Question

Consider the curves in the first quadrant that have equationsy=Aexp(7x), where A is a positive constant. Different valuesof A give different curves. T

Other
ANSWERED
asked 2021-05-16
Consider the curves in the first quadrant that have equationsy=Aexp(7x), where A is a positive constant. Different valuesof A give different curves. The curves form a family,F. Let P=(6,6). Let C be the number of the family Fthat goes through P.
A. Let y=f(x) be the equation of C. Find f(x).
B. Find the slope at P of the tangent to C.
C. A curve D is a perpendicular to C at P. What is the slope of thetangent to D at the point P?
D. Give a formula g(y) for the slope at (x,y) of the member of Fthat goes through (x,y). The formula should not involve A orx.
E. A curve which at each of its points is perpendicular to themember of the family F that goes through that point is called anorthogonal trajectory of F. Each orthogonal trajectory to Fsatisfies the differential equation dy/dx = -1/g(y), where g(y) isthe answer to part D.
Find a function of h(y) such that x=h(y) is the equation of theorthogonal trajectory to F that passes through the point P.

Answers (2)

2021-05-18
image
0
 
Best answer
2021-09-08

Consider the curve and point,

\(y=Ae^{7x};P(6,6)\)

a) Find the value of f(x) as follows:

The curve passes through the point P(6,6).

That implies,

\(6=Ae^{7(6)}\)

\(A=\frac{6}{e^{42}}\)

Substitute \(A=\frac{6}{e^{42}}\) in the curve \(y=Ae^{7x}\).

\(y=\frac{6}{e^{42}}e^{7x}\)

Thus, the required value is \(f(x)=\frac{6}{e^{42}}e^{7x}\)

b) Find the slope at P of tangent line C as follows:

Slope of the tangent line is,

\(\frac{dy}{dx}=\frac{d}{dx}[\frac{6}{e^{42}}e^{7x}]\)

\(=\frac{6}{e^{42}}\frac{d}{dx}[e^{7x}]\)

\(=\frac{6}{e^{42}}(7e^{7x})\)

\(=\frac{42}{e^{42}}e^{7x}\)

At the point \(P(6,6),\)

\(\frac{dy}{dx}|_{(6,6)}=\frac{42}{e^{42}}e^{7(6)}\)

\(=\frac{42}{e^{42}}e^{42}\)

\(=42\)

Thus, the slope of the tangent line at the point \(P(6,6)\) is,

\(\frac{dy}{dx}|_{(6,6)}=42\)

c) Find the slope of the curve at the curve D perpendicular to C at the point P.

As D is perpendicular, slope of D is,

\(=\frac{1}{dy/dx}\)

\(=-\frac{1}{42}\)

Thus, the required slope is \(-\frac{1}{42}\)

d) From the above part (b),

\(\frac{dy}{dx}=\frac{42}{e^{42}}e^{7x}\)

\(=\frac{6\cdot7}{e^{42}}e^{7x}\)

\(=7(\frac{6}{e^{42}}e^{7x})\)

\(=7y\)

Therefore, the slope of the tangent line in terms of y is,

\(\frac{dy}{dx}=7y\)

e) Find the function \(x=h(y)\) as follows:

Consider,

\(\frac{dy}{dx}=-\frac{1}{g(y)}\)

From the above part \((d),g(y)=7y\)

Substitute \(g(y)=7y\) in the differential equation \(\frac{dy}{dx}=-\frac{1}{g(y)}\)

\(\frac{dy}{dx}=-\frac{1}{7y}\)

Apply the separation of variables,

\(-7ydy=dx\)

Integrate on the both sides,

\(\int-7ydy=\int dx\)

\(-7(\frac{y^2}{2})+c=x\)

\(x=-\frac{7y^2}{2}+c\)

\(h(y)=-\frac{7y^2}{2}+c\)

Thus, the required values is,

\(h(y)=-\frac{7y^2}{2}+c\)

