5

1- Find the average rate of change fory = % Xo = 1 Xp = 2 2 - Find the instantaneous rare of change for the above function at3- Show that the function y = x? + 3x +...

Question

1- Find the average rate of change fory = % Xo = 1 Xp = 2 2 - Find the instantaneous rare of change for the above function at3- Show that the function y = x? + 3x + 3 satisfy they- +xy= 2y- = 4- Find dy/dx for: a- y sec? x tan? x b- y = sec? 6 ~X) c y = sin(tan(3x)) d-y = V1 + sin" (x") e- y = 2(* 1)6+ f:y = cos '{sin(Zx")} 5-Find the equation of the tangent to the function y sinX atx = 0/4 Show that y = xsinx is solution toy" +y = Zcosx

1- Find the average rate of change fory = % Xo = 1 Xp = 2 2 - Find the instantaneous rare of change for the above function at 3- Show that the function y = x? + 3x + 3 satisfy they- +xy= 2y- = 4- Find dy/dx for: a- y sec? x tan? x b- y = sec? 6 ~X) c y = sin(tan(3x)) d-y = V1 + sin" (x") e- y = 2(* 1)6+ f:y = cos '{sin(Zx")} 5-Find the equation of the tangent to the function y sinX atx = 0/4 Show that y = xsinx is solution toy" +y = Zcosx



Answers

A function $y=f(x)$ and values of $x_{0}$ and $x_{1}$ are given.
(a) Find the average rate of change of $y$ with respect to $x$ over the interval $\left[x_{0}, x_{1}\right]$
(b) Find the instantaneous rate of change of $y$ with respect to $x$ at the given value of $x_{0}$.
(c) Find the instantaneous rate of change of $y$ with respect to $x$ at a general point $x_{0}$.
(d) Sketch the graph of $y=f(x)$ together with the secant and tangent lines whose slopes are given by the results in parts (a) and (b).
$$y=\frac{1}{2} x^{2} ; x_{0}=3, x_{1}=4$$

So I go for this problem is to find the average rate of change. So given F of X equals X cubed, we know that Um the average rate of change is going to be f of two -F of one. I had to buy 2 -1, which is just once, that's seven. Now, if we take the derivative, so we have, the instantaneous rate of change will have three X squared. Um So the derivative is three X squared evaluated at one and we end up getting three, then we find the instantaneous rate of change of why with respect to act at a general point it's not. And if we graph these together, we see that this average rate of change is much greater than this instantaneous rate of change. However, this one here is much more accurate. I mean that's the one that we're going to end up starting to use. If we just had a random X not, it would be three X not squared.

For this problem here we have F of X Equals one over acts. Um so then we evaluate the average rate of change, will have FF three -F of two, All divided by 3 -2, which is one. We get a negative 1/6 as our rate. And then if we want to take f prime of X, we know the f prime of X is going to be a negative one over X squared. In this case, that would be X not squared. So when we evaluated at X, not that's going to be negative 1/2 squared, which is negative 1/4. So we see that it's it's close but not quite in terms of the rate. Um So this would be our final result and we see the similarities between the slopes but we also see some major differences, which is why it's better to calculate instantaneous rate of change as opposed to average rate of change.

Yeah. For the given exercise, we want to um graph or solve the one over X squared here, we want to find the average rate of change. So we take F of two minus F of one and then we divide that by two minus one. So it's just gonna be divided by one is about a negative three quarters negative 3/4. However, we want to find the instantaneous rate too. So we see the f. Prime of X is going to equal negative two over Act two III. Um So then when we're making X equal to X not, it'll be this value right here and in this case X not is going to be one. So we'll get a negative to um as our answer, Which we see is very different from a negative .75.

