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Chapter 8: Problem 23

Find the equations of the tangent lines at the point where the curve crosses itself. $$ x=t^{2}-t, \quad y=t^{3}-3 t-1 $$

Short Answer

Expert verified
The equations of the tangent lines at the points where the curve crosses itself are \( y = x - 3 \) and \( y = -2x - 1 \) respectively.

Step by step solution

01

Find the Intersection Point

To locate the intersection point, set \( x(t) = t^{2}-t \) and \( y(t) = t^{3}-3t-1 \) then solve for common solutions. After solving, the intersection point comes out to be \( t = 1, -1 \). Since the solution has two points, the curve crosses itself.
02

Find the Derivative

To find the tangent lines, we need to first find the slope of the curve at the intersection point. This can be calculated by the derivative \( y'(t) / x'(t) \). So, calculate \( x'(t) = 2t - 1 \) and \( y'(t) = 3t^{2} - 3 \).
03

Calculate the Slope

After calculating \( x'(t) \) and \( y'(t) \), we insert \( t = 1, -1 \) into \( y'(t) / x'(t) \) to calculate the slope. After calculating, the slopes at \( t = 1, -1 \) are 0 and -2 respectively.
04

Write the Equation for Tangents

Now, knowing the slope and the point at which the tangents intersect, we obtain the equations of the tangents using the following form: \( y - y_1 = m(x - x_1) \). Thus, this gives us \( y = x - 3 \) and \( y = -2x - 1 \), which are the equations of the tangents at \( t = 1, -1 \) respectively.

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Most popular questions from this chapter

Use a graphing utility to graph the curve represented by the parametric equations (indicate the orientation of the curve). Eliminate the parameter and write the corresponding rectangular equation. $$ x=4 \sin 2 \theta, y=2 \cos 2 \theta $$

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Use a graphing utility to graph the polar equation over the given interval. Use the integration capabilities of the graphing utility to approximate the length of the curve accurate to two decimal places. $$ r=\sin (3 \cos \theta), \quad 0 \leq \theta \leq \pi $$

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Use the results of Exercises \(31-34\) to find a set of parametric equations for the line or conic. $$ \text { Ellipse: vertices: }(\pm 5,0) ; \text { foci: }(\pm 4,0) $$
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