A function of three variables is given by,
f (x,y,t) = x3y2sin t + 4x2t + 5yt2 + 4xycos t
Find ft (3.44,0.58,8.1) giving your answer to 3 decimal places.
To find the partial derivative of a function with respect to one of its variables, follow these steps:
Step 1: Differentiate the function term by term, treating the variable of interest as the main variable and keeping the other variables constant.
Step 2: Simplify the resulting expression.
In this case, we want to find the partial derivative of f(x, y, t) with respect to t, denoted as ft(x, y, t).
Starting with the given function:
f(x, y, t) = x^3 * y^2 * sin(t) + 4 * x^2 * t + 5 * y * t^2 + 4 * x * y * cos(t)
Step 1: Differentiate each term with respect to t.
To differentiate x^3 * y^2 * sin(t) with respect to t, we treat x, y, and t as constants:
∂/∂t (x^3 * y^2 * sin(t)) = x^3 * y^2 * cos(t)
To differentiate 4 * x^2 * t with respect to t, we treat x as a constant:
∂/∂t (4 * x^2 * t) = 4 * x^2
To differentiate 5 * y * t^2 with respect to t, we treat y as a constant:
∂/∂t (5 * y * t^2) = 10 * y * t
To differentiate 4 * x * y * cos(t) with respect to t, we treat x, y, and t as constants:
∂/∂t (4 * x * y * cos(t)) = -4 * x * y * sin(t)
Step 2: Simplify the resulting expression.
Adding up the differentiated terms gives us:
ft(x, y, t) = x^3 * y^2 * cos(t) + 4 * x^2 - 4 * x * y * sin(t) + 10 * y * t
Now, substituting the provided values into the function ft(x, y, t), we have:
ft(3.44, 0.58, 8.1) = (3.44^3) * (0.58^2) * cos(8.1) + 4 * (3.44^2) - 4 * 3.44 * 0.58 * sin(8.1) + 10 * 0.58 * 8.1
Evaluating this expression will give us the final answer, rounded to 3 decimal places.