In today's video, we apply the Quotient Rule to explore the behavior of the curve y equals one over x squared plus one, known as the witch of Maria Agnesi. Our main example is a curve of fundamental importance in theory of rational functions, that is ratios of polynomials, and involves reciprocating the quadratic x squared plus 1, which has a special place in mathematics. Being the simplest polynomial you can think of that does not have any roots over the real number system since it's always greater than or equal to one for any real number x, so it cannot possibly be zero. In fact, this innocent and unassuming quadratic leads to the construction of the arithmetic of complex numbers C, with quite remarkable properties which I hope you will say but take more advanced courses in mathematics, that follow one from this course. Consider the function with the rule y equals 1 over x squared plus 1. It's domain is all the real number line, since x squared plus 1 is always nonzero. One can ask what its graph looks like. The resulting curve is known as the witch of Maria Agnesi named after an Italian mathematician, Maria Agnesi who lived and worked in the 18th century, and wrote one of the first treatises on calculus. The word witch in fact comes about by a linguistic accident in translating texts from Latin, though happily the name of the curve survives, and is evocative of a witch's hat, as you'll see in a moment. Let's work through our usual checklist for sketching a curve. In this case, for y equals 1 over x squared plus 1. The y-intercept is the value of y when x equals 0, which is 1. There are no x-intercepts as the reciprocal of x squared plus 1 is always positive. For asymptotic behavior, it's clear that the limit of 1 over x squared plus 1 is 0 when x gets large, and positive or large and negative so that the x-axis becomes a horizontal asymptote. We then investigate the derivative which gives us information about when the curve is increasing or decreasing, and for finding any turning points. To set up the chain rule, we write y as x squared plus 1 to the minus 1, which becomes u to the minus 1 where u equals x squared plus 1. So, the derivative y dash which is dy/dx becomes dy/du times du/dx by the chain rule, which is negative u to the minus 2 times 2x, which we can rewrite as negative 2x over u squared, and then express everything in terms of x to get y dashed equals negative 2x divided by x squared plus 1 squared. Notice that the denominator is always positive and y dash equals 0 precisely when the numerator is 0, that is when x equals 0. We can now easily build the sign diagram. We have y dash equals 0 when x equals 0, and either side of 0 has the opposite sign of x, because the denominator is positive, so the sign is determined by the numerator giving the pattern positive-negative corresponding to increasing, decreasing with a global maximum occurring when x equals o. We can put all of this information together locating the y-intercept, the fact that the x-axis is an asymptote and the turning point of x equals 0. The qualitative behavior parallels what we saw when we analyzed the Gaussian curve, y equals e to the minus x squared in an earlier video. One expects to be able to join the pieces together to create a smooth curve that also looks bell shaped. There appear to be two points of inflection where concavity changes. To be sure about the behavior with regards to concavity and to locate the infections, we need to investigate the second derivative y double dash. Because of first derivative y dash is a rational function, we should expect to apply the quotient rule. To investigate the second derivative, we write y dashed as u over v where u is the numerator minus 2x, and v is the denominator x squared plus 1 all squared. So, u-dashed is minus 2, and v dashed is dv/dx in Leibniz's notation, which is dv/dw times the dw/dx by the chain rule, if we make the substitution w equals x squared plus 1, which becomes 2w times 2x which is 4x times x squared plus 1. We now have all the ingredients for the quotient rule, and inserting all the pieces gives this complicated looking expression which simplifies after some effort to the rational function 2 times 3x squared minus 1, divided by x square plus 1 cubed as you can check yourself, and please do it. It's good exercise in algebraic manipulation. Note that the denominator ends up being the third power of x squared plus 1, because of cancellation with a copy of x squared plus 1 in the numerator, thus we end up with this very nice expression for y double dash. Note that the denominator is always positive being the cube of x squared plus 1 which is positive. Also, y-double-dash is zero precisely when the numerator is zero, which occurs when x is plus or minus one over the square root of 3. We can now build the sign diagram with y-double-dash being zero for x equal to plus or minus root 3, the pattern of positive, negative positive determined by the sine of 3x squared minus 1, corresponding to concave up, concave down, concave up within inflections occurring when x equals plus or minus 1 over root 3. We can add this extra information to the previous compilation when we worked through the curve sketching checklist. This completes the sketch of the curve y equals 1 over x squared plus 1 affectionately known as the witch of Maria Agnesi. The curve is also in the shape of a bell though it certainly is not the same as a Gaussian curve y equals e to the minus x squared even though they share many similar qualitative features. I would like to finish by establishing a remarkable connection between the witch Maria Agnesi and trigonometry. Recall the single branch of the tan curve that results by restricting the domain to the interval between plus or minus pi on two, sandwiched between two vertical asymptotes, and satisfies the horizontal line test, certified as suitable for inversion with inverse function obtained by reflecting in the line y equals x to get the inverse tan function which is increasing and sandwiched in between two horizontal asymptotes. We can see how the slopes of miniature tangent lines behave as we move from left to right along the inverse tan curve. Notice how the slopes are all positive and the steepest slope appears to be at the origin and takes the value one, and the shallowest slopes appears to the far left and far right and appear to be approaching zero. The slopes of tangent lines are just the derivatives and this behavior suggests that the derivative of the inverse tangent function behaves qualitatively like the function that gives rise to the witch Maria Agnesi. Maybe, if y equals inverse tan of x, then y dashed equals 1 over x squared plus 1. It turns out to be true. The proof emerges from the magic of Leibniz's notation, watch. Start with a circular density, sine squared x plus cos squared x equals 1. Divide through by cos squared x eventually to get tan x somehow into the game. This becomes sine x of cos x all squared plus 1 equals 1 over cos squared x, which can be re-written as tan squared x plus 1 equals sec squared x. Since tan x equals sine x and cos x and from last time sec x is the new name for 1 over cos x. So, we have this very nice trig identity, closely related to the circular identity which says tan squared x plus 1 equals sec squared x. Put y equal to tan x so that x equals inverse tan of y, and dy/dx is sec squared x from the previous video, which is tan squared x plus 1 from the identity we have just established, and this is just y squared plus 1. Now, we tip the derivative upside down using Leibnitz's notation, the trick we used in an earlier video to get dx/dy equal to the reciprocal of dy/dx which is 1 over y squared plus 1. But x is inverse tan of y, so this says that d/dy of inverse tan of y must be 1 over y squared plus one. We usually prefer to use the symbol x as a typical input. So, replacing y by x gives d/dx of inverse tan of x equals 1 over x squared plus one, which is the rule for the witch of Maria Agnesi, confirming I guess before based on visualizing slopes of tangent lines. This result is one of the main ingredients in the theory of integration of rational functions, some aspects of which we will touch on in the final module of this course. In today's video, we applied the quotient rule to help us understand the second derivative of the rule for the function whose graph is the so-called witch of Maria Agnesi. The rule for this curve is y equals one over x squared plus 1 which turns out to be the derivative of the inverse tan function. Please read the notes, and when you're ready please attempt the exercises. Thank you very much for watching, and I look forward to seeing you again soon.