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Trace the curve $y$=$x^{3}$ Discuss the following curves for extence,symmetry,Asymptotes and loops

1 Answer

  • The strategies for curve tracing
  • (1) Domain-The values of x for which the function is defined
  • (2)Extent -Horizontal (vertical ) extent is determined by the intervals for $x(y)$ for which the curve exists
  • (3) $x=0$ gives the y intercept and y=0 gives the x -intercept
  • (4) If $(0,0)$ satisfies the equation, the curve passes through the orgin
  • Symmetry:
  • (5) Find out whether the curve is symmetrical about any line. The curve is symmetric about
  • (i) x-axis if the equation is unaltered when y is replaced by -y
  • (ii) y-axis if the equation is unaltered when x is replaced by -x
  • (iii)the orgin if the equation is unattened when x is replaced by -x abd y by -y
  • (iv) the line $y=x$ if the equation is uttered when x and y are interchanged
  • (v) the line $ y=-x$ if the eauation is unchanged if x and y are replaced by -y and -x
  • (6) Asymptotes ($\parallel$ to coordinate axes)
  • If $y \to c$ (finite) when $x \to \pm \infty$
  • (or $ x \to k $ (finite ), when $y \to \pm \infty$
  • the line $y=c$(or $x=k$ ) is an asymptole parallel to x-axis (or y-axis)
  • (7)Monotsnicity : Determine the intervals for which the curve is increasing or decreasing, using the first derivative test
  • (8) Special points: Determine the intervals of concavity/convexity and the points of inflection, using the second derivative test.
Step 1:
(i) Domain, extent, intercepts and orgin
The function is defined for all real values of x and hence the domain is the entire interval $(-\infty, \infty)$.
The horizontal extent is $-\infty < x < \infty $ and the vertical extent is $-\infty < y <\infty$ and the vertical extent is $ -\infty < y < \infty$
The curve passes through the orgin.
Since when$ x=0,y=0$. The curve does not intersect the coordinate axes anywhere else
Step 2:
(ii) Symmetry Test : The test for symmetry shows that the curve is symmetric about the origin.
The equation does not change when x is replaced by -x by -y simultaneously
Step 3:
(iii) Asymptotes: There are no constant c,k for which $ y \to \pm \infty$ when $ x \to c$ or $ x \to \pm \infty $ then $ y \to k$
Therefore, the curve does not have any asymptotes
Step 4:
Monotonicity: The first derivative test shows that the curve is increasing throughout $(-\infty, \infty)$
Step 5:
Special points : The curve is concave downward in $(-\infty,0)$ and concave upward in $(0, \infty)$. Since $y^x=6 x < 0$ for $ x < 0 $ and $y''=6x > 0$ for $ x > 0$ and $y''=0$ for $x=0,$ yielding $(0,0)$ as a point of inflection
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