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Unformatted text preview: Study Guide for the 2nd Exam MTH241
Exam Date: 28th October 2011. In class
Exam Covers Sections 14.1 − 14.8 You must know the followings
1. Domain of the function.
2. Limit and Continuity
Prove or disprove the existance of limit. Do all the problems from the handout which I distributed
3. Partial Derivatives. Find fx , fy , fxx , fyy , fxy etc
4. Tangent plances and linearizations.
Equation of the tangent plane of the curve z = f (x, y ) at (a, b) is given by
fx (a, b)(x − a) + fy (a, b)(y − b) − (z − f (a, b)) = 0
Linearization of f (x, y ) = 0 around (a, b) is given by
fx (a, b)(x − a) + fy (a, b)(y − b) − (z − f (a, b)) = 0.
5. Chain Rule.
∂ f dx
∂ f dy
∂ x dt + ∂ y dt
z = f (x, y ), x = g (s, t), y = h(s, t) ⇒ dz = ∂ f ∂ x + ∂ f ∂ y
⇒ dz = ∂ f ∂ x + ∂ f ∂ y
Implicit Diﬀerentiation: F (x, y ) = 0, y = f (x) ⇒ dx = − Fx
Fy z = f (x, y ), x = g (t), y = h(t) ⇒ dz
dt = 6. Directional Derivatives, Gradiant, Maximum rate of change. Finding the equations of Tangent
plane and Normal lines.
7. Maximum and Minimum:
Suppose the second partial derivative of f are continuous on a disk with center (a.b) and suppose
that fx (a, b) = 0 and fy (a, b) = 0 i.e (a, b) are the co-ordiantes of the critical point. Let
D = D(a, b) = fxx (a, b)fyy (a, b) − [fxy (a, b)]2
a. If D > 0 and fxx (a, b) > 0 ⇒ f (a, b) is a Local Minimum Point
b. If D > 0 and fxx (a, b) < 0 ⇒ f (a, b) is a Local Maximum Point
c. If D < 0 then f (a, b) is not local max or min point. It is called Saddle Point
If D = 0 then no information about the point (a, b)
c. Absolute maxima and minima in a closed bounded set.
Find the values of f at the critical points of f in D.
Find the extreme values of f on the boundary of D.
The largest of the values from steps 1 and 2 is the absolute maximum value etc... 9. Lagrange Multipliers. Sample problems
These problems are not the exact problems but by solving these you can understand the hardness
of the original problems. Do the homework problems and the problems which I discussed
1. Find the limit 4 it exists or show that 2 limit doesn’t exist.
a. lim(x,y)→(0,0) x2 −y2 b. lim(x,y)→(0,0) x2x2sin 2y
2. Determine the set of points at which the following function is continuous.
f (x, y, z ) = √ y x2 −y 2 +z 2 3. Find the indicated partial derivative of the function ω = .
2y +z Find ∂3ω
∂ x∂ y ∂ z and ∂3ω
∂ x2 ∂ y . 4. Find the equation of the tangent plane at the point (2, −2, 12) to the surface z = 3(x − 1)2 +
2(y + 3)2 + z .
5. Find the linearization L(x, y ) of the function f (x, y ) = sin(2x + 3y ) at the point (−3, 2).
6. Use chain rule to ﬁnd ∂R
∂x and ∂R
∂y where R = ln(u2 + v 2 + w2 and u = x + 2y, v = 2x − y, w = 2xy .
7. Find the directional derivative of the function f (x, y ) = 1 + 2x y at (3, 4) along the vector
v = < 4, − 3 > .
8. Find the equation of the plane and the normal line to the surface y = x2 − z 2 at the point (4, 7, 3).
9. Find the local maximum and minimum values and the saddle point (if any) of the function
f (x, y ) = 2x3 + xy 2 + 5x2 + y 2 .
10. Use Lagrange multipliers to ﬁnd the maximum and minimum values of the function f (x, y, z ) =
2x + 6y + 10z subject to the given condition x2 + y 2 + z 2 = 35. Rules of Exam
1. Exam is closed book and closed notes.
2. You can carry a 4 × 6 Formula card which can be used to write only important formulae and
results. You’re not allowed to write any problem there. After the exam, you turn it back along with
your exam packet.
3. You’re not allowed to use any kind of Calculator.
4. You’re not allowed to go outside the exam hall without the permission of the proctor. ...
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- Fall '08