Mathematically, what is the third derivative useful for in functions?
Well technically, the third derivative would tell you the rate of change of the concavity/ convexity of the curve but this is not the mathematical use of the third derivative.The 2 common usages of the third derivative mathematically, which I can think of are:1) Third order partial differential equations arise in the study of dispersive wave motion, including water waves, plasma waves, waves in elastic media, and elsewhere.It models linear dispersion in a wave: [math] u_t + u_{xxx} = 0 [/math]2) The third derivative, or higher derivatives for that matter, are generally used to improve the accuracy of an approximation to the function.[math]f(x_0 + h) = f(x_0) + f'(x_0)h + f''(x_0)\frac{h^2}{2!} + f'''(\xi)\frac{h^3}{3!}[/math]Taylor's expansion of a function around a point involves higher order derivatives, and the more derivatives you consider, the higher the accuracy. This also translates to higher order finite difference methods when considering numerical approximations.3) Another use would be for creating cubic splines for interpolating polynomials or regression models, but since the third derivative doesn't give a lot of information about the function itself, so usually it's not worth the effort.
Find the vector (i+j) X (i-j) without using determinants?
Find the vector, not with determinants, but by using properties of cross products. I've gotten three different "answers" using my vague understanding of these properties so I'm obviously stabbing in the dark. I know i x j =k and j x k =i and all those from the table but I don't understand how to apply them in this problem and it's an even problem so I can't check my answers. Thanks for any help!
Is there a name given for a line that touches a parabola (in geometry) only once but isn't a tangent?
The answer to this question depends on how you want this line to ‘touch’ the parabola. Usually the world ‘touch’ is interpreted to mean “not surpass a given border”, or in this scenario; “not to intersect the parabola”.I will however briefly answer the latter ultimatum (encase you were not strictly suggesting that there could be no intersection).A: Vertical Intercept.For a line to intersect a parabola only once, and not be tangent to any point on the parabola, it must be a strait vertical line. e.g. x=-1.As you can see there are infinate solutions to this. I could have a line intersect at any location on the x-axis, and it would satisfy your condition - given that it is completely vertical.Now, let’s say you did indeed intend for no intersection. Then the answer would be a range of different equations.For example:This image shows how y=-x^2 ‘touches’ the parabola at x=0. Similar results can be produced with all parabolas.Say you had a parabola; f(x). Then, I can tell you that the parabola that ‘touches’ the parabola at it’s turning point is given by g(x) such that:[math] g(x) = -f(x)+2c [/math]Where c is the distance the parabola is from the x-axis. See if you can work out how & why this works.Here is another parabola which only ‘touches’ y=x^2 once:It’s equation: y=-(x-1)(x-3)+1We are not just limited to parabolas either:It’s equation: y=cos(x-pi)+1There are infinitely many solutions to this problem. I also just realized that we can have cubic polynomials (and certainly other functions) intersect the parabola only once - which may be considered a solution, depending on your question.Regards,Daniel
What is something that almost nobody knows about triangles?
Given lateral heights of isosceles triangle with minimum perimeter then lateral angles = acos(phi-1)=51.827°…Note: Angles formula in radians: COS(A)+COS(1)=COS(B)+COS(C)Given lateral angles = acos(phi-1) of isosceles triangle with base=4 then lateral heights = 4/√phi = 3.1446… (good approx of π)Given lateral angles = acos(phi-1) of isosceles triangle with base=2 then lateral sides = phi and height = √phiGiven lateral angles = acos(phi-1) of isosceles triangle with lateral heights=180 m then Great Pyramid of Giza transversal section measures approx
In a quadrilateral, the measues of the angles are in the ratio 2: 3: 4: 6. How do you find the measure of the largest angle?
The measures of the angles of a quadrilateral are in the ratio 2:3:4:6Since we have ratio and not direct values let the common multiple be xSo, we have the angles as 2x,3 x,4x,6xThe sum of all angles of a quadrilateral is 360°So, 2 x +3 x+4x+6 x= 360°15 x= 360°x = 360/15x= 24Now the largest angle is6x= 6×24= 144There fore the largest angle is 144°
Why Is Math so Difficult for Those With ADHD?
It’s not that math is hard for people with ADHD….its that math is hard for some people.We are not defined by ADHD, everything negative or difficult is not attributed to ADHD. People have strengths and weaknesses.I remember one time My husband went with me to get my sons meds. He started discussing with the dr issues even after I told him we are just going to tell the dr no changes, no issues and get his script (because there was none).The dr. Cracked me up when after intently listening to my husband discuss my sons behaviors and why those indicated he needed a higher dose he says - well, Mr. Husband….the ADHD medicine isn’t a substitute for parenting. Those are all parenting issues, you need to still parent. HahahaSo….regarding math. Many people just suck at math. People with and without ADHD.Try not to be narrow focused regarding issues or weaknesses that arise and attributing them to ADHD. ADHD isn’t a character flaw, it isn’t negative, it’s just a different way the mind works. It excels in many areas the non-ADHD brain struggles in. It is like stating some people learn best being shown through demonstration by someone else and some people learn best hands on through trial and error. Neither is right or wrong….it’s just a different way of learning. ADHD is just a different way of thinking.What can happen that will hurt you in the long run if you are narrow focused on ADHD being the cause for negative aspects of learning and life is you don’t need to look any further for a reason or resolution, so you won’t change the behavior. When attributing it to ADHD it leaves no reason to change or get better in that area because you know you can’t change having ADHD.For example, regarding math….if you attribute a struggle in math to ADHD your search for an answer and resolution is over. Knowing it is not ADHD you may discover math is difficult for you because you rush on the mundane parts. You can have a really complicated equation to solve and love the challenge and solve it well but get the wrong answer by adding or subtracting wrong because you rush on that part since now the challenge is solved. Now you found a problem you can take active steps to resolve and be better. That process doesn’t occur if you automatically attribute to ADHD. Resulting in missing an opportunity to grow and be better.
What is meant by entropy in machine learning contexts?
Entropy is a measure of randomness. In other words, its a measure of unpredictability. Let’s take an example of a coin toss. Suppose we tossed a coin 4 times, and the output of the events came as {Head, Tail, Tail, Head}. Based solely on this observation, if you have to guess what will be the output of the coin toss, what would be your guess?Umm..two heads and two tails. Fifty percent probability of having head and fifty percent probability of having a tail. You can not be sure. The output is a random event between head and tail.But what if we have a biased coin, which when tossed four times, gives following output: {Tail, Tail, Tail, Head}. Here, if you have to guess the output of the coin toss, what would be your guess? Chances are you will go with Tail, and why? Because seventy-five percent chance is the output is tail based on the sample set that we have. In other words, the result is less random in case of the biased coin than what it was in case of the perfect coin.We will take a moment here to give entropy in case of binary event(like the coin toss, where output can be either of the two events, head or tail) a mathematical face:Entropy = -(probability(a) * log(probability(a))) – (probability(b) * log(probability(b)))where probability(a) is probability of getting head and probability(b) is probability of getting tail.More at here: Entropy In Machine Learning