0

expert advice

Have a similar question?
We can deal with it in 3 hours

Relevant Questions

asked 2021-05-09
The dominant form of drag experienced by vehicles (bikes, cars,planes, etc.) at operating speeds is called form drag. Itincreases quadratically with velocity (essentially because theamount of air you run into increase with v and so does the amount of force you must exert on each small volume of air). Thus
\(\displaystyle{F}_{{{d}{r}{u}{g}}}={C}_{{d}}{A}{v}^{{2}}\)
where A is the cross-sectional area of the vehicle and \(\displaystyle{C}_{{d}}\) is called the coefficient of drag.
Part A:
Consider a vehicle moving with constant velocity \(\displaystyle\vec{{{v}}}\). Find the power dissipated by form drag.
Express your answer in terms of \(\displaystyle{C}_{{d}},{A},\) and speed v.
Part B:
A certain car has an engine that provides a maximum power \(\displaystyle{P}_{{0}}\). Suppose that the maximum speed of thee car, \(\displaystyle{v}_{{0}}\), is limited by a drag force proportional to the square of the speed (as in the previous part). The car engine is now modified, so that the new power \(\displaystyle{P}_{{1}}\) is 10 percent greater than the original power (\(\displaystyle{P}_{{1}}={110}\%{P}_{{0}}\)).
Assume the following:
The top speed is limited by air drag.
The magnitude of the force of air drag at these speeds is proportional to the square of the speed.
By what percentage, \(\displaystyle{\frac{{{v}_{{1}}-{v}_{{0}}}}{{{v}_{{0}}}}}\), is the top speed of the car increased?
Express the percent increase in top speed numerically to two significant figures.
asked 2021-05-14
Consider the accompanying data on flexural strength (MPa) for concrete beams of a certain type.
\(\begin{array}{|c|c|}\hline 11.8 & 7.7 & 6.5 & 6 .8& 9.7 & 6.8 & 7.3 \\ \hline 7.9 & 9.7 & 8.7 & 8.1 & 8.5 & 6.3 & 7.0 \\ \hline 7.3 & 7.4 & 5.3 & 9.0 & 8.1 & 11.3 & 6.3 \\ \hline 7.2 & 7.7 & 7.8 & 11.6 & 10.7 & 7.0 \\ \hline \end{array}\)
a) Calculate a point estimate of the mean value of strength for the conceptual population of all beams manufactured in this fashion. \([Hint.\ ?x_{j}=219.5.]\) (Round your answer to three decimal places.)
MPa
State which estimator you used.
\(x\)
\(p?\)
\(\frac{s}{x}\)
\(s\)
\(\tilde{\chi}\)
b) Calculate a point estimate of the strength value that separates the weakest \(50\%\) of all such beams from the strongest \(50\%\).
MPa
State which estimator you used.
\(s\)
\(x\)
\(p?\)
\(\tilde{\chi}\)
\(\frac{s}{x}\)
c) Calculate a point estimate of the population standard deviation ?. \([Hint:\ ?x_{i}2 = 1859.53.]\) (Round your answer to three decimal places.)
MPa
Interpret this point estimate.
This estimate describes the linearity of the data.
This estimate describes the bias of the data.
This estimate describes the spread of the data.
This estimate describes the center of the data.
Which estimator did you use?
\(\tilde{\chi}\)
\(x\)
\(s\)
\(\frac{s}{x}\)
\(p?\)
d) Calculate a point estimate of the proportion of all such beams whose flexural strength exceeds 10 MPa. [Hint: Think of an observation as a "success" if it exceeds 10.] (Round your answer to three decimal places.)
e) Calculate a point estimate of the population coefficient of variation \(\frac{?}{?}\). (Round your answer to four decimal places.)
State which estimator you used.
\(p?\)
\(\tilde{\chi}\)
\(s\)
\(\frac{s}{x}\)
\(x\)
asked 2021-04-25
A wagon with two boxes of Gold, having total mass 300 kg, is cutloose from the hoses by an outlaw when the wagon is at rest 50m upa 6.0 degree slope. The outlaw plans to have the wagon roll downthe slope and across the level ground, and then fall into thecanyon where his confederates wait. But in a tree 40m from thecanyon edge wait the Lone Ranger (mass 75.0kg) and Tonto (mass60.0kg). They drop vertically into the wagon as it passes beneaththem. a) if they require 5.0 s to grab the gold and jump out, willthey make it before the wagon goes over the edge? b) When the twoheroes drop into the wagon, is the kinetic energy of the system ofthe heroes plus the wagon conserved? If not, does it increase ordecrease and by how much?
asked 2021-02-25
We will now add support for register-memory ALU operations to the classic five-stage RISC pipeline. To offset this increase in complexity, all memory addressing will be restricted to register indirect (i.e., all addresses are simply a value held in a register; no offset or displacement may be added to the register value). For example, the register-memory instruction add x4, x5, (x1) means add the contents of register x5 to the contents of the memory location with address equal to the value in register x1 and put the sum in register x4. Register-register ALU operations are unchanged. The following items apply to the integer RISC pipeline:
a. List a rearranged order of the five traditional stages of the RISC pipeline that will support register-memory operations implemented exclusively by register indirect addressing.
b. Describe what new forwarding paths are needed for the rearranged pipeline by stating the source, destination, and information transferred on each needed new path.
c. For the reordered stages of the RISC pipeline, what new data hazards are created by this addressing mode? Give an instruction sequence illustrating each new hazard.
d. List all of the ways that the RISC pipeline with register-memory ALU operations can have a different instruction count for a given program than the original RISC pipeline. Give a pair of specific instruction sequences, one for the original pipeline and one for the rearranged pipeline, to illustrate each way.
Hint for (d): Give a pair of instruction sequences where the RISC pipeline has “more” instructions than the reg-mem architecture. Also give a pair of instruction sequences where the RISC pipeline has “fewer” instructions than the reg-mem architecture.
...