Provide a CLEAR and COMPLETE answer. Exercise No. 5. Differential equation of a family of curves.In each of the following problems, obtain the differential equation of the family of curveswith the properties indicated.5. All circles tangent to both the x- and y- axes.6. Parabolas with vertex and focus on the y-axis.

Answered: In each of the following problems,…

Algebra for College Students

10th Edition

ISBN:9781285195780

Author:Jerome E. Kaufmann, Karen L. Schwitters

Publisher:Jerome E. Kaufmann, Karen L. Schwitters

Chapter13: Conic Sections

Section13.1: Circles

Problem 45PS

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Find an equation of the tangent line to the parabola y=3x2 at the point 1,3.
Direction: Find the differential equation of the family of curves described. 5. Circles with fixed radius "r" and tangent to the x-axis.
3. Consider the curve which is the graph of the equation x³ + y³ − 3xy = 0. Graph this curve using your favourite software or graphing calculator to see what this looks like! (It will also help you check your work.) (a) dx Find an expression for dy in terms of x and y, justifying your steps with the appropriate rules. (b) Find all points on the curve whose tangent lines are horizontal. You may use the fact, without justification, that the derivative, dy, does not exist at the point (0,0) (convince yourself of this by looking at a graph, but no need to comment on it in your solution here).
Q.5 Find the equation of the tangent to the graph of the function +2xy+3xy +6=0 at (2,3)
(1) Families of Curves a. Obtain the differential equation of the family of straight line with the algebraic sum of the intercept fixcd as k. b. Find the differential equation of the family of ellipses having their major axis parallel on the x-axis.
Consider the curve described by the following equation. e12x+2y=8y-4xy+1 Use this information to answer the questions to the right. (a) At a general point on this curve, we have that dy/dx= (b) The slope of the line tangent to the curve at the point (2,−12) is= (c) There is a point on the curve close to the origin with coordinates (0.06, b), and the line tangent to the curve at the origin is given by y=2x. An estimate of b using linear approximation or differentials is b≈
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2. Consider the equations y=r.y=r, and y=2. (a) Carefully sketch a graph of each of the three fuctions on the same set of axes (graph paper will be provided). Show at least 3 points on cach curve (b) Which function appears to be increasing the fastest? Support your answer using 1-2 sentences and the graph you created. (c) Which function appears to be increasing the slowest? Support your answer using 1-2 sentences and the graph you created.
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16. The equation of motion of a particle is =t-2t + t-t, where s is in meters and t is in %3D seconds. (a) Find the velocity and acceleration as functions of t. (b) Find the acceleration after 1 s. (c) Graph the position, velocity, and acceleration functions on the same screen.
Answer the following Questions. Show Solution. • Find the equation of the tangent line to the curve x + ey = 1 at the point (0, 0). • Find the points on the ellipse x?+ 2y? = 8where the slope of the tangent line is horizontal. • Find the equation of the tangent line to the graph of y4 + 3y – 4x3 = 5x+ 4 at the point P(0, 1).

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In each of the following problems, obtain the differential equation of the family of curves with the properties indicated. 5. All circles tangent to both the x- and y- axes. 6. Parabolas with vertex and focus on the y-axis.