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"\\iint" y/x^2+1da where d = (x,y) 0<x<4 and 0<y< X
Find the greatest and smallest values that the function f(x,y) = 4xy takes on the ellipse x^2 + 2^y2 = 1 by using the Method of Lagrange Multipliers.
Evaluate the double integral (9 - x^2 - y^2)^(1/2) where
R = {(x,y) : x^2 + y^2 < or = to 9 , x > or = to 0 }
Use double integration to find the volume of the solid bounded by the paraboloid z= 9x^2 + y^2, below by the plane z = 0, and laterally by the planes x = 0, y = 0, x = 3, and y = 2.
y=sec^(−1) (x) ⇔ secy=x,f or y∈[0,π/2)∪[π,3π/2).
With this choice of principal branch the derivative of sec^(−1) (x) is
d/dx*sec^-1 (x)= 1/x*√(x^2 -1)
However, as mentioned in the side remark in Section 4.9, there is another common
definition of sec^−1(x), which is y=sec^(−1) (x), which is
y=sec^−1(x) ⇔ secy=x, for y∈[0,π/2)∪(π/2,π].
Show that, with this choice of principal branch to define the inverse secant function, its derivative becomes
d/dx*sec^(−1) (x)=1/|x|*√(x^2 -1)
Let x>0, and y be a positive function of x such that x^y=y^-x.
Find y′
Let f be a differentiable function that is positive everywhere and such that
f′(x)=−f(x)ln(f(x))for allx∈ℝ.
f′(x)=−f(x)ln(f(x))for allx∈R.
(a) Use logarithmic differentiation for the function g(x)=(f(x))^e^x to show that g′(x)=0 for all x∈ℝ
(b) Then, using the fact that any function whose derivative vanishes over some interval must be constant over that interval, deduce that f(x)=e^(e^-x)* ln C for some positive constant C
In the temperature range between 0C and 700C the resistance R [in ohms] of a certain platinum resistance thermometer is given by R = 10 + 0.04124T − 1.779 × (10^−5)T^ 2 where T is the temperature in degrees Celsius. Where in the interval from 0C to 700C is the resistance of the thermometer most sensitive and least sensitive to temperature changes? [Hint: Consider the size of dR/dT in the interval 0 ≤ T ≤ 700.
Find the mass and center of mass of the thin plate that occupies the triangle with vertices (0,0), (2,1), and (0,3) and has density function p(x,y) = x + y.