Question is saying find the indicated quantities for F X equals two three Xs were in discussion. There are total seven parts so we will solve when reverend in practicing the average rate of change of fx is exchanges from 1 to 4. So here it is saying exchanges from 1 to 4 so we can write it a food minus F one upon four minus mint. It means firstly we have to put the value of texas 4, 10 went. Now going to solve this so three can do 40 square minus three into venice square divided by four minus minus three. So it can be returned in three in 244 16 minus three divided by 3 16 to 3 48 48 minus three means 45.3. When we reduce it, we get 15. So we can say this is the answer of the first part. Now come to the second part. So for second part deeper in the B part, it is saying that the slope of the second line through the Voice one, F. One and four. F four on the graph Y. Is equal to fX. So solving second part exactly like first part. So F four minus everyone because points in both parts are completely scene. So here I also get the same answer which is 15. Now going to solve see part of the cushion. So in the super it is saying the slope of the second line through the points one F. 11 place H. F. One place it where it is not equal to zero. Simplify your answer. So solving the third part, it can be returned his F. One plus edge minus F. One divided by one plus edge minus one here one and one will be cancelled out. So it can be written as three because now we have to put the value of X. S. One plus in second time one. So it can be written as one plus edge hole is where pre existing because it is three in government Holy square divided by edge. So opening the bracket by using the identity A plus B. Holy square. So it can Britain is one plus two which plus edges square minus three into venice square which is only one divided package. Now going to solve it. So three plus multiplying the bread six. H. Plus three inches square minus three divided by edge. So finally six edge plus three edges square divided by age equals two edge six plus 38 divided by age. It means when I saw this we will write it six plus three H. Because H and H will be counselor. So six plus three H. Will be the answer of our C. Part. Now come to the fourth depart. So according to depart it is asking for the slope of graph at one end F one. It means in the part we can write it limit yes one plus edge limit will be evidence to zero, the same minus F. One divided by one plus edge minus one. So here we can put the limit and solve it because our C. Part and depart are exactly seem. So here we can take another three plus six plus three age which is from third c part. Now I will put the value of edge zero so it will be six plus three in 20 means our answer is 60 so we can see this is an answer of depart. Now come to the E. Part in the IT park. It is see that the instantaneous rate of change of Y equals two fx with respect to X at X equals stupid. It means this time we have given in a part exactly same as G. We can say limit. H tends to zero. If one plus H minus F one upon one plus edge minus one equals 26 Just three. It means in place of H. I will put zero. So the answer will be only six. So we can say this is the answer of E. Part. Now we have to do next part. Next part means F. Part in F. Part the slope of the tangent light F one and F. One. I mean this is also exactly same as D. And E. It means it's answer is also six plus three in 20 which is equals to six only. So we can see our F. Part answer is also F sexually. Now our last part will be G. The creation of the ancient line at 0.1 and F. One. So in G. Part we can see Dad we have to use one. F one will be equals to one entry. N. M will be equal to six. So we can say using point slope equation of a line. We know the equation of the line. Why minus violent equals to em X minus excellent. It means y minus three will be equal to because these are the points of X and drive an excellent raven and value is six x minus. We're not solving this so y minus three equals 266 minus six. So finally we can write it. Y equals 26 x minus three. So we can say this is our final answer of the third last part.


