now let's say if you're given the graph of F ofx and it looks something like this if that's F ofx what is the graph of fime the first derivative of f how would you graph it well for one thing if you know the parent function it can be easy to find frime for example the parent function for this graph is x^2 so that's F ofx the derivative of x^2 is 2X and so we should have the derivative is basically a linear function with a slope of two so it looks something like that now let's analyze the slope of f because the derivative gives you the slope at any point x on the left side notice that the function is decreasing which means that the slope is negative and at the origin or when X is zero we have a horizontal tangent line so at that point the slope is equal to zero and on the right side the function is increasing so the slope is positive we can see that at the origin the Y value of fime is equal to zero and on the left side of the graph notice that the Y values are negative because the slope is negative and on the right side because the slope is positive the graph of fime is above the x- axis the Y values are positive so what you need to know as you go from F to fpre the slope of f becomes the Y values of fime now what about FP Prime the second derivative how can we graph that particular function now if FR Prime is 2x the derivative of 2x is simply two so we're going to have a horizontal line at two now it makes sense because the slope of this line is constant it's always increasing and the slope is always positive two so therefore the Y value of the derivative will always have a value of two so whenever you need to find the derivative of a function the slope of that function becomes the Y value of the derivative of that function let's move on to another example so let's say if f ofx is this particular function using this information go ahead and sketch a graph for fime and F Prime so just by looking at it we know that the parent function is x^2 because it's a parabola that opens in the downward Direction so therefore fime must be -2X which looks something like this and fpre is equal to -2 so we have a horizontal line that has a yalue of -2 now let's analyze the slope of f ofx on the left side the function is increasing which tells us that the slope is positive therefore for the derivative that portion of the function has to be above the xaxis so that the y value is positive at that region now at zero we have a horizontal tangent line which means the slope is zero so at that point the Y value has to be zero which means that it must be passing through the xaxis now on the right side the function is decreasing which means that the slope is negative so on the right side for fime the function has to be below the x-axis the Y values have to be Nega negative now if we compare fime and fpre for fime the function is always decreasing which means that the Y value of fpre must always be negative or below the xaxis which we do see that so given the graph of f ofx go ahead and sketch the derivative of that function find F Prime and FP Prime now the first thing that we should do is identify the function that we have the parent function is X Cub the derivative of x Cub is 3x^2 so therefore F Prime is going to look something like this it's going to be an upward Parabola and the derivative of 3x^2 is 6X so we're going to have a linear function with a very high slope and now let's analyze it so on the left side of the graph the function is increasing which means the slope is positive and then we have a horizontal tangent line around zero so the slope is zero then the function continues to increase so the slope is positive notice that the slope is never negative the function is never decreasing and as you can see F Prime is never below the x-axis it's always above it now notice that anytime you have a horizontal tangent line where the slope is zero the derivative of the function will have a yvalue of zero it's going to touch the x- axis so basically that's going to be your x intercept now if we consider FP Prime fime is decreasing on the left side the slope is negative and it's increasing on the right side the slope is positive so as we can see on the left the function is below the xaxis because the slope is negative on the right side the function is above the xaxis because fime is increasing the slope is positive so let's say this is f ofx how would you sketch the graph of frime of X this time we don't know what the parent function is now let's assign values to this let's say the maximum is at1 and the function becomes horizontal at -2 let's say it passes through the origin and at positive one it has a minimum at two it touches the xais and at three it's horizontal so the first thing I would do is identify where the slope is zero Beyond -2 the slope is zero and at negative 1 that we have a horizontal tangent line the slope is zero there as well and that positive one and Beyond three so -2 and anything less than that the Y value of the derivative will be equal to zero at -1 it should be zero again and at one it's going to be zero and Beyond three is going to be zero again so keep in mind this is just a rough sketch now notice that the function is increasing between -2 and one which means a slope is positive and then beyond -1 to 1 the function is uh decreasing so the slope is negative but let's focus on -2 to one which is in this region so it goes from zero to positive back to zero so it's going to go up and back down now between1 and one the slope changes from zero and then it becomes negative back to zero so it's going to decrease it's going to be negative and then it's going to go back to zero now between one and three the slope changes from zero and then the function is increas and so the