Exercises

Sketch the level curves of the following functions at the given levels.

1. f( x,y) = x2+y2 2. f( x,y) = y-x2 k = 1,4,9,16 k = 0,1,2,3,4 3. f( x,y) = xy 4. f( x,y) = x2+4y2 k = 1,2,3,4 k = 0,4,16,100 5. f( x,y) = ( x-y) 2 6. f(x,y) = xy k = 1,4,9,16 k = -2,-1,0,1,2

Find the gradient of the function implied by the level curve, and then show that it is perpendicular to the tangent line to the curve at the given point (in 15-17, you will need to use implicit differentiation to find the slope of the tangent line).

7. y = x3+x, at ( 1,2) 8. y = x2,  at (2,4) 9. x2+y = 2, at ( 1,1) 10. x2+xy = 10,    at ( 2,3) 11. x-y = 1 at ( 2,1) 12. x+y = 5, at (1,-2) 13. xy = 1, at ( 1,1) 14. x2y = 2 at (1,2) 15. xsin( xy) = 0, at ( 1,p) 16. xcos( x2y) = 2, at ( 2,p) 17. 2ex+yex = y2, at ( 0,2) 18. sin(x2y) = x+1, at ( -1,p)

Find the directional derivative of f in the direction of the given vector at the given point.

19. f( x,y) = x3+y3 at ( 1,1) 18. f( x,y) = x4+y2 at ( 1,1) in direction of v = á 2,1 ñ indirection of v = á -3,4 ñ 21. f( x,y) = cos( xy) at ( p/2,0) 20. f( x,y) = cos( xy) at ( p/2,0) in direction of v = á 1,0 ñ indirection of v = á 2,1 ñ 23. f( x,y) = xy at ( 1,1) 24. f(x,y) = xsin( xy) at ( p,0) in direction of v = á 1,1 ñ indirection of v = á 1,-1 ñ

Find the direction of fastest increase of the function at the given point, and then find the rate of change of the function when it is changing the fastest at the given point.

25. f( x,y) = x3+y3 at ( 1,1) 26. f( x,y) = x4+y2 at ( 1,1) 27. f( x,y) = cos( xy) at ( p/2,0) 28. f( x,y) = cos( xy) at ( p/2,0) 29. f( x,y) = xy at ( 1,1) 30. f(x,y) = xsin( xy) at ( p,0)

31. Show that ( 1,2) is a point on the curve g( x,y) = 2 where g( x,y) = x^-2y. Then find the slope of the tangent line to g( x,y) = 2 in two ways:

1. By finding using the gradient to find slope.
2. By showing that y = 2x² and applying the ordinary derivative.

Then sketch the graph of the curve, the tangent line, and the gradient at that point.

32. Show that ( 3,2) is a point on the curve g( x,y) = 15 where g( x,y) = x²+xy. Then find the slope of the tangent line to g( x,y) = 15 in two ways:

1. By finding using the gradient to find slope.
2. By solving for y and applying the ordinary derivative.

Then sketch the graph of the curve, the tangent line, and the gradient at that point.

33. Use (1) to determine the level curves with levels k = 179^°F, 217^°F, 233^°F, and 250^°F. Compare your results to the curves in (4).

34. In what direction is the temperature field

g( x,y) = 70+180e-( x-3) 2/10-( y-2)2/10

increasing the fastest at the point ( 1,0) ? At the point ( 0,2) ? Compare your result with the temperatures in (2).

35. Show that the gradient of the temperature function

g( x,y) = 70+180e-( x-3) 2/10-( y-2)2/10

always points toward ( 3,2) . What is the significance of this result?

36. What is the rate of change of g( x,y) in (1) at ( 1,1) in the direction of

u =  1 Ö2 ,  1 Ö2

Why would we expect it to be less than 48. 82^°F  per  inch?

37. A certain "mountain'' shown below, where units are in thousands of feet: Use the "mountain" plot above to estimate the levels of each of the level curves below:

38. In exercise 37, sketch the path of least resistance from the highest point on the mountain to its base.

39. Write to Learn: Show that the curve r(t) = á 4cos( t) ,4sin( t) ñ is a parametrization of f( x,y) = x²+y² with level 16. Then find T and N at t = p/6. How is N at t = p/6 related to Ñf at the point ( 2,2Ö3) ? Write a short essay explaining this relationship.

40. *Write to Learn: Obtain a city map and fix a street corner on that map to be the origin. Define a function f( x,y) to be the driving distance from that street corner to a point (x,y) , where the driving distance is the shortest possible distance from the origin to ( x,y) along the roads on the map. What are the level curves of the driving distance function? Sketch a few of them. What is the path of least resistance at a given point? Write an essay describing the f( x,y) function, its level curves, and its paths of least resistance.