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 | 1. For
implicit differentiation, we assume that y is a function of x: we write y(x) to remind ourselves of this. However, for the circle x2+y2 = 1, it is not true that y is a
function of x. Since  there are actually (at least) two functions of x defined implicitly. Explain why this is not really a contradiction; that is, explain exactly what we are assuming when we do
implicit differentiation. |
| | 2. To perform
implicit differentiation on an equation such as x2y2+3 = x, we start by differentiating all terms. We get 2xy2+x22yy' = 1. Many students learn the rules this way: take “regular” derivatives of all terms, and tack on a y' every time you take a y -
derivative. Explain why this works, and rephrase the rule in a more accurate and understandable form. |
| | 3. In
implicit differentiation, the derivative is typically a function of both x and y; for example, on the circle x2+y2 = r2, we have y' = -x/y. If we take the
derivative -x/y and plug in any x and y, will it always be the slope of a tangent line? That is, are there any requirements on which x's and y's we can plug in? |
| | 4. In each example in this section, after we differentiated the given equation we were able to rewrite the resulting equation in the form f (x, y)y' (x) = g(x, y) for some expressions f (x, y) and g(x, y). Explain why this can always be done; that is, why doesn't the chain rule ever produce a term like [y' (x)]2 or 
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| In exercises 5-8, compute the slope of the tangent line at the given point both explicitly (first solve for
y as a function of x) and implicitly.
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| 5. x2+4y2 = 8 at (2, 1) | 6. x3y-4 = x2y at (2,  ) |
| 7. y-3x2y = cos x at (0, 1) | 8. y2+2xy+4 = 0 at (-2, 2).
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| In exercises 9-20, find the
derivative y' (x) implicitly.
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| 9. x2y2+3y = 4x | 10. 3xy3-4x = 10y2 |
| 11.  | 12. sin xy = x2-3 |
| 13.  | 14. 3x+y3-4y = 10x2 |
| 15.  | 16. xey-3ysin x = 1 |
| 17.  | 18. cos y-y2 = 8 |
| 19. e4y-ln y = 2x | 20. 
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| In exercises 21-28, find an equation of the tangent line at the given point. If you have a CAS that will graph implicit curves, sketch the curve and the tangent line.
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| 21. x2-4y2 = 0 at (2, 1) | 22. x3-4y2 = 4 at (2, 1) |
| 23. x2-4y3 = 0 at (2, 1) | 24. x2y2 = 4x at (1, 2) |
| 25. x2y2 = 4y at (2, 1) | 26. x3y2 = -3xy at (-1, -3) |
| 27. x3y3 = 9y at (1, 3) | 28. x2y3 = 9y at (1, -3)
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29. A baseball player stands 2 feet from home plate and watches a pitch fly by. In the diagram, x is the
distance of the ball from home plate and  is the angle indicating the direction of the player's gaze. Find the rate ' at which his eyes must move to watch a fastball with x' = -130 ft/s as it crosses home plate at x = 0.
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| 30. In the situation of exercise 29, humans can maintain focus only when  (see Watts and Bahill's book Keep Your Eye on the Ball). Find the fastest pitch that you could actually watch cross home plate. |
| 31. A camera tracks the launch of a vertically ascending spacecraft. The camera is located 2 miles from the launchpad. If the spacecraft is 3 miles up and traveling at 0.2 miles per second, at what rate is the camera angle changing? |
| 32. Repeat exercise 31 for the spacecraft at 1 mile up (assume the same velocity). Which rate is higher? Explain in commonsense terms why it is larger. |
| 33. Assume that the infected area of an injury is circular. If the radius of the infected area is 3 mm and growing at a rate of 1 mm/hr, at what rate is the infected area increasing? |
| 34. For the injury of exercise 33, find the rate of increase of the infected area when the radius reaches 6 mm. Explain in commonsense terms why this rate is larger than that of exercise 33. |
| 35. A plane is located x = 40 miles (horizontally) away from an airport at an altitude of h miles. A radar at the airport detects that the
distance s(t ) between the plane and airport is changing at the rate of s' (t ) = -240 mph. If the plane flies toward the aiport at the
constant altitude h = 4 , what is the speed x' (t ) of the airplane? |
| 36. Repeat exercise 35 with a height of 6 miles. Based on your answers, how important is it to know the actual height of the airplane? |
| 37. Rework
Example 8.5 if the police car is not moving. Does this make the radar gun's measurement more accurate? |
| 38. Show that the radar gun of
Example 8.5 gives the correct speed if the police car is at the
origin. |
| 39. Show that the radar gun of
Example 8.5 gives the correct speed if the police car is at x =  moving at a speed of (-1)50 mph. |
| 40. Find a position and speed for which the radar gun of
Example 8.5 has a slower reading than the actual speed. |
| 41. Suppose that the average yearly cost per item for producing x items of a business product is  If the current production is x = 10 and production is increasing at a rate of 2items per year, find the rate of change of the average cost. |
| 42. Suppose that the average yearly cost per item for producing x
items of a business product is  The three most recent yearly production figures are given in the table. |
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| Year | 0 | 1 | 2 |
| Prod. (x) | 8.2 | 8.8 | 9.4 |
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| Estimate the value of x' and the current (year 2) rate of change of the average cost. |
| 43. For a small company spending $ x thousand per year in advertising, suppose that annual sales in thousands of dollars equal s(x) = 60-40e-0.05x. The three most recent yearly advertising figures are given in the table. |
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| Year | 0 | 1 | 2 |
| Adver. | 16,000 | 18,000 | 20,000 |
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| Estimate the value of x' and the current (year 2) rate of change of sales. |
| 44. For a small company spending $ x thousand per year in advertising, suppose that annual sales in thousands of dollars equal s(x) = 80-20e-0.04x. If the current advertising budget is x = 40 and the budget is increasing at a rate of $ 1500 per year, find the rate of change of sales. |
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45. Suppose a 6 - ft tall person is 12 ft away from a 18 - ft tall lamppost (see the figure). If the person is moving away from the lamppost at a rate of 2 ft/s, at what rate is the length of theshadow changing? Hint: Show that 
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Figure for exercise 45.
