Exercises
Sketch the level curves of the following functions at the
given levels.
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 1517, you will need to use implicit
differentiation to find the slope of the tangent line).

    
    
    
    
   
xcos( x^{2}y) = 2, at ( 2,p) 


2e^{x}+ye^{x} = y^{2}, at ( 0,2) 
  
sin(x^{2}y) = x+1, at ( 1,p) 




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


f( x,y) = x^{3}+y^{3} at ( 1,1) 
  
f( x,y) = x^{4}+y^{2} at ( 1,1) 


in direction of v =
á 2,1
ñ 
  
indirection of v =
á 3,4
ñ 


f( x,y) = cos( xy) at ( p/2,0) 
  
f( x,y) = cos( xy) at ( p/2,0) 


in direction of v =
á 1,0
ñ 
  
indirection of v =
á 2,1
ñ 

   
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.


f( x,y) = x^{3}+y^{3} at ( 1,1) 
  
f( x,y) = x^{4}+y^{2} at ( 1,1) 


f( x,y) = cos( xy) at ( p/2,0) 
  
f( x,y) = cos( xy) at ( p/2,0) 

   
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^{2}y. Then find
the slope of the tangent line to g( x,y) = 2 in two ways:
 By finding using the gradient to find slope.
 By showing that y = 2x^{2} 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) = x^{2}+xy. Then
find the slope of the tangent line to g( x,y) = 15 in two ways:
 By finding using the gradient to find slope.
 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  (y2)^{2}/10



 (6) 
which for integer pairs (x,y) in [ 0,10] ×[ 0,5] is given by
and which has the isotherms
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) = x^{2}+y^{2} 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
v_{inc} = 


f_{x}( p)
Ñf( p) 

, 
f_{y}( p)
Ñf( p) 

,Ñf( p)   
, v_{level} = 


f_{y}( p)
Ñf( p) 

, 
f_{x}( p)
Ñf( p) 

, 0 



v_{perp} =
á f_{x}( p),f_{y}( p) ,1
ñ 

at a point p = ( p,q) with the assumption
that Ñf( p) exists and is nonzero.
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
v_{perp} = v_{inc }× v_{level} 

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
then there are numbers a and b such that
r' (0) = av_{inc }+ bv_{level} 
