*You must show sufficient detail to support your work to earn credit for your calculations.
This can be hand-written work, typed calculations, or excel formulas. To avoid round-off error,
retain at least six decimal places in all of your calculations. Complete
all trigonometric calculations in radians.
Assume the function f is defined as
f
(
x , y
)
=2 sin xtan y
Use differentiation rules to find the exact partial derivatives
∂ f
∂ x ∧∂ f
∂ y
evaluate those exact partial derivatives at (-3.1 , 1.56) .
Use the finite difference formulas to estimate and at .
∂ f
∂ x ∧∂ f
∂ y at (−3.1 ,1.56)
Use your calculated values to fill in this table:
Estimated partial derivatives using finite difference formulas:
h
finite difference approx. to
∂ f
∂ x
Exact
∂ f
∂ x
finite difference approx. to
∂ f
∂ y
Exact
∂ f
∂ y
0.01 -18526.1 -134838.6
0.001 -1854243 -194696.3
0.0001 -18542412 -1861069.7
1. h= 0.01
∂ f
∂ x
∂ f
∂ x =
(
xi, yi
)
≅f
(
xi, yi
)
−f
(
xi, yi
)
(
−3.1 ,1.56
)
≅
f
(
−3.1+h , 1.56
)
−f(−3.1 ,1.56)
h
(
−3.1 ,1.56
)
≅
f
(
−3.1+0.01 ,1.56
)
−f(−3.1 ,1.56)
0.01
(
−3.1 ,1.56
)
≅
f
(
−3.09 ,1.56
)
−f(−3.1 ,1.56)
0.01
(
−3.1 ,1.56
)
≅
2sin
(
−3.09
)
−tan
(
1.56
)
−2 sin
(
−3.1
)
−tan
(
1.56
)
0.01
∂ f
∂ x
(
−3.1,1.56
)
≅−185 26.1
