Enrichment, Extension, and Applications

Question 1

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Exercises

    1. Find the equation of the tangent to the curve \( y = 2x^2 - 3x - 6 \) that is parallel to the line \( y = 5x + 4 \).
    2. Find the equation of the tangent to the curve \( y = 2x^2 - 3x - 6 \) that is perpendicular to the line \( y = 5x - 14 \).
  1. Let \( C \) be a point on the positive \( x \)-axis and \( D \) be a point on the positive \( y \)-axis such that the line \( CD \) is tangent to the curve \( y = \dfrac{8}{x} \), \( x \gt 0 \), at point \( P \) as shown in the diagram. Prove that any such point \( P \) is the midpoint of line segment \( CD \). Diagram of situation described in question
    1. Determine the coordinates of the two points on the curve \( y = x^3 - 9x^2 + 24x - 17 \) at which tangents to the curve have zero slope.
    2. Sketch the curve from part a). What is the relationship between the points determined in part a) and the graph of the curve?
  2. Find the equation of the line that is tangent to both \( y = x^2 \) and \( y = x^2 - 4x + 24 \).
  3. The flight path of a spacecraft is described by the curve \( y = 36x - x^3 \), for \( 0 \lt x \lt 6 \), where \( y \) is the height of the spacecraft (in km), and \( x \) is the horizontal distance from the launch pad (in km). If the engine is shut down at any point \( P \) on the curve, the spacecraft will fly off along the tangent line at \( P \). If the spacecraft is moving from left to right, and the engine is shut down when \( x = 4 \), how far from the launchpad will the spacecraft land?
  4. Find all positive values of \( k \) such that the circle with equation \( x^2 + y^2 = k^2 \) is tangent to the circle with equation \( (x - 5)^2 + (y + 12)^2 = 49 \).
  5. Find the area under the tangent to the curve \( y = \sqrt{x} \) at \( x = 4 \), that is above the \( x \)-axis and between the lines \( x = 0 \) and \( x = 8 \).
    1. Show that the slope of the tangent line to the circle \( x^2 + y^2 = 1 \) at the point \( P~(u, v) \) is \( \dfrac{-u}{v} \). Hence, find the equation of the tangent line in terms of \( u \) and \( v \).
    2. Determine the equations of the lines through \( P~(2, 2) \) that are tangent to the circle \( x^2 + y^2 = 1 \).
    3. Show that the distances from \( P~(2, 2) \) to the points of tangency are equal, and find those distances.
  6. Define \( g(x) = \dfrac{\sin(x + 0.01) - \sin(x)}{0.01} \), for \( x \in \mathbb{R} \).
    1. Evaluate \( g(x) \) for the \( 17 \) values: \( x = -2\pi, -1.75\pi, -1.5\pi, \ldots, 1.5\pi, 1.75\pi, 2\pi \) (where \( x \) is given in radians).
    2. Plot the \( 17 \) pairs of points \( \big( x, g(x) \big) \). On the same graph, plot the graph of \( h(x) = \cos(x) \). Comment on a plausible value for the slope of the tangent to \( y = \sin(x) \) based on the plot.
  7. The circle with equation \( (x - 5)^2 + (y - 3)^2 = 25 \) intersects the \( x \)-axis at points \( A \) and \( B \). Find the equations of all parabolas (having vertical axis of symmetry) whose only points in common with the circle are both \( A \) and \( B \).
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