that anytime you take the square root of a square you have to introduce the positive and negative roots Plus or minus 10 because when you square root when you squirt end or -10 you end up with positive 100 okayNow we have our critical values left list them are critical values are x equals negative 10 and positive 10 we are done with part 1 of the procedure now it's Advanced 2 part 2 which involves creating a number line sign chart let me partition my workspace here part two and three are basically combined in one okay so we have our number line sign chart we're going to grab the critical values on a number line this will be an open interval because we do not have any cut strings on the domain okay so we're going to grab a -10 and positive 10 now let's pick up some partitions so we can have a visual distinction as to what's going on where because we have two