Question

For each of the following distributions,find μ = E( X ), E[ X(X - 1)], a σ2 = E[X(X -1)] + E(X) - μ²:

f(x)= 3! (1/4)x (3/4)3-x x = 0,1,2,3.
x!(3 - x)!

Answer 1

To find the slope of the tangent line to the parametric curve at t=π/3, we calculate the derivatives dx/dt and dy/dt, then plug in t=π/3 to compute the slope as dy/dx, which is -√3/4 as an exact value.

Step-by-step explanation:

To find the exact slope of the tangent line to the parametric curve {x=8cos(t), y=6sin(t)} at the point where t=π/3, we first calculate the derivatives of x and y with respect to t.

The derivative of x with respect to t is dx/dt = -8sin(t), and the derivative of y with respect to t is dy/dt = 6cos(t). The slope of the tangent line can be found using the formula dy/dx = (dy/dt) / (dx/dt).

At t=π/3, we plug in the value of t to get dx/dt = -8sin(π/3) and dy/dt = 6cos(π/3). Simplifying, dx/dt = -8(√3/2) = -4√3 and dy/dt = 6(1/2) = 3.

The slope of the tangent line is therefore dy/dx = 3 / (-4√3) = -√3/4. We give this answer as an exact value, without converting to decimal form.