Two lines are perpendicular if they are para el and cross
1 answer:
You might be interested in
Answer:
The Line integral is π/2.
Step-by-step explanation:
We have to find a funtion f such that its gradient is (ycos(xy), x(cos(xy)-ze^(yz), -ye^(yz)). In other words:
we can find the value of f using integration over each separate, variable. For example, if we integrate ycos(x,y) over the x variable (assuming y and z as constants), we should obtain any function like f plus a function h(y,z). We will use the substitution method. We call u(x) = xy. The derivate of u (in respect to x) is y, hence
(Remember that c is treated like a constant just for the x-variable).
This means that f(x,y,z) = sen(x,y)+C(y,z). The derivate of f respect to the y-variable is xcos(xy) + d/dy (C(y,z)) = xcos(x,y) - ye^{yz}. Then, the derivate of C respect to y is -ze^{yz}. To obtain C, we can integrate that expression over the y-variable using again the substitution method, this time calling u(y) = yz, and du = zdy.
Where, again, the constant of integration depends on Z.
As a result,
if we derivate f over z, we obtain
That should be equal to -ye^(yz), hence the derivate of K(z) is 0 and, as a consecuence, K can be any constant. We can take K = 0. We obtain, therefore, that f(x,y,z) = cos(xy) - e^(yz)
The endpoints of the curve are r(0) = (0,0,1) and r(1) = (1,π/2,0). FOr the Fundamental Theorem for Line integrals, the integral of the gradient of f over C is f(c(1)) - f(c(0)) = f((0,0,1)) - f((1,π/2,0)) = (cos(0)-0e^(0))-(cos(π/2)-π/2e⁰) = 0-(-π/2) = π/2.
Answer:
Yes we can conclude.
Step-by-step explanation:
The sampling distribution of can be approximated as a Normal Distribution only if:
np and nq are both equal to or greater than 10. i.e.
- np ≥ 10
- nq ≥ 10
Both of these conditions must be met in order to approximate the sampling distribution of as Normal Distribution.
From the given data:
n = 50
p = 0.80
q = 1 - p = 1 - 0.80 = 0.20
np = 50(0.80) = 40
nq = 50(0.20) = 10
This means the conditions that np and nq must be equal to or greater than 10 is being satisfied. So, we can conclude that the sampling distribution of pˆ is approximately a normal distribution