Exercises

Find the linearization L( x,y) of f( x,y) at the given point in the xy-plane. Then graph L( x,y) and f( x,y) to illustrate that z = L( x,y) is tangent to z = f(x,y) .

1. f( x,y) = x2+y3 at ( 1,2) 2. f( x,y) = x2y+y2 at ( 1,2) 3. f( x,y) = x2+xy+3x  at  ( 0,0) 4. f( x,y) = ( x+2y) 2  at  ( 1,3) 5. f( x,y) = ln( x2+y2) at (1,1) 6. f( x,y) = xsin( xy) at ( 1,p) 7. f( x,y) = x2sin( y)   at  ( 0,p) 8. f( x,y) = exln( y2+1)   at( 1,0) 9. f( x,y) = 3x+7y+1 at ( 1,2) 10. f( x,y) = 3x+7y+1 at ( 1,2) 11. f( x,y) = xtan( xy) at ( 1,p) 12. f( x,y) = csc( x+y) at æ è    p 2 ,  p 3 ö ø   13. f( x,y) =  -1 ( x2+y2) 3/2   at ( 0,1) 14. f( x,y) =  1 1-xy   at  ( 0,0)

Find the Hessian matrix and the quadratic approximation of f at the given point.  

. 15. f( x,y) = x2+y3  at  ( 1,2) 16. f( x,y) = x2+2xy+y3  at  ( 2,2) 17. f( x,y) = x2+xy+3x  at  ( 0,0) 18. f( x,y) = ( x+2y) 2  at  ( 1,3) 19. f( x,y) = 3x-2y+1 20. f( x,y) = 5 21. f( x,y) = ln( x2+y2) at (1,1) 22. f( x,y) = xsin( xy) at ( 1,p) 23. f( x,y) = x2sin( y)   at  ( 0,p) 24. f( x,y) = exln( y2+1)  at ( 1,0)

Find Dz and dz at the given point in the xy-plane.

25. z = x2+2y at ( 1,2)    26. z = x2y+y2 at ( 1,1) dx = Dx = 0.1,    dy = Dy = 0.01 dx = Dx = 0.1,   dy = Dy = -0.01 27. z = xtan( xy) at ( 1,p) 28. z = ( x+2y) 2 at ( 1,1) dx = Dx = 0.01,    dy = Dy = 0.01 dx = Dx = 0.01,   dy = Dy = -0.1 29. z = tan-1 æ è    y x ö ø at ( 1,1) 30. z = e-x2-y2 at ( 0,0) dx = Dx = 0.01,  dy = Dy = 0.01 dx = Dx = 0.1,  dy = Dy = -0.2

31. A rectangular box has a height of 4 feet to within an accuracy of 1 in and a square base with width 2 feet to within an accuracy of 1 in. Find the volume V and the approximate error dV in the volume of the can.

32. The length of a pendulum is measured to be l = 5 feet to within 1 in of accuracy. The period of the pendulum as it swings is measured to be T = 2.5 seconds to within 0.1 seconds. The acceleration due to gravity is related to the swinging of the pendulum by

g =  4p2l T2

What is the acceleration due to gravity with respect to these measurements and about how accurate is the computed value of the acceleration?

33. The diameter of the top of a soup can is measured to be 3 inches to within 1/16 of an inch. The height of the can is measured to be 4 inches to within 1/16 of an inch.  

a.

Write the surface area S as a function of the height and diameter of the can.  

b.

Find S when the height is 4^¢¢ and the diameter is 3^¢¢.

c.

Find dS when the height is 4^¢¢, the diameter is 3^¢¢, and the differentials of height and diameter are both 1/16^¢¢.  About how much variation in the area computation is possible given that height and diameter are accurate only to a sixteenth of an inch?

34. Find the volume V and the differential dV for the can in exercise 33.  About how much variation in the volume computation is possible given that height and diameter are accurate only to a sixteenth of an inch?

35. Show that the tangent plane to any plane of the form z = ax+by+c is the plane itself.

36. Show that the quadratic approximation at (0,0) of

Q( x,y) = ax2+bxy+cy2

is Q( x,y) itself.

37. The mirror for a telescope should be in the shape of a paraboloid with focus ( 0,0,p) , which is the graph of a function of the form

Q( x,y) =  x2+y2 4p

(4)

For example, a telescope mirror with focus at ( 0,0,1) is shown below: However, if the telescope mirror is small enough, it more economical to manufacture a spherical cross-section that approximates (4). The graph of a spherical cross-section of radius R is given by

f( x,y) = R- R2-x2-y2

Find the Quadratic approximation of f at ( 0,0) and show that it is of the form (4). What is R in terms of p? How does this relate to the manufacture of spherical mirrors as approximations to parabolic mirrors?

38. In problem 37, what value of R leads to its quadratic approximation having a focus of ( 0,0,2) ?

39. By letting x be of the form x = [ p,q+h] ^t where h is approaching 0, use the definition of the total derivative to show that if Ñf( p) = [ a,b] , then b = f_y( p,q) .

40. Show that if f( x) is differentiable at p, then there exists a neighborhood of p on which

Ñf( p) ( x-p) -e| | x-p| | < f( x) -f( p) < Ñf( p) ( x-p) +e| | x-p| |

Use this to explain why f( x) is continuous at p . (that is, why

lim x® p  f( x) = f( p)  )

41. Show that the tangent plane to the graph of f(x,y) = x²-y² is of the form

z = 2px-2qy-( p2-q2)

(7)

Then show that the intersection of z = x²-y² with its tangent plane (7) results in a pair of straight lines. How many lines does the surface z = x²-y² contain?

42. Write to Learn: Write a short essay which explains why if f( x,y) is a polynomial such as f( x,y) = x²y+y², then f_x and f_y are the coefficients of h and k in the expansion of f( x+h,y+k) . How does this relate to the concept of differentiability?

43. Write to Learn: In a short essay, explain why the function f( x,y) = (x²+y²)^1/2 is not differentiable at ( 0,0) . (Hint: consider the graph of f( x,y) and also consider the definition of differentiability).

44.  Show that f_x( 0,0) exists but f_y( 0,0) does not exist when

f( x,y) = x4+y2

Does the total derivative of f( x,y) exist at (0,0) ?  Explain.

45.  Explain how Definition 5.1 applies to functions of 3 variables U( x,y,z) and then mimic the derivation which follows to show that

ÑU( p) = á Ux( p),Uy( p) ,Uz( p) ñ

46.  For f( x) differentiable at a point p,let us define a function A( Dx) by

A( Dx) = Ñf( p) ·Dx

Show that A( Dx) is linear (i.e., that A( Dx+Dy) = A( Dx) +A( Dy) and A( kDx) = kA( Dx) for any scalar k ).  Also, explain the significance of vectors Dx for which A(Dx) = 0.