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.

Exercises 33 - 36 continue the exploration of the scalar field

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

(6)

which for integer pairs (x,y) in [ 0,10] ×[ 0,5] is given by

(7)

and which has the isotherms

(8)

33. Use (6) to determine the level curves with levels k = 179^°F, 217^°F, 233^°F, and 250^°F. Locate the point (6, 3) on a level curve and calculate the gradient at that point.  What is significant about the gradient at that point? Which direction is it pointing in?  

34. What is the rate of change of g( x,y) in (6) 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?

35.  In what direction is the temperature field (6) increasing the fastest at the point ( 1,0) ? At the point ( 0,2) ? Compare your result with the temperatures in (7).

36. Show that the gradient of the scalar field (6) always points toward ( 3,2) . What is the significance of this result?     

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: Sketch the path of least resistance from the highest point on the mountain to its base.

38.  Show that if F(x,y) = y - f(x) and if f(x) is differentiable everywhere, then the curve

r( t) = á t, f(t) + k, k ñ,  t in ( -¥,¥)

 is a level curve of z = F(x,y) with level k.  What is r' (t) · ÑF  when x = t and  y = f( t) +k?  Explain.

39. Write to Learn: Show that the curve r(t) = á 4cos( t) ,4sin( t) ñ is a parametrization of the level curve 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.

In exercises 41 - 44, we consider the vectors

vinc =  fx( p) ||Ñf( p) || ,  fy( p) ||Ñf( p) || ,||Ñf( p) || ,  vlevel =  -fy( p) ||Ñf( p) || ,  fx( p) ||Ñf( p) || ,   0

vperp = á -fx( p),-fy( p) ,1 ñ

at a  point p = ( p,q) with the assumption that Ñf( p) exists and is non-zero.

41. Show that v_level ^ v_perp and that v_level is a unit vector.  What is ( D_u f) ( p) when u = v_level  ?  What is significant about these results about v_level ?

42. Explain why v_level and v_inc are tangent to the surface z = f( x,y) at the point ( p,q,f( p) ) .  Then show that

vperp = vinc  ×  vlevel

What is the significance of this result?

43.  What is the equation of the plane through (p, q, f( p) ) with normal v_perp?  What is the significance of this result?

44.  Write to Learn: In a short essay, explain why if r( t) is a curve on the surface z = f( x,y)  and if

r( 0) = ( p, q, f( p) )

then there are numbers a and b such that

r' (0) = avinc  +  bvlevel