p.1
MATH 2202
Check List for Mid-Term 3
I) Lagrange multiplier method. Maximize/minimize f with constraint g = c
• What equations that we have to solve?
• Different tricks in solving the equations.
• What is the geometrical meaning behind the Lagrange Multiplier method? That is
why we have the condition ∇f = λ∇g ?
• After solving the equations, how can we tell if the critical points are max or min
points? For example,
f (1,1) = 5, f (2,2) = 0, f (−1,−1) = −10
• How can we use the value of 𝜆 to estimate a new optimal value with a change of the
constraint constant?
II) Taylor Theorem and Approximation
a) 1st order, 2nd order, and higher order Taylor Polynomials.
L(x, y) = f (a,b) + fx (a,b)(x − a) + fy (a,b)(y − b)
Q(x, y) = f (a,b) + fx (a,b)(x − a) + fy (a,b)(y − b)
(
1
fxx (a,b)(x − a)2 + 2 fxy (a,b)(x − a)(y − b) + fyy (a,b)(y − b)2
2
b) Approximation with Taylor Polynomial.
Approximate f (1.31,−0.91) using L(x,y) or Q(x,y) at (1, -1)
+
)
III Integration of scalar function.
i) Try the review exercises 3 to practice integration
ii) Double integral and Fubini Theorems.
• Understand the geometrical idea behind the Fubini Theorem.
• You should be very familiar with the dydx, dxdy or r drd θ region.
Please don’t write
Please don’t write
1
1
0
0
π
1
0
0
∫ ∫
∫ ∫
f (x, y) dy dx unless the region is really a square [0,1] × [0,1] .
f (r cosθ ,r sin θ ) rdrdθ unless you are sure the boundary is a
circular arc of radius 1 with center at the origin.
p.2
Please don’t write
π /4
1
cosθ
0
0
∫ ∫
x rdrdθ ,
iii) Convert between dydx, dxdy, and drd θ . Don’t forget to draw the picture first. You cannot
just make the changes by looking at the limits of the integral.
Note: The left boundary curve (for dxdy) does not necessary correspond to the lower
boundary curve (for dydx).
iv) Triple integral. Understand which shadow to use (project along z axis or x axis, or y-axis) .
For example, some regions are y-simple but not z-simple, so you have to make sure that
you have the correct description. In some cases, you may be asked to describe the same
solid multiple times by using different projections.
v) Cylindrical and Spherical coordinates -- rdrdθ dz and ρ 2 sin φ dρ dφ dθ , don't forget the
scaling factor.
• Cylindrical coordinates = polar coordinates + z axis.
• Spherical coordinates – Convert all equations of x, y, z into 𝜌, 𝜙, 𝜃, then draw a halfcross section , ρ = ? to ρ = ? , φ measured from the positive z axis, then rotate the
horizontal angle θ .
• Remember that 𝑥 ! + 𝑦 ! + 𝑧 ! = 𝜌! and 𝑥 ! + 𝑦 ! = 𝜌! sin! 𝜙.
• Usually, the vertical angle φ is obtained from the formula of a cone, eg 𝜙 = tan"# 1√24
or by finding the intersection point of two boundary curves, eg solve for 𝜙 from 𝜌 =2 and
𝜌=
#
$%& '
.