MATH 107 PRACTICE
Please work out each of the given problems.
Credit will be based on the steps that you show towards the final answer.
Show your work.
PROBLEM 1 Please answer the following true or false.
If false, explain why or provide a counter example.
If true, explain why.
(11 Points) Suppose that
is defined at x
then f '(x)
is also defined
at x = 3
(12 Points) Let x
= x(t), y = y(t)
be parametric equations for a differentiable curve such that x''(-1)
= y''(-1) = 3
, then the curve is concave up at the point
(12 Points) If f(x,y)
is a differentiable function at the point P, then Dgradf(P)(P)
the directional derivative in the
direction of gradf(P) cannot be negative.
Test the following series for convergence.
If applicable, determine if the series converges absolutely or
(35 Points) Determine
the Maclaurin series for the function
PROBLEM 4 (35
Points) Find the interval of
PROBLEM 5 (35 Points) Find the length of one of the petals of the graph of the curve
r = 4 sin(3q)
PROBLEM 6 (35 Points) Find a unit vector that is perpendicular to the two vectors
3i + 4j - k
and 2i + j + 2k
Consider the paraboloid
z = x2 + y2
(35 Points) Let
f(x,y) = x cos(y2)
. Use the chain rule to determine
the angular rate of change of f given that x = r cos q, y = r sin q
PROBLEM 9 (35 Points) Find the equation of the tangent plane to the surface
at the point (2,1,1).
The temperature of a room in your factory can be modeled by the equation f(x,y)
. There is a round table of radius
centered at the origin.
Use the method of Lagrange multipliers to determine the hottest points on