Review tangent approximation
In this section, we are going to see how to use the partial derivatives. When we have a function of two variables, x and y, then we have, actually, two different derivatives: partial f, partial x, called f sub x. This is the derivative off with respect to x while keeping y constant. And we have the partial derivative f, also called † prime, where we vary y and keep x as a constant. There is a formula for approximating what happens to both x and y when you vary either one. Partoal derivatives f(x,y)–>δf/δf=f_x (vary x; y=const) δf/δy=f_y (vary y; x=const) The function f(x) tells us what happens when x a is changed by a small amount, delta X. The function f(y) tells us how f changes when y is changed by small amount, delta y. When both functions are combined with each other, the effects of changing both variables simultaneously will add up with each other because you can imagine that first x will be changed and then y will be changed. And then you will change your mind, or the other way around, it doesn’t really matter. So if we change x by a certain amount, delta x, and if we change y by the amount delta y, and let’s say that we have equals f of x, y, then that changes by an amount which is approximately f sub x times delta X plus sub y times delta y. So, the intuition here is that the two effects add up. If x changes by small amount and then y changes, then first changing x will modify f. How much does it modify f? The answer is that it depends on how quickly f changes with respect to x, which we call f sub x. and if we change then the rate of change of “f’ when we change y is its derivative with respect to y, which we call f sub y. So, we get this change in the value of f. Approximation formula If we change x–>x+Δx z=f(x,y) then Δz~~f_xΔx+f_y Δy And of course, that’s only an approximation formula. Actually, there would be higher order terms involving second and third derivatives, and so on. One way to justify the formula is to consider its effect on the tangent plane approximation. The formula allows us to calculate the tangent plane through a graph of the function f. [Graph] To find the partial derivative off with respect to x, we must examine the graph of f. We can see that if we slice the graph off by a plane that is parallel to the xz-plane, we will obtain a function g whose derivative is fx. If x changes, z changes. The slope of that relationship is the derivative with respect to x. If y changes, z changes. The slope of that relationship is the derivative with respect to y. Therefore in each case, we have a line. Those lines are tangent to the surface. So now if we have two lines tangent to the surface, well then together they determine for us the tangent plane to the surface. [Graph] So we know that the partial derivatives off with respect to x and y are slopes, and we can write down the equations of these lines in terms of xyz coordinates. For example, if a is equal to the coefficient of partial f with respect to x for a given point, then we have a line given by the following conditions. Suppose that the line z=0+a(x-x0) is drawn in green to represent the line on a plane that intersects the parallel slice of the x-z plane. Then z will vary with x at a rate that depends on a if y remains constant at yO. The line tangent to the graph of fat (x0, y0) has slope b. If we call this tangent line T(x,y), and if we call the other line S(x,y), then together they determine the plane containing f(x,y). The equation of a plane is given by the formula z equals z0 plus a times × minus x0 plus b times y minus yO. If you look at what happens when you hold y constant and vary x, you can recover the first line. Justify this formula: tangent plane z=f(x,y)? know: f_x,f_y are slopes of z tangent lines of δf/δx (x_0,y_0)=a–>L_1={Z=Z_0+a(x-x_0); y=y_0 δf/δy(x_0,y_0)=b–>L_2={Z=Z_0+b(y-y_0); x-x_0 Li, Lz are both tangent to the graph z-f(x,y). Together they determine the plane. Z=Z_0+a(X-x_o)+b(y-y_o). Approx. formula says: the graph off is close to its tangent plane. If the value of x is held constant and the value of y varies, the graph of f will approach its tangent plane. The approximation formula can be used to estimate how the value off changes if we change x and y at the same time.
