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C4 function and suppose its 3rd Taylor polynomial at (0,0,0) is given by
P3(x , y, z) = 1+ 2x + 2y + 3z + x2 − 5x y + 2y2 − 7yz + z3.
Let P0, P1, P2, and P4 be the 0th, 1st, 2nd and 4th Taylor polynomials of f at (0,0,0).
No justification is necessary.
(1a) (1 point) Estimate the value f (0,0,0.2) with a quadratic approximation of f at (0,0,0).
f (0,0,0.2)≈
(1b) (4 points) If possible, determine each of the following quantities. If it is not possible, write “N/A".
f (0,0,0) =
f (0,0,1) =
∂ f
∂ x
(0,0,0) =
∂ 3 f
∂ x∂ y∂ z
(0,0,0) =
H f (0,0,0) =
lim
(x ,y,z)→(0,0,0)
f (x , y, z)− P3(x , y, z)
||(x , y, z)||2 =
lim
(x ,y,z)→(0,0,0)
f (x , y, z)− P3(x , y, z)
||(x , y, z)||3 =
MAT237 Term Test 2 - Page 2 of 11 January 23, 2020
2. (5 points) For each part below, select the BEST answer. Fill in EXACTLY ONE circle. (unfilled filled )
No justification is necessary.
(2a) For a C3 function f : !2→ !, you plot the quadratic form at each of its critical points a, b, c, d ∈ !2.
quadratic form of f at a quadratic form of f at b quadratic form of f at c quadratic form of f at d
If you apply the second derivative test to each critical point, at most one of them gives an inconclusive
answer. Identify which one (if any) gives the inconclusive answer.