Name
MATH 107 PRACTICE
FINAL 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. A.
(11 Points) Suppose that
B.
(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
(x(-1),y(-1))
. C.
(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. PROBLEM 2
Test the following series for convergence.
If applicable, determine if the series converges absolutely or
conditionally. A.
(17 Points)
B.
(18 Points)
PROBLEM 3
(35 Points) Determine
the Maclaurin series for the function
3 PROBLEM 4 (35
Points) Find the interval of
convergence 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 PROBLEM 7
(35 Points)
Consider the paraboloid
z = x2 + y2
PROBLEM 8
(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
x2 at the point (2,1,1). PROBLEM 10
(35 Points)
The temperature of a room in your factory can be modeled by the equation f(x,y)
= exy
. There is a round table of radius
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