Final answer:
The gradient of the function f(x,y)=8y−3x at the point (10,3) is (-3, 8). To sketch the gradient and the level curve passing through (10,3), plot the gradient as an arrow starting from (10,3) and the function f(x,y) = 8y−3x in the xy-plane.
Step-by-step explanation:
To find the gradient of the function f(x,y)=8y−3x, we need to find the partial derivatives with respect to x and y. The partial derivative with respect to x is -3, and the partial derivative with respect to y is 8. Therefore, the gradient is given by ∇f(x,y) = (-3, 8).
To sketch the gradient and the level curve that passes through the point (10,3), we can plot the gradient as an arrow starting from the point (10,3) in the direction of (-3, 8). The level curve passing through (10,3) can be obtained by plotting the function f(x,y) = 8y−3x in the xy-plane.