Final answer:
To find the gradient of a function at a given point, calculate the partial derivatives. Sketch the gradient as a vector in the plane. To sketch the level curve, find the equation of the curve by setting f(x,y) = k and plot it on a graph.
Step-by-step explanation:
To find the gradient of a function at a given point, we need to calculate the partial derivatives of the function with respect to each variable. The gradient of a function f(x,y) is given by the vector (∂f/∂x, ∂f/∂y). Once we have the gradient, we can sketch it as a vector in the plane.
To sketch the level curve that passes through the given point, we need to find the equation of the level curve. This equation will be of the form f(x,y) = k, where k is a constant. We can then plot this equation on a graph to visualize the level curve.
Let's say the given function is f(x,y) = x^2 + y^2 and the point is (1,2). To find the gradient, we calculate the partial derivatives: ∂f/∂x = 2x and ∂f/∂y = 2y. At the point (1,2), the gradient is (2,4). We can sketch this vector in the plane. To find the equation of the level curve passing through (1,2), we set f(x,y) = k and solve for x and y. In this case, the equation is x^2 + y^2 = 5, which is a circle centered at the origin with radius sqrt(5).