Finding Tangents to Curves
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Questions and Answers

What is the equation of the tangent to the curve $2(1 + y) = x^2$ at the origin, if it exists?

  • $y = 2x$
  • $y = -2x$
  • $y = x$
  • $y = -x$ (correct)
  • At which point does the curve $2(1 + y) = x^2$ intersect the y-axis?

  • (0, -1) (correct)
  • (0, 1)
  • (0, 0)
  • (0, 2)
  • What is the gradient of the curve $2(1 + y) = x^2$ at the origin?

  • $0$
  • $1$
  • $-1$ (correct)
  • $2$
  • Which of the following represents the correct transformation of the curve $2(1 + y) = x^2$ into standard form?

    <p>$y = \frac{x^2}{2} - 1$</p> Signup and view all the answers

    What is the nature of the curve given by the equation $2(1 + y) = x^2$?

    <p>Parabolic</p> Signup and view all the answers

    What is the form of the equation representing the tangent line to the curve at the origin?

    <p>y = mx + b</p> Signup and view all the answers

    What is the necessary condition for the tangent to exist at the origin for the curve?

    <p>The curve must be differentiable at that point.</p> Signup and view all the answers

    What can be inferred if the derivative of the curve at the origin is zero?

    <p>The curve has a horizontal tangent at the origin.</p> Signup and view all the answers

    Which of the following characterizes the slope of the tangent line to the curve at the origin if it exists?

    <p>It can be negative or positive depending on the curve.</p> Signup and view all the answers

    What type of geometric figure does the equation $2(1 + y) = x^2$ represent?

    <p>A parabola</p> Signup and view all the answers

    Study Notes

    • The given equation is 2(1 + y) = x2 + x.
    • To find the equation of the tangent at the origin (0,0), we need to determine if the origin is a point on the curve.
    • Substituting x = 0 and y = 0 into the equation, we get 2(1 + 0) = 02 + 0, which simplifies to 2 = 0. This is not true, so the origin is not on the curve.
    • Therefore, there is no tangent to the curve at the origin.
    • The text mentions "the equation of the tangent to the curve..." implying a different goal, potentially finding the tangent at a different point.
    • The existence of a tangent at the origin depends crucial on whether the origin lies on the curve.
    • If the origin was on the curve, various methods to find the tangent would be possible, such as implicit differentiation or finding a point nearby.

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    Description

    This quiz explores the concept of tangents to curves through a specific equation. You will learn how to determine if points are on the curve and analyze the implications of these findings. Test your understanding of these fundamental concepts in calculus.

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