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Calculus
A fence must be built to enclose a rectangular area of 45,000ft^2. Fencing material cost $1 per foot for the two sides facing north and south and $2 per foot for the other two sides. Find the cost of the least expensive fence.
Calculus
A right triangular plate of base 3.0 m and height 1.5 m is submerged vertically in water, with top
vertex 3.5 m below the surface. Find the force on one side of the plate.
vertex 3.5 m below the surface. Find the force on one side of the plate.
Calculus
Find the force on one side of a cubical container 6.0 cm on an edge if the container is filled with
mercury. The weight density of mercury is 250 kN/m³.
mercury. The weight density of mercury is 250 kN/m³.
Calculus
HardyWeinberg equilibrium problem
if the three genotypes AA, Aa, and aa have respective frequencies pAA =θ^2, pAa = 2θ(1 − θ), and paa = (1 − θ)^2,
n! / (divide) n1! n2! n3! PAA^n1 PAa^n2 Paa^n3
where n = n1 + n2 + n3. This probablity depends on θ.
There is a method called the maximum likelihood method, that can be used to estimate θ. The principle
is simple: We find the value of θ that maximizes the probability of the observed data. Since the coefficient n! / (divide) n1! n2! n3!
does not depend on θ, we need only maximize
L(θ) = PAA^n1 PAa^n2 Paa^n3
(a) Show that if L(θ) is maximal for θ = ˆθ then ln(L(θ)) is maximal for θ = ˆθ.
(Note that L is strictly positive and twice differentiable.)
(b) Use the result in (a) to find the value ˆθ that maximizes L(θ)
if the three genotypes AA, Aa, and aa have respective frequencies pAA =θ^2, pAa = 2θ(1 − θ), and paa = (1 − θ)^2,
n! / (divide) n1! n2! n3! PAA^n1 PAa^n2 Paa^n3
where n = n1 + n2 + n3. This probablity depends on θ.
There is a method called the maximum likelihood method, that can be used to estimate θ. The principle
is simple: We find the value of θ that maximizes the probability of the observed data. Since the coefficient n! / (divide) n1! n2! n3!
does not depend on θ, we need only maximize
L(θ) = PAA^n1 PAa^n2 Paa^n3
(a) Show that if L(θ) is maximal for θ = ˆθ then ln(L(θ)) is maximal for θ = ˆθ.
(Note that L is strictly positive and twice differentiable.)
(b) Use the result in (a) to find the value ˆθ that maximizes L(θ)
Calculus
Which of the following statements are true? Give reasons for your answers, in the
form of a short proof or a counterexample.
i)
2
2
2
dx
dy
dx
d y
=
ii) The inverse function of 3x
y = e is ln x
3
1
y = .
iii) If f is increasing and 0 )x(f > on an interval I, then
)x(f
1
)x(g = is decreasing on I.
iv) An equation of the tangent line to the parabola 2
y = x at )4,2 (− is
y − 4 = 2 x(x + )2 .
v) If f is one-one onto and differentiable on R , then
f )6(
1
f( )6()'
1
′
=
−
.
form of a short proof or a counterexample.
i)
2
2
2
dx
dy
dx
d y
=
ii) The inverse function of 3x
y = e is ln x
3
1
y = .
iii) If f is increasing and 0 )x(f > on an interval I, then
)x(f
1
)x(g = is decreasing on I.
iv) An equation of the tangent line to the parabola 2
y = x at )4,2 (− is
y − 4 = 2 x(x + )2 .
v) If f is one-one onto and differentiable on R , then
f )6(
1
f( )6()'
1
′
=
−
.
Calculus
Trace the curve y=(x^2-1)^(1÷3) and state all the properties you use to trace it.
Calculus
integrate between limits 0 and d50〖1/(√2π *dp* ln GSD) exp[-〖{ln(dp )-ln(MMD)}〗^2/(2〖{ln(GSD)}〗^2 )]ddp 〗
Calculus
Find the dimensions of a right circular cylinder that is open on the top, is closed on the bottom, has volume V , and uses the least amount of material.
Calculus
Find first-order partial derivatives of F(x)= cosx²y²z²/Arc sinxy²
Calculus
F(x)= Arc cotx² + Arc sinx² . x²
