#
MTH 243

####
University of Rhode Island

Practice problems for Exam
#1
**1. **Find an equation of the sphere having
the points A(2,-1,1) and
B(-2,1,5) as the opposite points of the diameter.

**2. **Which of the tables below represent
a linear function f(x,y)?
For those that represent a linear function find an equation for
the function.

**(a)**

x \ y |
-1 |
0 |
1 |
2 |

0 |
1.5 |
1 |
0.5 |
0 |

10 |
3.5 |
3 |
2.5 |
2 |

20 |
5.5 |
5 |
4.5 |
4 |

30 |
7.5 |
7 |
6.5 |
6 |

**(b)**

x \ y |
-3 |
-2 |
-1 |
0 |

-3 |
18 |
13 |
10 |
9 |

-2 |
13 |
8 |
5 |
4 |

-1 |
10 |
5 |
2 |
1 |

0 |
9 |
4 |
1 |
0 |

**3. **Find an equation of the linear function
represented by a contour map below. Make sure that your equation is as
accurate as possible by choosing points where values of the function are
shown on the diagram exactly.

**4. **Consider two vectors **a
**=
(-2, 1, 2) and **b
= **(1, c, 4).

**(a) **For what value(s) of c
is the angle between **a** and **b**
equal to 45^{o } ?

**(b)** For what value(s) of c
is the area of parallelogram formed by these two vectors equal to 19
?

**5.** **(a) **
Find an equation of the plane containing the points A(4,1,1),
B(-2,2,-1) and C(2,1,5).

**(b)** Find an equation of the
line through the points A and C.

**6.** Let f(x,y) = ln(1 + y^{2
}+ x^{3 } ). Find f_{x}** **
and f_{y}** ** and the differential
df = f_{x}dx**
+ **_{ }f_{y}** **dy.** **Using
this and the linear approximation formula find the approximate value of
f(0.2, 0.1).

**7.** **(a) **Find the
directional derivative of f(x,y) = 4x sin(
x + y) + cos( x - y) at (0,p/4)
in the direction toward (p/4,p/2).

**(b)** Find an equation for the
tangent plane to the surface given by the equation z
= f(x, y) at the point (p/4,
p/4).

**(c) **In what direction does
f increase the fastest at the point
(2, 2) ? Give your answer as a unit vector.
Find the maximum rate of change at this point.

**8.** **(a)**
Find the gradient of the surface x y^{5
}+ y z^{5 }+ z x^{5 }- 3 = 0
at the point (1,1,1).

**(b) **Find an equation for the
tangent plane to this surface at this point. What angle does this gradient
vector make with this tangent plane ? Explain.