Did you know?
Did you know that the equation f(x,y)=3xy^2-x-y represents a mathematical function?
In part (a) of the problem, we are asked to find the equation of the tangent plane to the graph of f(x,y) at the point (2,1,3). This involves using calculus-based techniques to determine the slope of the tangent plane at that particular point.
Moving on to part (b), linear approximation is used to estimate the value of f(2.1,0.9). Linear approximation involves approximating a function by its tangent line at a given point, allowing us to estimate the function's value nearby. In this case, we approximate f(2.1,0.9) using the tangent plane at (2,1,3).
Lastly, in part (c), we need to find the directional derivative of f at (2,1) in the direction of î+j. This directional derivative represents the rate at which the function changes along the direction specified, in this case, along the sum of the î and j unit vectors.
These three parts explore different mathematical concepts such as tangent planes, linear approximation, and directional derivatives.
