Sign in to jmaths to print this page or save it as a PDF: jmaths-8ce638778fd6.herokuapp.com/maths/ai-sl/differentiation/12-9-10/problems/b/

jmaths

IB Mathematics: Applications and Interpretation SL · 12.9: Differentiation · 12.9.10

Name Date

Harder tangent and normal problems Problem set B

Show your working. Use a letter for any unknown point, and use the calculator for derivatives you cannot find by hand. Give calculator answers to 3 significant figures.

Each question matches the question with the same number in set A. Do the ones you got wrong there.

Difficulty: first skills applying them stretch

Skills

  1. Use technology to answer each part. Give each answer in the form \(y=mx+c\), with m and c to 3 significant figures.

    1. (a) Find the equation of the tangent to \(y=x\sqrt{x+3}\) at \(x=5\).
    2. (b) Find the equation of the normal to \(y=\frac{1}{x+7}\) at \(x=-5\).

  2. Use technology to find the equation of the normal to each curve at the given point.

    1. (a) \(y=x^2e^{-4x}\) at \(x=0\)
    2. (b) \(y=(x-3)^2e^{x/2}\) at \(x=3\)

  3. Give each answer in the form \(y=mx+c\).

    1. (a) Find the equation of the tangent to the curve \(y=x^2-5x\) which is parallel to the line \(y=3x-7\).
    2. (b) Find the equation of the normal to the curve \(y=x^2-2x\) which is parallel to the line \(x+6y=2\).

  4. Find the equations of the normals to the curve \(y=x^3-2x\) which are perpendicular to the line \(y=10x-3\). Give each in the form \(y=mx+c\).

  5. Find the coordinates of the points on the curve \(y=x^3-3x^2-2x\) at which the tangent is parallel to the line \(y=7x+5\).

  6. Find all the points on the curve \(y=x^2+4x-4\) where the gradient of the tangent equals the y-coordinate.

  7. Find the coordinates of the point where the normal to each curve at the given point meets the curve again.

    1. (a) \(y=x^2+2x\) at \(x=1\)
    2. (b) \(y=\frac{4}{x}\) at \(x=2\)

In context

  1. A rope is pulled tight from the point \((2, -2)\) so that it just touches the curved surface of a bowl with cross-section \(y=x^2+3\) (the rope lies along a tangent to the curve).

    1. (a) Find the coordinates of the two points where the line can touch the curve.
    2. (b) Write down the equation of the tangent at the point with the larger x-coordinate, in the form \(y=mx+c\).

  2. A curved road is modelled by the curve \(y=\frac{a}{x}\), where a is a positive constant. The normal to the curve at the point where \(x=4\) is parallel to the line \(4x-y=9\). Find the value of a.

Stretch

  1. The tangent to the curve \(y=\frac{10}{x}\) at the point where \(x=a\) (\(a>0\)) meets the y-axis at P and the x-axis at Q. O is the origin. Show that the area of triangle OPQ does not depend on a.

  2. Show that the tangent to the curve \(y=x^3+4x-7\) at the point with x-coordinate k meets the curve again at the point with x-coordinate \(-2k\).

jmaths.xyz · AI SL · 12.9.10 Harder tangent and normal problems