Implicitly Defined Curves and Their Tangents

8/10/2019 Implicitly Defined Curves and Their Tangents

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2/5

2 1 0 1 2

2

1

0

1

2

Plot ofh x

6 4 2 0 2 4 6

6

4

2

0

2

4

6

Plot ofg x

6 4 2 0 2 4 6

6

4

2

0

2

4

6

Plot off x

Exercises

1. For the following lemniscates

1(a). Make 3 labeled plots; one plot for eachindividual implicit function and a joint plot

including both implicit functions.

f[x]: -2(x^2+y^2)^2=3x^2-3y^2,y], with x and y

values from -6 to 6

Solved for y =

G[x]: 2(x^2+y^2)^2=3x^2-3y^2, with x and y values from -6 to 6

H[x]: Joint plot including both f[x] and g[x] with x and y values from -2 to 2

I used the CountourPlot command of Mathematica and used the given functions to plug in.

1(b). Use your plot to estimate the points where the vertical and horizontal tangents are and thenuse the implicit derivative to find these points analytically with the help ofMathematica

The problem asks to estimate where the vertical and horizontal tangents are and use the implicit

derivative to find these points analytically withMathematica.. In the graph of f(x), there

will be two horizontal tangents from around the x values of -.5 to .5 In addition, at around

the x values of .25 and 0.25 there will be two vertical tangents. I found this through that

a horizontal tangent will occur when the slope is 0, and wherever the plot is a horizontal


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1.0 0.5 0.0 0.5 1.0 1.5 2.0

1.5

1.0

0.5

0.0

0.5

1.0

1.5

line, on the other hand, a vertical tangent line will occur wherever there is a vertical line.

In the graph of g(x), there will be horizontal tangents from the y values of -1.4 and 1.4

and .5 to 1.75, there will be vertical tangents at the x values -1.25 and 1.25.

[

]

1(c) How are the lemniscates related to one another? How are the vertical and horizontal

tangents of the two lemniscates related to one another?

In this problem, the question asks how the two lemniscates are related to each other and how are

their tangents related to one another. Through examining the graph, both graphs share the same

origin, at (0, 0). Similarly, the distance from the origin to the maximum on the y axis for one

lemniscate is the same from the origin to the highest x value on the horizontal lemniscate. The

tangent values of the horizontal and vertical are switched for the two lemniscates.

2(a) Make a plot of this equation for x between -1 and 2 and y between -1.5 and 1.75The problem accounted for a plot of the equation for x between -1 and 2

and y between -1.5 and 1.75. From completing the practice exercises

using the command ContourPlot, I used that function defining the

specified x values and y values.


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2(b) Does the name of the curve seem appropriate? Is this curve a function? Explain.

The problem asks whether or not the name seems appropriate for how the function appears and if

the requirements of a function are met. I looked at the top portion of the plot and it seemed to be

a cover of a wagon. In addition, the bottom portion which ranged from around the y values of -1.5 looked like the portion of the wagon which included the wheels. In order answer whether if

this curve is a function, I used the book to determine the definition of a function. The definition

is as follows, A function, f, is a rule that assigns to each element x in a set D exactly one

element, called f(x), in a set E. This means that every x value has only one y value. As such, I

used the vertical line test on the curve and it does not pass the test, which means that this curve is

not a function.

2(c) Use your plot to estimate the points where the vertical and horizontal tangents are and then

use the implicit derivative to find these points analytically with the help of Mathematica.

The question asks to estimate the points where there will be vertical and horizontal tangents. Inaddition, to using the implicit derivative to find these points analytically with the help of

Mathematica. There will be a horizontal tangent lines at y = 1.5 and y= -1.0. In addition, there

will be a vertical tangent line at the x value around -1 and 1.5. In order to find the horizontal

tangent lines, I used the information that horizontal tangent lines are finding where y=0.

Horizontal Tangents:

( ) () { ()()

() }

Vertical Tangents:


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2(d) Is there only one tangent line at x = 1.5? Are there more than one? Find the slopes and the

equations of the tangent line(s) when x = 1.5 (Hint, substitute x = 1.5 into the implicit function

and solve for y as done in the discussion above)

The problem asks whether if there is one tangent line at the x value of 1.5 in addition to finding

the slopes and the equations of the tangent line(s) when x = 1.5. I substituted x = 1.5 into solving

the implicit function. Afterwards the input showed that there were 5 values, which means that

there are 5 tangent lines. No, there are several tangent lines at x =1.5 y - .05407847381625697=1.24552(x-1.5)

y +1.220672111142222558=10.6958(x-1.5)

y1.5781449376722834= -0.232298(x-1.5)

Implicitly Defined Curves and Their Tangents

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