Long description: fige

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A graph showing a parametric curve and its tangent line at a specific point.

Alt text: A graph showing a parametric curve and its tangent line at a specific point.

This description was generated by AI and has not been verified by a human. If you spot an error, please use the feedback link below.

  • The image displays a two-dimensional Cartesian coordinate system with the horizontal axis labeled \(x\) and the vertical axis labeled \(y\).
  • A smooth, curved parametric curve is plotted on the graph, passing through the origin area.
  • A straight line is drawn through a point on the curve, representing the tangent line.
  • Mathematical expressions related to derivatives are shown near the graph: \(\frac{dy}{dt} = 2t + 1\) and \(\frac{dx}{dt} = 2t - 1\).
  • The calculation suggests that when the parameter \(t=1\), the slope \(\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{2t+1}{2t-1}\) equals \(\frac{3}{1} = 3\).
  • The text indicates that for \(t=1\), the point \((x, y)\) is \((0, 2)\), and the equation of the tangent line is \(y = 3x + 2\).

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