Chapter 3: Worksheet 9 Jack K. Cohen Colorado School of Mines

Implicit Differentiation and Related Rates; Applications

Suggested Problems Section 3.8: 2, 4, 6, 16.

1. x³ = 4x is an equation valid only for special values of x.
   1. Find these special x values.
   2. Differentiate both sides of the equation to get another equation.
   3. Is the new equation valid for all x or just special x's? Are the special x's the same as those for the original equation?

2. (x + 1)³ = x³ +3x² + 3x + 1 is an identity in x.
   1. Verify that the stated relation really is an identity.
   2. Differentiate both sides of the identity and show that the result is another identity.
   3. Differentiate both sides of the identity in (b) and show that the result is yet another identity.

3. cos 2x = cos²x - sin²x is an identity in x.
   1. Verify informally that the stated relation is an identity (eg. you could substitute several values on both sides by hand or Plot or Table both sides).
   2. Differentiate both sides of the identity and show that the result is another identity.

4. can be used to compute implicit derivatives. Equations in are denoted by using double = = in contrast to the single = used for naming objects (observe that both of these symbols are used in the example below). To get to treat y as a function of x, we consistently write it as y[x]. Though it isn't hard to do by hand, we use a new statement, Solve, to isolate dy/dx = y'[x]. As an example, consider (3.8.9):

   diffeqn = D[x^2 (x - y[x]) == y[x]^2 (x + y[x]), x]
                         2
       2 x (x - y[x]) + x  (1 - y'[x]) == 
    
                                       2
         2 y[x] (x + y[x]) y'[x] + y[x]  (1 + y'[x])
         
   Solve[diffeqn, y'[x]] //Simplify
                     2                  2
                  3 x  - 2 x y[x] - y[x]
       {{y'[x] -> -----------------------}}
                   2                    2
                  x  + 2 x y[x] + 3 y[x]
   

   Find dy/dx both by hand and with for

   1. (3.8.9) x²(x - y) = y²(x + y)
   2. (3.8.10) (x² + y²)² = 4xy

5. A comparison of new and old approaches:
   1. (3.8.19) For the equation 1/x³ +1/y³ = 2, find dy/dx by implicit differentiation and also find the tangent line at (1, 1).
   2. Check your answer to (a) by first solving for y and then computing dy/dx.
   3. In general, you would not be able to take the approach of part (b) since solving for y in closed form might be impossible. Even when you can do it, the implicit differentiation approach might be easier. In your opinion, which approach was better for this problem? Why?

6. (3.8.22) Show that the graph of xy⁵ + x⁵y = 1 has no horizontal tangents.

7. For the equation x³ + y³ = 3xy - 1 (note similarity to the equation x³ + y³ = 3xy in Example 3.8.2), show
   1. (3.8.23) There are no points where the graph is horizontal.
   2. (3.8.63) Show that the graph consists of a straight line and an isolated point. hints: use Factor on the original expression and Expand and Simplify on the expression (x - y)² + (x - 1)² + (y - 1)².

8. can be used to plot implicit functions. The routine to do this is called ImplicitPlot, but it is not part of the standard distribution and has to be loaded in as shown below. Also, be aware that plotting implicit functions is a hard problem, so this routine often fails or might have to be ``coaxed'' by asking for plots in restricted regions that you can then piece together. Also, it usually takes some experimentation to get the ranges right. The commands below produce the plot shown in Figure 1.

   <<Graphics`Master`
   ImplicitPlot[x^3 + y^3 == 3 x y, {x, -3, 3}]
   

   Figure: Problem 8

   

   Use ImplicitPlot to reproduce the following figures in the text:

   1. (fig 3.56) x² - xy + y² = 9 (be careful to put a space between x and y and don't forget about the double equal sign).
   2. (fig 3.57) (x² + y²)² = x² - y²

9. (3.8.31) Sand is being emptied from a hopper at the rate of 10 m³/sec forms a conical pile whose height is always twice its radius. note: the volume V of a cone of height h and radius r is given by V = ${\frac{{\pi}}{{3}}}$ r²h.
   1. Using related rates, find the rate at which the radius of the pile is increasing when its height is 5 m.
   2. Find the same rate by first eliminating the height from the formula for the volume.
   3. Which approach is easier for this particular problem?

10. (3.8.42) The volume V of water in a partially filled spherical tank of radius a is V = ${\frac{{1}}{{3}}}$ πy²(3a - y), where y is the maximum depth of the water. Suppose that water is being drained from a spherical tank of radius 5 ft at the rate of 100 gal/min.
    1. Find the rate at which the depth y of water is
       1. y = 7 ft
       2. y = 3 ft
       (note: One gallon of water occupies a volume of approximately 0.1337 ft³.)
    2. Generalize the statement and solution of this problem. Check the dimensions.
    3. (3.8.56) Show that your general solution can be used to solve problem # 56: When a spherical tank with a radius of 10 ft contains water with a maximum depth of y feet, the volume of the water in the tank is V = ${\frac{{\pi}}{{3}}}$ (30y² - y³) cubic feet. If the tank is being filled with water at the rate of 200 gal/min, how fast is the water level rising when y = 5 ft? (note: One gallon of water occupies a volume of approximately 0.1337 ft³.)

11. (3.8.45) A ladder 41 ft long has been leaning against a vertical wall. It begins to slip, so that its top slides down the wall while its bottom moves along the ground; the bottom moves at a constant speed of 10 ft/s. How fast is the top of the ladder moving when it is 9 ft above the ground? Generalize—preferably before solving the specific problem. Check the dimensions.

12. (3.8.53) At time t = 0, a single-engine military jet is flying due east at 12 mi/min. At the same altitude and 208 mi directly ahead of it, still at time t = 0, is a commercial jet, flying due north at 8 mi/min. When are the two planes closest to each other? What is the minimum distance between them? Show that the dimensions of your answers are correct.