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| 46. Rework exercise 45 if the person is 6 ft away from the lamppost and is walking toward the lamppost at a rate of 3 ft/s. |
| 47. Suppose that a raindrop evaporates in such a way that it maintains a spherical shape. Given that the volume of a sphere of radius r is V =   r3 and its surface area is A = 4 r2, if the radius changes in time, show that V' = Ar'. If the rate of evaporation ( V' ) is proportional to the surface area, show that the radius changes at a
constant rate. |
| 48. Suppose a forest fire spreads in a circle with radius changing at a rate of 5 feet per minute. When the radius reaches 200 feet, at what rate is the area of the burning region increasing? |
| 49. Boyle's Law for a gas at
constant temperature is PV = c, where P is pressure, V is volume and c is a
constant. Assume that both P and V are functions of time. Show that P' (t )/V'(t ) = -c/V2.
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| 50. In exercise 49, solve for P as a
function of V. Treating V as an independent variable, compute P' (V). Compare P' (V) and P' (t )/V' (t ) from exercise 49. |
| 51. A dock is 6 feet above water. Suppose you stand on the edge of the dock and pull a rope attached to a boat at the
constant rate of 2 ft/s. Assume that the boat remains at water level. At what speed is the boat approaching the dock when it is 20 feet from the dock? 10 feet from the dock? Isn't it surprising that the boat's speed is not constant? |
| 52. Sand is poured into a conical pile with the height of the pile
equaling the diameter of the pile. If the sand is poured at a constant rate of 5 m3/s, at what rate is the height of the pile increasing when the height is 2 meters? |
| In exercises 53 and 54, find the locations of all horizontal and vertical tangents.
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| 53. x2+y3-3y = 4 | 54. xy2-2y = 2
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| In exercises 55 and 56, find the second
derivative y''
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| 55. x2+y2 = 4 | 56. x2+y3-2y = 3
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| 57. In
Example 8.1, it is easy to find a y - value for x = 2, but other y - values are not so easy to find. Try solving for y if x = 1.9. Use the tangent line found in
Example 8.1 to estimate a y - value. Repeat for x = 2.1.
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| 58. Use the tangent line found in
Example 8.2 to estimate a y - value corresponding to x = 1.9; x = 2.1.
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| 59. For elliptic curves, there are nice ways of finding points with
rational coordinates (see Ezra Brown's article Three Fermat Trails to Elliptic Curves in the May 2000 College Mathematics Journal for more information). If you have access to an implicit plotter, graph the elliptic curve defined by y2 = x3 - 6x + 9. Show that the points ( - 3, 0) and (0, 3) are on the curve. Find the line through these two points and show that the line intersects the curve in another point with
rational (in this case, integer) coordinates. |
| 60. For the elliptic curve y2 = x3-6x+4, show that the point (-1, 3) is on the curve. Find the tangent line to the curve at this point and show that it intersects the curve at another point with
rational coordinates. |
| 61. Use
implicit differentiation to find y' (x) for x2y-2y = 4. Based on this equation, why would you expect to find vertical tangents at x =  and horizontal tangents at y = 0 ? Show that there are no points for these values. To see what's going on, solve the original equation for y and sketch the graph. Describe what's happening at x = and y = 0.
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| 62. Show that any curve of the form xy = c for some
constant c intersects any curve of the form x2-y2 = k for some
constant k at right angles (that is, the tangent lines to the curves at the intersection points are perpendicular). In this case, we say that the families of curves are orthogonal.
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 | 63. Suppose a slingshot (see
Section 2.1) rotates counterclockwise along the circle x2+y2 = 9 and the rock is released at the point (2.9, 0.77). If the rock travels 300 feet, where does it land? [Hint: find the tangent line at (2.9, 0.77) and find the point (x, y) on that line such that the
distance is 
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| | 64. A landowner's property line runs along the path y = 6-x. The landowner wants to run an irrigation ditch from a reservoir bounded by the ellipse 4x2+9y2 = 36. The landowner wants to build the shortest ditch possible from the reservoir to the closest point on the property line. We explore how to find the best path. Sketch the line and ellipse, and draw in a tangent line to the ellipse that is
parallel to the property line. Argue that the ditch should start at the point of tangency and run
perpendicular to the two lines. We start by identifying the point on the right side of the ellipse with tangent line
parallel to y = 6-x. Find the slope of the tangent line to the ellipse at (x, y) and set it equal to -1. Solve for x and substitute into the equation of the ellipse. Solve for y and you have the point on the ellipse at which to start the ditch. Find an equation of the (normal) line through this point
perpendicular to y = 6-x, and find the intersection of the normal line and y = 6-x. This point is where the ditch ends. |
