solve-the-given-problems-find-values-of-x-for-which-the-following-curves-have-horizontal-tangents--3

Question

Solve the given problems.
Find values of $x$ for which the following curves have horizontal tangents: (a) $y=x+\sin x$ (b) $y=4 x+\cos \pi x$

Carson Merrill · Accepted Answer

for this problem we want to find values of experts to following curves have horizontal tangents. Um So we have Y equals X plus some X and Like was four x plus Cosan kayaks. So looking at this first curve right here, we see, there&#x27;s gonna be a horizontal tangents right at these points right here. So whenever we see that um the values and mm pie. So if we have Y equals M. Pie and we let and E three actually 1 chu Uh two won&#x27;t be there, but three will be so and high where N is an odd integer. That&#x27;s how we would reflect that. And then um we would also see for these values here, this graph, it would be whenever we have um Sine Inverse Sine four over pie over pie.

Theresa Nguyen · Answer

since we are looking for a tangent line to why is equal to four Sign of X close and attacks at X is equal to pi over three. The first thing we want to do is to find a tangent point. To do that, we plug in X into our equation. This will look like four. Sign of pi over three. No sign Pi over three. This comes out to be square root of three. Now that we have this, we&#x27;re going to work on our slope and to find our slope we&#x27;re going to find the derivative of our equation above so d over DX those four sine x co Scitex. To find this, we first want to simplify our derivative so we know the identity of co sign of X. So two packs is equal to sign of two acts multiplied by 1/2. If we plug based identity back in, we will get D over DX of four times 1/2 sign two x simplified further. You will get d over DX times two sign two x Now that we&#x27;ve gotten to this point, we want to take your constant out. This will look like two D over DX of sign two acts. So now that we&#x27;ve gotten to this point, we can take our derivative to do that we apply the chain Rule de Fufu over De X is equal to d f over de you multiplied by D you over d x Here are f is sign of you while so our f time isco cited you Our you is two x so you prime will be too Putting all of these values together our derivative will look like to co sign you Times two simplified We will get for co sign of you and plugging Are you back in? We will get four co sign two x This is our derivative So now we can plug in our pi over three to find the actual slope for co sign of pi over three times two turns out to be negative too. This is our slope so we know that are slope which is m is negative two and our tangent point is at pi over three and swear it of three We can now use the line Why is equal to M X plus B to form our attention line putting all of our values together we get Why this equal to negative two x plus squared if three plus two pi over three. This is our line, and if we want to plot it into a calculator, the graph we&#x27;ll look like this.

Joseph Liao · Answer

our problem 60 to 1 equation with line tended to the following curves and user graphing utility to put the curve and change the line. So function why equals four sine x go sign X at X equals three So we take the dirt of this using the product wool get for sine x primeco sine x plus for sine x co sign Ex prime It&#x27;s comes out to four sine x negative sine x plus co sign X Co sign X Okay, which is equal to four times co sine squared X minus signs Great X goes on to x and then we could go this slope Just you i d x. That&#x27;s a goto four co sign two x plug in X equals power three you get and vehicles for a co Siam Pi over three nets equal to negative two Know that we have the slope. Let&#x27;s find the y coordinate at for our executed power Three So why is a good A for sun pi over three times coasts on pirate three which could a square to three. So the point No, we want to find the slow about Is power three coming squared or three green for a line do why minus why one goes M times X minus x. One question for the line becomes that now. Graphing this, you know Go sue four Ghost or two to the lying says two comma 00 Call four with the changing curve where this is tangent at X equals pi over three. This line&#x27;s Why equals negative two. X squared at 33 This curve is why equals for sine x Co sign X All right?

Vikash Ranjan · Answer

The question is, why was Sinus into? Got six. So we have to find the value of X between, you know, to buy with a 1,000,000 lying to the half off. One was Sinus because it&#x27;s it&#x27;s a young girl. So the graph off my waas Sinus in tow. Koszics age son here on DDE and this graph, you find it the point between General Dubai Yeah, The graph is for younger. So let me Judah points. These points become This is what this is by before this is our home. This is treat by my four. This is fine by my food. This is seven by and by Fuller. So we obtained this four points. These are by a four treat by my food. 554 and seven by before the sister is what end to jump on. This is that answer

Joseph Liao · Answer

are a problem. 64 with a function y Eichel&#x27;s CSC ex Find the change in line at X equals pi over four. Right de y d X Sequel two Derivative of CSC Ex, which is equal to negative CSC co Attention X Plugging in power four for X we find the slope Jizz *** CSC Power four times co tangent Popover for radicals. Negative squared, too now Ah, X equals pi over four. We have why Ziegel too. She is C however, four that kiss us square to two, then using the equation of a line we have y minus square to to is to go to a negative skirt 2 to 3 times X minus pi over four. Why might discredit two equals negative Scored a two x plus square to two times pi over to discredit to square medicals Why equals negative for two x plus route to plus pi over to root pie. Now the graph dysfunction the lying Have a pi over four right here. So your graph should look approximately like that with them intersecting at X equals pi Over four. This is going to be your line. Why equals recovery two x plus route to plus pi over two to and this part&#x27;s gonna Weichel CSC ex aren&#x27;t

Theresa Nguyen · Answer

looking for the quotidian line of why is equal to co sequence of X where X is equal to pi over four. The first thing what to do is to find the tangent point. To do that, we&#x27;re plugged our X back into the equation. Co sequence of pi over four is equal to square roots of two. Now we know that the tangent point is pi over four square roots of two. Where is Pi? Over four is X and square root of two is why the next thing we want to do is to look for a slope. To do that, we&#x27;re taking be derivative d over dx of co sequent of acts. To find this, we&#x27;re going to use our common denominator de over t x of co sequence of X is equal to negative co tension X co Sequent X plugging our answer it we get negative coated Agent X co seeking tax as the derivative to Cosi connects. You know that we have this derivative. We went to plug in pirate for into X to get her actual slope So negative co tangent of pi over four co sequent of pi over for this turns out to be negative square root of two. So this is our slope. We are going to use the equation of why is equal to M X plus B to find attention line. We know that em is equal to negative squared of to and we have the points pi over four and square root of two as accent. Why plugging on all the values we will get Square root of two is equal to negative square root of two multiplied by pi over four plus b to solve for B we&#x27;re going toe isolate it so b is equal to square it too open print the seas four plus pi close parentheses All of that over four And now that we have our be let&#x27;s plug everything back into the equation, we will get why is equal to negative square root of two times X plus square root of two open parentheses for plus pi close parentheses over for this is our tangent line. Now we&#x27;re going Teoh, plot this in the calculator. It could bear it to our co sequence of X curve. It will look like so

Problem

Solve the given problems. Use differentials to es…

Problem 48 Easy Difficulty

Solve the given problems.
Find values of $x$ for which the following curves have horizontal tangents: (a) $y=x+\sin x$ (b) $y=4 x+\cos \pi x$

Answer

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

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