slope is going to be positive and then it's going to be zero again so it's going to look something like this and then it stays Zer after that so as you can see just by looking at whether the function is increasing or decreasing you can draw a nice rough sketch but it helps if you identify where the slope is zero and plot the X intercepts first and then you can graph it so let's try another example let's say if this is our function let's start over so if this is f ofx what is frime of x so let's identify where the slope is zero so we have a slope of zero in this region Beyond -2 between negative 1 and one the slope is zero at two it's zero and at four we have a horizontal tangent line so beyond -2 the y value will be zero between -1 and 1 it will be zero and at two it's going to be zero and at four it's going to be zero as well now between -2 and-1 the function is increasing but at a constant rate which means the slope is constant it's going to be a straight line but notice that the derivative is discontinuous at this point so we should probably put like an open circle because the slope changes instantaneously from zero to let's say positive one whenever the slope changes instantaneously that means that it's not differentiable at -2 and at 1 so for this graph at that point it doesn't connect so we could put an open circle here as well now notice that we have another notice that it's not differentiable at xal 1 the slope changes instantaneously from zero to some negative value so how can we graph that how can we show what's Happening Here notice that the slope changes from a negative value and increases towards zero which means that it's concave up so therefore we should have we should start at a very high negative value and it's going to increase towards zero after that that the function continues to increase the slope is positive so it's going to be above the x-axis at three and then at four is going to be back at zero and then after that the function continues to decrease with a negative slope so this is going to decrease as well so this is a rough sketch of our graph so we have a lot of points of discontinuity because it's not differentiable at -2 negative 1 or at one now keep in mind whenever the function is not differentiable at a particular point that means that the derivative is not continuous at that point so whenever the slope changes instantaneously from one value to another then when you graph the derivative of the function you're going to have some sort of jump discontinuity or some other type of discontinuity now let's say if this is the function f ofx which looks like the absolute value of x how would you sketch the derivative for this function let's not worry about the second derivative the absolute value Val of X can be broken down into two functions positive X and negative X on the left side the slope is Nega one and on the right side the slope is positive one and it's constant so we're going to have a horizontal line so that's how you can graph the derivative of the absolute value function so notice that even though the function is continuous at zero it's not differential by zero because the slope changes instantaneously from - 1 to 1 it skips zero so whenever the slope changes instantaneously whenever you have a sharp term a sharp term will always cause the derivative to be discontinuous at that point so as you can see we have a jump discontinuity now given the graph of f ofx go ahead and sketch frpr of X so how can we begin what would you do this problem now if you notice if you continue this function this looks like a sine wave so this graph is positive sin x the derivative of s is positive cosine x cosine starts from the top and then it looks something like this but we don't need the whole graph just this portion so that's the portion of the cosine graph that we need now let's see if it makes sense so first let's identify where the slope is zero so we have a horizontal tangent line at these two points and as you can see it touches the x-axis at those two points so whenever the slope is zero the Y value has to be zero now as we can see from let's call this negative 1 if it's sign is really like uh pi two but let's just use one and Nega 1 just to keep things simple between1 and one the function is increasing and we can see that the derivative is above the the function for the derivative is above the xaxis now between -2 and 1 the function is decreasing and as we can see it's below the x-axis and from 1 to two it's decreasing as well and it's below the x- axis so as we could see uh it makes sense these functions agree with each other consider this function let's say this is five and this value is ne4 now sometimes you might be given the graph of F ofx and you may have to answer questions like when is the function increasing when is it decreasing what did it critical points what is the relative extrema what is the uh where's the inflection points what are the intervals where the function is concave up and when it's concave down so let's work on that stuff so first when is the function increasing write the intervals when the function is increasing so the function is increasing from -2 to 1 it's going up and it's increasing from 3 to Infinity that's when it's going up so we can say starting from -2 to 1 Union 3 to Infinity now what about when the function is decreasing when is it decreasing so starting from -4 it's decreasing from4 to -2 and from 1 to 3 it's also decreasing so we could say -4 to -2 Union 1 to 3 now at what x values do we have a horizontal tangent line so basically what are the critical points of the function the critical points occur at -2 at 1 3 and also -4 so those are the critical points that's where we have a horizontal tangent line now I identify the relative extrema so we have a relative Max at one and a relative Min at xal 3 another relative or local minimum value at x = -2 and we can't really say anything about um4 because it appears as if the function is horizontal at this point if it was decreasing then4 would be a local Max but we can't really say anything it appears to be horizontal now when is the function concave up and when is it concave down whenever you have an upward u shape it's concave up and it's concave down when it looks like this so to find the intervals where it's concave up look for the upward u-shaped portion of the graphs so notice that from3 to one it's concave up and it's also concave up from to this appears to Infinity it doesn't appear to change direction like this so we could say it's concave up for the remaining portion of the graph so it's concave up starting AT3 to1 Union 1 to infin inity now when is it concave down so notice that in this region it's concave down let me use a different color we have this portion of the graph here it's concave down as well so we could say starting from4 to -3 it's concave down and also from -1 to one it's concave down now where are the inflection points the inflection points occur whenever the concavity changes so notice that the concavity changes at this point here and here as well so the inflection points occur at X values of -3 -1 and two whenever the concavity changes Direction it's an inflection point so that's it for this particular function now let's say if we're given the graph of fime instead of f and let's say it looks something like this so if this is the first derivative when is the function inre inreasing when is it decreasing when is it concave up concave down everything so let's take it one at a time let's start with the intervals where the function that is f of x not F Prime when it's increasing so f is increasing when the first derivative is positive meaning that it's above the x- axis so the function is increasing in this region and in this region because fime is above the x- axis so it's increasing between -1 and one Union and then starting from three to Infinity now what about when the function is decreasing so the opposite is true the function function is decreasing whenever the first derivative is negative meaning when it's below the x axis so in this region so therefore we could say it's decreasing starting from well I guess eventually this could go past the x-axis if it continues to go up so let's say it touches the xaxis atg -3 so it's decreasing between -3 to -1 Union 1: 3 that's when it's below the xaxis now since we've extended the graph we may have to adjust our answer for when the function is increasing because it's increasing in this region so it's increasing from negative Infinity to -3 because it's above the xaxis union -1 to 1 and also 3 to Infinity so now how can we determine the critical points of the function so the critical points for f is whenever the first derivative is equal to zero and since we have the graph of f Prime the first derivative is zero whenever it touches the X axxis so the critical points are x = -3 -1 1 and 3 now what about the relative Max and the relative Min so let's analyze the relative Max verse in a graph of F this is the shape of the relative Max the slope changes from positive to negative now a relative minimum looks like this the slope changes from negative to positive so let's analyze the critical points -3- 1 1 and 2 not two but 1 and 3 at -3 going from left to right the slope which is the sign of the first derivative it changes from positive to negative keep in mind the yvalue of f Prime is the slope for f so if the yvalue changes from positive to negative the slope of the function f changes from positive to negative which means that three ne3 is the maximum now what about 1 do we have a Max or a Min at1 the Y value of f Prime changes from negative to positive which means that the slope of the original function f changes from negative to positive so we have a minimum at one fime changes from positive to negative that's the Y value of fime so therefore we we have a Max and at three the Y value of fime changes from negative to positive which indicates that's a Min so that's how you can identify the maximums and minimums if you're given the graph of f Prime they're located on the x axis and if the yv value changes from positive to negative it's a maximum and if the yv value changes from negative to positive then the function f has a minimum at the x value now what about the concavity when is it concave up and when is it concave down the function is concave up when the second derivative is positive and it's concave up when the first derivative is increasing keep in mind the second derivative is the slope of the first derivative so when frime is increasing the slope is positive which means that F Prime is positive so the function fime is increasing here and here so that's when it's concave up so it's concave up from -2 to zero that's when it's increasing and it's also concave up from two to Infinity now when is it concave down it's concave down when fime is decreased in inv value so it's concave down from negative Infinity to -2 Union 0 to 2 now the last thing that we need to do is find the inflection points now I'm going to highlight in yellow when it's concave down so it's concave down when it's decreasing and it's also concave down in this region in blue I'm going to highlight when it's concave up it's concave up when fime Prime is increasing which is in this region so as we can see the concavity changes at -2 at zero and at xal 2 so therefore those are the inflection points so that is it for this video hopefully you found it to be useful thanks for watching and have a great day