Practice Test II
1. For f(x,y) below, calculate all four second-order partial derivatives and check that
. Assume the variables are restricted to a domain on which the function is
defined.
2. If
find
and
variables are restricted to domains on which the functions are defined.
. The
3. If
find
using the chain rule. Assume the variables are restricted to domains on
which the functions are defined.
4. Consider the function
. Suppose
is the surface
a) Find a vector which is perpendicular to the level curve of through the point
in the direction in which decreases most rapidly.
b) Find the equation of the plane tangent to f at the point (2,6).
5. Find the projection of vector 𝑣⃗ = 2i – 3j +k onto the vector 𝑤
⃗⃗⃗ = i – j +k .
6. Find the directional derivative at the point (1,0) on
direction of 𝑤
⃗⃗⃗ = i – j + k.
7. Prove or disprove, via the integral test: ∑∞
𝑛=1
1
𝑛1/2
converges.
in the
Practice Test II
8. Prove or disprove using the comparison test: ∑∞
𝑛=1
1
𝑛2 +10
9. Prove or disprove using the limit comparison test:
converges.
converges.
10. A function f(x,y) is illustrated in the table below. Find a linear estimate for the
value of the function f(2.1,2.1)
Y
X
1
2
3
1
2
4
8
2
2
4
8
3
2
4
8
11. Find an estimate of grad f at (x,y) = (2,2), if f(x,y) is illustrated by the table above.
12. Extra credit question: Prove that the base 10 repeating decimal converges, where
a,b, and c are digits that are not all zero: .abcabcabc…