Free response · Unit 3, Unit 4 Beta

A cubic curve: approximation, no horizontal tangents, and a particle

Implicit differentiation and related rates · Part B: no calculator · 9 points

The question

Name: ______________________   Date: ____________   A cubic curve: approximation, no horizontal tangents, and a particle (9 points)

Diagram

The curve y3−xy=6y^3 - xy = 6

Consider the curve given by y3−xy=6y^3 - xy = 6. The points (1,2)(1, 2) and (7,3)(7, 3) are on the curve.
The curve y cubed minus x y equals 6 The graph of the curve y cubed minus x y = 6 on axes with x from minus 2 to 14 and y from minus 4 to 4. It has two pieces: an upper piece rising through (1, 2) and (7, 3), and a lower piece below the x-axis. The curve never meets the x-axis. The points (1, 2) and (7, 3) are marked. −2 2 4 6 8 10 12 14 −4 −3 −2 −1 1 2 3 4 x y (1, 2) (7, 3) y3 − xy = 6The curve has two pieces,one above and one belowthe x-axis; it never meets it.

  1. (a) Show that dydx=y3y2−x\frac{dy}{dx} = \frac{y}{3y^2 - x}. [2 points]

  2. (b) Write an equation for the line tangent to the curve at (1,2)(1, 2). Use it to approximate the yy-coordinate of the point on the curve near (1,2)(1, 2) with x=1.2x = 1.2. [2 points]

  3. (c) Show that there is no point on the curve at which the line tangent to the curve is horizontal. [2 points]

  4. (d) A particle moves along the curve. At the instant it passes through (7,3)(7, 3), its xx-coordinate is decreasing at 2 units per second. Find dydt\frac{dy}{dt} at that instant. Is the yy-coordinate increasing or decreasing? Justify your answer. [3 points]