Your inclass final exam will consist entirely of problems similar to the following, with numbers/functions changed but worded the same way. It is assumed you will thoroughly master all the problems on this handout and errors in understanding on your inclass final will earn less part credit than on the previous exams. MATH253  FormulasLinearization: L(x, y) = f_{ x} (x_{ 0} , y_{ 0}) (x – x_{ 0}) + f_{ y} (x_{ 0} , y_{ 0}) (y – y_{ 0}) + z_{ 0}Total Differential: df = f_{ x} (x_{ 0} , y_{ 0}) dx + f_{ y} (x_{ 0} , y_{ 0}) dy Chain Rule for Functions of Two Independent Variables, w = f(x,y): Tangent Line to a Level Curve: f_{ x} (x_{ 0} , y_{ 0}) (x – x_{ 0}) + f_{ y} (x_{ 0} , y_{ 0}) (y – y_{ 0}) = 0 Tangent Plane (explicit): f_{ x} (x_{ 0} , y_{ 0}) (x – x_{ 0}) + f_{ y} (x_{ 0} , y_{ 0}) (y – y_{ 0}) – (z – z_{ 0}) = 0 Tangent Plane (implicit): f_{ x} (P_{ 0}) (x – x_{ 0}) + f_{ y} (P_{ 0}) (y – y_{ 0}) + f_{ z} (P_{ 0}) (z – z_{ 0}) = 0 Normal Line: x = f_{ x} (x_{ 0} , y_{ 0}) t + x_{ 0} , y = f_{ y} (x_{ 0} , y_{ 0}) t + y_{ 0} , z =  t + z_{ 0} D_{ u} f = Ñf • u Error in Standard Linear Approximation:  E(x, y)  < ½ M(  x – x_{ 0}  +  y – y_{ 0}  )^{ 2} where M is any upper bound for  f_{ xx}  ,  f_{ yy}  ,  f_{ xy}  on R. dy/dx =  F_{ x }/ F_{ y}
Length of a curve: Center of Mass: 
If A = áa_{ 1}, a_{ 2}, a_{ 3}ñ , B = áb_{ 1}, b_{ 2}, b_{ 3}ñ then A·B = ABcos θ = a_{ 1}b_{ 1} + a_{ 2}b_{ 2} + a_{ 3}b_{ 3} = unit vector in the direction of A Direction cosines: cos α = , cos β = , cosγ = distance between a point (x_{ 0}, y_{ 0}) and a line ax + by + c = 0 Work =  F cos θ  A × B = A × B = area of the parallelogram determined by A and B. A·(B × C) = volume of a parallelpiped Parametric equations of a line: x = at + x_{ 0}, y = bt + y_{ 0} , z = ct + z_{ 0} Equation of a plane: ax + by + cz = d Green’s Theorem: Stoke’s Theorem: Divergence Theorem:

Write a triple integral which represents the volume of the first octant solid bounded by the coordinate planes and the graphs of  
An equation of the surface of a mountain is
 
Ike, the inchworm, travels along
 
Ike’s cousin, Izzy, is moving in space along the path given by
,
 
Let C be the curve with equations x = 2 – t^{ 3} , y = 2t – 1, z = ln t.
 
A particle starts at the origin with initial velocity  
Suppose z = f(x, y), where x = g(s, t), y = h(s, t), g(1, 2) = 3, 
Let
 
A painting contractor charges $4 per square meter for painting the four walls and ceiling of a room. The dimensions of the ceiling are measured to be 4 m and 5 m, the height of the room is measured to be 3 m, and these measurements are correct to 2.0 cm. Use differentials to approximate the greatest error in estimating the cost of the job from these measurements.  
Use polar coordinates to combine the sum below into one integral. Then evaluate this integral.  
It is known that a certain curve f(x, y) = c, in two dimensions has slope:
. It is also known that (1,0) satisfies  
Find the most general M(x,y) for which M(x,y) dx + (2ye^{ x} + 4x) dy will be an exact differential.  
Evaluate , showing all steps.  
Evaluate each line integral on the specified path.
 
This sum of two double integrals may be written as one double integral. What is this one double integral?  
Find every point on the given surface at which the tangent line is horizontal. z = 3x^{ 2} + 12x + 4y^{ 3}  6y^{ 2} + 5 
A surveyor wants to find the area in acres of a certain field (1 acre is 43560 ft^{ 2}). She measures two different sides, finding them to be a = 500 feet and b = 700 feet, with a possible error of as much as one foot in each measurement. She finds the angle between the two sides to be 30° with a possible error of as much as .25° . The field is triangular, so its area is given by A = ½ ab sin θ. Use differentials to find the maximum resulting error, in acres, in computing the area of this field using this formula.  
Evaluate the given integral by first converting it to cylindrical coordinates.  
Let  
Suppose that f(x,y) = e^{ x – y} and f (ln 2, ln 2) = 1. Use the technique of linear approximation to estimate  
Suppose u = á 1, 0 ñ, v =
, D_{u}f(a,b) = 3 and D_{v}f(a,b) =
.
 
Consider the two surfaces ρ = 3 csc φ and r = 3. Are they the same surface, or are they different surfaces? Explain your answer.  
Evaluate
, where R consists of the set of points such that  
Consider the triple integral
representing a solid S. Let R be the projection of S onto the plane z = 0.
 
Compute the work done by the vector field  
Consider the vector field F(x,y,z) = 2xi + 2yj + 2zk. If C is any path from (0, 0, 0) to  
Find the volume of the region bounded above by the sphere 
Express the vector  
Evaluate along the straight line joining these points. Use any method that works.  
Find the x coordinate of the center of mass of the triangular lamina with vertices  
Suppose it is known that
along the line segment joining  
Evaluate where C is one counterclockwise trip around the circle x^{ 2} + y^{ 2} = 4.  
The gravitational constant G is often measured at different points on the surface of the earth with a simple pendulum and the formula (L in feet, T in seconds). Suppose it is known that the length L of a pendulum = 2 feet and that one period T = 2 seconds. A technician is able to measure L with error of ± one inch and T with an error of ± one tenth of a second. Approximate the maximum percentage error in the measurement of G which might be caused by the technician’s errors. Use differentials.  
The point (2, 3, 6) lies on the surface
. Compute the numerical value of at (2, 3, 6).  
Find all the local maxima, local minima, and the saddle points of the function .  
Find unit tangent and unit normal vectors to the curve  
A particle moves through 3space in such a way that its velocity is  
Let F (x, y, z) = (z + y^{ 2})i + 2xyj + (x + y)k
 
For what value of the constant b is the vector field F = bxy^{ 2}i + x^{ 2}yj irrotational? 
Find the area of the portion of the surface x^{2} – 2z = 0 that lies above the triangle bounded by the lines  
Let S be the cylinder x^{2} + y^{2} = a^{2} , 0 ≤ z ≤ h, together with its top,  
Use the Divergence theorem to find the outward flux of the field 
a) á30, 150ñ b) ascending c)  
a) 15/4 cm b) 17/4 cm/min  
a) 2π cm b) max acc = 2 cm/min^{2} when t = 0 or π min  
a) yes (15/8, 0,  ln 2) b) x = 3t + 1, y = 2t + 1, z = t  
r =  
47  
(a) (b)  
$3.12  
sin xy – xe^{2y} – y^{2} = 1  
M(x, y) = y^{2}e^{x} + 4y + G(x)  
245.925  
a) 1.9 b)  4 c) 0  
(2, 0, 7) and (2, 1, 9) 
.022 acres  
= (π/2)×(ln 17)  
r ^ v when t = 0  
1.06  
a) á 3, 1ñ b) , max value = Ñf(a, b) c)  
same surface, cylinder of radius 3  
(sin 4 – sin 1)π ≈ 5.02  
a) b)  
0  
Hint: let r = áa_{1}t, a_{2}t, a_{3}tñ, 0 < t < 1 (the straight line path)  
5π/12  
5  
1/3  
6  
16π  
14.17%  
9  
saddle (0, 0); local max (2, 2) 
(3, 4, 7)  
a) i b) 2  
1  
7/3  
2πa^{2}  
8/3 