Similar Solved Questions

5 answers
Find the G.S. of the system Y' = AY
Find the G.S. of the system Y' = AY...
5 answers
Formation E of a hard of the prevention destruction of sperm receptors of the shell sperm egg of more around by plasma than the on enzymes the one egg egg nemperane plasma released by fertilizing the membrane egg an egg achieved by
formation E of a hard of the prevention destruction of sperm receptors of the shell sperm egg of more around by plasma than the on enzymes the one egg egg nemperane plasma released by fertilizing the membrane egg an egg achieved by...
5 answers
The following are Ilmiting " reactant problems. Use the balanced reactions shown in prelab question #1_Calculate the mass (grams) Ba(NH SOzhi 148 Ba(NOsh reacts with 2.4 & NH,SO,HCalculate the mass (grams) of BaSOa 14 & Ba(NOsh reacts with 2.4 &NH;SO,H.redcts with 2.4 & NH_SO,H: Calculate the mass (grams) of Ba(NHzhz if 1.4 g Ba(NOzh
The following are Ilmiting " reactant problems. Use the balanced reactions shown in prelab question #1_ Calculate the mass (grams) Ba(NH SOzhi 148 Ba(NOsh reacts with 2.4 & NH,SO,H Calculate the mass (grams) of BaSOa 14 & Ba(NOsh reacts with 2.4 &NH;SO,H. redcts with 2.4 & NH_SO...
5 answers
Use 2x + VI graphical 16 2x =Z 4xt5y methods solve the Jeauii programming L
Use 2x + VI graphical 16 2x =Z 4xt5y methods solve the Jeauii programming L...
5 answers
You are testing the null hypothesis that there is no relationship between two variables Xand Y From your sample ofn = 22, you determine that SSR = 60 and SSE = 20. Complete parts through belowr
You are testing the null hypothesis that there is no relationship between two variables Xand Y From your sample ofn = 22, you determine that SSR = 60 and SSE = 20. Complete parts through belowr...
5 answers
[Retetences]The Ko for siker sulfate (Ag,80 ) is 12x 1-' Calculate the solubility of silver sulfte in cach of de following watcSolubilitymoll6. 0.060 MAgNOsSolubilitymolll0.15 MK SO4SolubilitymolLSubrnit AnswntTry Anothet Vcrelon10 Ilemn atterpts remainino
[Retetences] The Ko for siker sulfate (Ag,80 ) is 12x 1-' Calculate the solubility of silver sulfte in cach of de following watc Solubility moll 6. 0.060 MAgNOs Solubility molll 0.15 MK SO4 Solubility molL Subrnit Answnt Try Anothet Vcrelon 10 Ilemn atterpts remainino...
5 answers
2. (18 pts:) For the double integraldydx3 -_Sketch the region of integration b. Set up an equivalent double integral in polar coordinates_ Evaluate the integral.
2. (18 pts:) For the double integral dydx 3 -_ Sketch the region of integration b. Set up an equivalent double integral in polar coordinates_ Evaluate the integral....
5 answers
Moles of Hz produced? (Show work wlth units and - verbal; Whzt Is the number descriptions of numbers ) Joss in mass of the anade? (Show work with units and vcrbal b) What was the descriptions ot numbers }Question (20 points} the molur mass ol the- metal? (Show work with unlts ard verbal What descriptions ot numbers )
moles of Hz produced? (Show work wlth units and - verbal; Whzt Is the number descriptions of numbers ) Joss in mass of the anade? (Show work with units and vcrbal b) What was the descriptions ot numbers } Question (20 points} the molur mass ol the- metal? (Show work with unlts ard verbal What descri...
5 answers
__________ move materials between the organelles of the endomembrane system.a. Vesiclesb. RNAc. Ribosomesd. Nuclear pores
__________ move materials between the organelles of the endomembrane system. a. Vesicles b. RNA c. Ribosomes d. Nuclear pores...
5 answers
Write a parameterization for the curves in the $x y$ -plane.An ellipse centered at the origin and crossing the $x$ -axis at ±5 and the $y$ -axis at ±7.
Write a parameterization for the curves in the $x y$ -plane. An ellipse centered at the origin and crossing the $x$ -axis at ±5 and the $y$ -axis at ±7....
5 answers
Which measuireCanta landency balter doscrbes hours woired?motle inedinnKe
Which measuire Canta landency balter doscrbes hours woired? motle inedinn Ke...
5 answers
HndhtheLexact length of the curve Ix# et ti yi4et/z 0 < t< 2
HndhtheLexact length of the curve Ix# et ti yi4et/z 0 < t< 2...
5 answers
Image file(s) Write the time (minutes} 0n the [irst pugc (lirst image) The elements ], M,Land Rare fromn the same group = trom the same pcriod J histhe highcst tomic radius Lhas lower electon Allitty" lhan J, but higher than Rand M.Mis least metallic and has the configuration 15? %s? ?ph 35? Rhas tWo dots in its dot suucture Identify the four pmpnts
image file(s) Write the time (minutes} 0n the [irst pugc (lirst image) The elements ], M,Land Rare fromn the same group = trom the same pcriod J histhe highcst tomic radius Lhas lower electon Allitty" lhan J, but higher than Rand M.Mis least metallic and has the configuration 15? %s? ?ph 35? Rh...
5 answers
Given that = is a function of x and that satisfies cos(xz) + In(y2 + 22) = 0,use partial derivatives to findFind an equation for the tangent plane to the surface given by x + y' + 22 = tan(ryz) at the point (4-1,0).
Given that = is a function of x and that satisfies cos(xz) + In(y2 + 22) = 0,use partial derivatives to find Find an equation for the tangent plane to the surface given by x + y' + 22 = tan(ryz) at the point (4-1,0)....
5 answers
Question 12'balanced coin rollcd 260 tlmos_ fost 0.0154 At least 0.0154 At least 0.0167 most 0,0167(Use Chcbyshe Wncoramni
question 12 'balanced coin rollcd 260 tlmos_ fost 0.0154 At least 0.0154 At least 0.0167 most 0,0167 (Use Chcbyshe Wncoramni...
5 answers
(w + 7)2 (y _1)2 + = 1 2, 6 5
(w + 7)2 (y _1)2 + = 1 2, 6 5...

-- 0.022071--