MATH 200 Lecture Notes - Lecture 11: Tangent Space, Farad, Quotient Rule
Math 200 Problem Set IV
1) Show that the function z(x, y) = x+y
x−yobeys
x∂z
∂x +y∂z
∂y = 0
2) Let fbe any differentiable function of one variable. Define z(x, y) = f(x2+y2). Is
the equation
y∂z
∂x −x∂z
∂y = 0
necessarily satisfied?
3) Four positive numbers, each less than 50, are rounded to the first decimal place and
then multiplied together. Estimate the maximum possible error in the computed
product.
4) The total resistance Rof three resistors, R1,R2,R3, connected in parallel is determined
by
1
R=1
R1+1
R2+1
R3
If the resistances, measured in Ohms, are R1= 25Ω, R2= 40Ω and R3= 50Ω, with
a possible error of 0.5% in each case, estimate the maximum error in the calculated
value of R.
5) The specific gravity Sof an object is given by S=A
A−Wwhere Ais the weight of
the object in air and Wis the weight of the object in water. If A= 20 ±.01 and
W= 12 ±.02 find the approximate percentage error in calculating Sfrom the given
measurements.
6) A rectangular beam that is supported at its two ends and is subjected to a uniform
load sags by an amount
S=Cp`4
wh3
where p=load, `=length, h=height, w=width and Cis a constant. Suppose p≈100,
`≈4, w≈.1 and h≈.2. Will the sag of the beam be more sensitive to changes in
the height of the beam or to changes in the width of the beam.
7) Find the equations of the tangent plane and normal line to the graph of the specified
function at the specified point.
a) f(x, y) = x2−y2at (−2,1) b) f(x, y) = exy at (2,0)
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8) Find a vector of length √3 which is tangent to the curve of intersection of z2=
4x2+ 9y2and 6x+ 3y+ 2z= 5 at (2,1,−5).
9) Find all horizontal planes that are tangent to the surface with equation
z=xye−(x2+y2)/2
What are the largest and smallest values of zon this surface?
10) Find the distance from the point (1,1,0) to the circular paraboloid with equation
z=x2+y2.
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MATH 200 Full Course Notes
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Document Summary
Math 200 problem set iv: show that the function z(x, y) = x+y x y obeys x z. Y = 0: let f be any di erentiable function of one variable. Y = 0: four positive numbers, each less than 50, are rounded to the rst decimal place and then multiplied together. Estimate the maximum possible error in the computed product: the total resistance r of three resistors, r1, r2, r3, connected in parallel is determined by. W = 12 . 02 nd the approximate percentage error in calculating s from the given measurements: a rectangular beam that is supported at its two ends and is subjected to a uniform load sags by an amount. 4, w . 1 and h . 2. 4x2 + 9y2 and 6x + 3y + 2z = 5 at (2, 1, 5): find all horizontal planes that are tangent to the surface with equation z = xye (x2+y2)/2.
