solve-the-given-problems-by-finding-the-appropriate-derivatives-the-deflection-y-in-mathrmm-of-a-5-3

Question

Solve the given problems by finding the appropriate derivatives.
The deflection $y$ (in $\mathrm{m}$ ) of a $5.00-\mathrm{m}$ beam as a function of the distance $x$ (in $m$ ) from one end is $y=0.0001\left(x^{5}-25 x^{2}\right)$. Find the value of $d^{2} y / d x^{2}$ (the rate of change at which the slope of the beam changes) where $x=3.00 \mathrm{m}.$

Manisha Sarker · Accepted Answer

in this problem, we need to find the value of v two, I. D X two when X is equal to three. Now, the value of why is given to be equal to 0.1 times X to the power five minus 25 X square. But let&#x27;s first find the first derivative. So that will be dy dx is equal to 0.1 times the derivative of actually power five, which is five times actually power four minus 25 times the derivative of X square which is two X. So this will be equal to 0.1 times five X. To the power four minus 50 X. Next let us determine the second derivative. That will be the derivative of this expression over here. So that will be equal to 0.1 times five times the derivative of X. To the power for which is four times X to the power four minus one. So that&#x27;s xq minus 50 times the derivative of X, which is one. So this will be 0.1 times 20 X cubed minus 50. Now we need to find the value of the second derivative at the point X is equal to three, so we substitute X equals to three, so we have 0.1 times 20 times three Q minus 50 and the value of this is equal to 0.49 and the answer is 0.49

Patrick Vaughn · Answer

So this is a multi part question in the first part. Ah, were given the equation. Well, we aren&#x27;t, but conceptually were given the equation for a tangent line at the point. A common half of a on the curve. Um, why equals f index? So how do we find f Prime of a So suppose that tangent line takes the form l of X is equal. Teoh M ax plus B. So then we know that the&#x27;s slope of the tangent line em is gonna be equal Teoh the value of the derivative at that point? A Because this is the changes. This is the change in line that satisfies hell of a equals f of A and the slope M is going to be equal to f prime of a Okay, So for Barbie, given that the tangent lines of the graph of y equals f of X at the point to come a five as the equation three X minus two. So we want to find out too. So after I moved to you So we&#x27;re given that, um, were given that l of X is equal to three x minus two l of two people&#x27;s five which week f of to and we want to find f prime of to So that&#x27;s just equal to the slope of L, which is gonna be equal. That three. So now, for a function y equals f of X Ah, we want to know the instantaneous rate of change For why, with respect to X at X equals Dio Just pretend I wrote instituted three change So it&#x27;s kind of the same concept. So we, um we know that Why wait? We know that the derivative is the instantaneous rate of change and we already found that f prime of two was Eagle the three, yeah.

Petros Markopoulos · Answer

this is a continuation of the previous exercise will calculate the deflection of the same beam if you flip it to its side. So it was two inch wide and eight inch deep, which now turns into a being that is eight inches wide and two inches deep. Variation models said that d equals K over W cubed with me being the deflection w being the with an L being that that&#x27;s and we had calculated K to be 281.6. So this is our formula now. We will plug in these new values and calculate the new value of the reflection, which is 281.6 over eight times to cubed, which is 281 point 6/8 times eight, which is then equal to 181 point 6/64 which gives us a final unser of 4.4 inches. You will note this is significantly bigger than the previous deflection we had calculated because death is cubed and switching these two numbers made the depth significantly smaller and because it is tubed, this affects their answer in a significant way

Petros Markopoulos · Answer

we&#x27;re going to solve the following problem Using a model variation. A deflection of a beam is inversely proportional to its with and the two of its depth. If the deflection of a four inch wide by four inch deep beam is 1.1 inches, find the deflection of a two inch wide by eight inch deep beam positioned us in the figure. This is our figure. You see that this is the depth and this is the width, and that&#x27;s in code the information given to us. So the deflection D is inversely proportional to the width of the beam in the cube of its depth. So it&#x27;s gonna the equation is gonna be a former K, which is a constant offer. W times hell or W is the width and l is the depth and l is gonna be cube because, he said, is inversely proportional to the cube of the death. And this again, we&#x27;ll know that this is a constant variation. Now in her first beam, have D one equals K over W 11 cube, and the numbers given to us for D one, w one and l one is 1.1 equals k over four times four cubed, which we can rewrite as K equals 1.1 times four the fourth power, which is equal to 1.1 times 256 which is our final answer of two 181.6. Now for another beam, we want to calculate D two, which is equal to K over W 202 cubed which, with the numbers given tow us is K over. Oh, two inch wide times A to the cube in Steve and we calculated kay here. So we have 281.6 over two times 512 which is 200 81.6 over 1024 which is our final answer. 0.27 fine inches.

Bobby Barnes · Answer

they want us to prove the formula for the derivative of Why is it too inverse? Hyperbolic. Seeking by differentiating X is equal to carbolic seeking of why And they give us a hand by using a hyperbolic I didn t toe help. So let&#x27;s first just go ahead and take the derivative Oh, what they ask us to So take it the derivative of this year we&#x27;d get one is equal to the derivative Uh, Sequent or hyperbolic Siegen is going to be negative seeking hyperbolic Why times tangent, hyperbolic Why? And then we multiply by. Why prime since remember why depends on X. Now we can go ahead and divide everything over to the other side. And we&#x27;d get why prime is equal to negative one over seeking hyperbolic of why times tangent hyperbolic of why now seeker seeking hyperbolic attention of why is just x so we can replace that negative one over X And now tangent hyperbolic Why this is equal to the square root Oh, one minus seeking hyperbolic squared. Why? And this is by the identity tangent hyperbolic squared you plus go seeking hyper bullet squared You is equal to one. So you just rearranged this equation to solve for tangent, hyperbolic to get that and again, we said that this year is just acts. So we get a negative one over X square root of one witness X squared, which is exactly what they give us in the book for this trip. So since we finished proving it, but proponents say in spite of these

Bobby Barnes · Answer

you want to find instantaneous? Try to change for F of X is he would X squared plus two at X is equal to zero. So we can just go ahead and follow the equation right above here. So are a is supposed to be zero. So the first thing I like to do is to figure out so is going to be f of a plus h says gonna be f of zero plus h, which is just f of age. So we&#x27;re gonna replace all the exes with nature&#x27;s. We have a, uh H and this is gonna be to h squared, plus two. Then the next thing I like to do is to go ahead and do so the f of H minus f of a, which is zero capital of. And when we find what f of zero is, that&#x27;s going to be zero squared plus two, which gives us two. So we&#x27;re going to subtract at all. So it is going to be eight squared plus two minus two. So the twos here just counts out with each other, and we&#x27;re left with h. So now for the last step, we&#x27;ll take the limit as h purchase zero, and then we divide by each as well so kind of as a pro tip. Once you get to this point, if for some reason you can&#x27;t evenly divide the H, then you need to go back in. Check your algebra, at least when you&#x27;re doing this with polynomial. But in this case, notice each word can be evenly divided by each. So that&#x27;s going to give us H. And then we can just supply the limit directly, which gives us zero. So the limit as H r. The instantaneous rate of change of this function at X 00 will just be zero.

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Allyn J. Washington, Richard S. EvansBasic Technical Mathematics with Calculus

Catherine R.

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus