Find the linearizations L(x, y, z) of the functions at the given points. f(x, y, z)=e^x+cos(y+z)

Find the linearizations L(x, y, z) of the functions at the...

Key Concepts

Multivariable Linearization

Linearization of a function of several variables is the process of approximating the function by its tangent hyperplane at a given point. This is achieved by using the function's value and its first-order partial derivatives at that point. The linearization provides a simple, linear function that approximates the behavior of the original function near the point of tangency.

Partial Derivatives

Partial derivatives measure the rate at which a function changes as one of its variables is varied, while the other variables are held constant. In the context of linearization, computing the partial derivatives with respect to each variable at the point of interest is crucial, as these derivatives form the coefficients of the linear terms in the tangent plane equation.

Gradient Vector

The gradient vector of a multivariable function is a vector composed of all its first-order partial derivatives. It points in the direction of the greatest rate of increase of the function. Within linearization, the gradient vector at the point of tangency provides the directional slopes and is used to construct the best linear approximation of the function near that point.

Chain Rule

The chain rule is a fundamental theorem used to differentiate composite functions. When the function includes compositions, such as a trigonometric function of a sum involving several variables, the chain rule helps to accurately compute the corresponding derivatives. This is especially important in determining the correct partial derivatives needed for the linearization process.

Evaluation at Specified Points

Evaluating both the function and its partial derivatives at specific points is an essential step in linearization. This process involves substituting the coordinates of the given point into the function and its derivatives to obtain numerical values that form the constant term and coefficients in the linearization formula. These evaluations yield the particular linear approximation tailored to the point of interest.

Transcript

00:03 We're given the function f of x, y, z equals e to the x plus the cosine of y plus z.

00:19 And we want to find the linearization of this function at several different points that we are given.

00:25 So let's find the formula for that linearization at our point.

00:30 And that formula is going to be the linearization equals the function, at the point p which i'm just going to write that like that for the moment and we will actually use the function when we evaluate these for the different parts plus the partial derivative of this with respect to each variable times that variable minus the points coordinate and when we do the partial derivative we're going to evaluate it at point p so with the partial derivative with respect to x we treat y and z as constants so this whole value cosine of y plus z is a constant.

01:14 So that partial derivative is e to the x, and we'll evaluate that at point p, times the quantity x minus x0.

01:26 And then we have to take the derivative of this with respect to y.

01:30 So derivative of e to the x will be zero, because there's no y's involved over there.

01:34 That's a constant.

01:36 So a derivative of cosine is negative sine of the same angle function, y plus z.

01:43 X times the derivative of the angle function which is just one and again that'll get evaluated at our point p times the quantity y minus the y coordinate of our point and then plus the derivative of that with respect to z which actually i'm going to change that because again derivative of cosine is negative sign so let's make that a minus sign of y plus z evaluated at our point p times the quantity z minus z sub 0.

02:26 So in part a, we are given the point 0, 0.

02:38 We need one more 0 there.

02:45 So there's our point p.

02:47 So we are going to plug that in for our x in our partial derivatives, and our y and our z in our partial derivatives, and for the x0.

03:00 Y0, and z0.

03:03 So f of p would be e to the zero power plus the cosine of 0 plus 0.

03:15 There's the f of p portion plus e to the zero power times x minus zero minus the sign of 0 plus 0 times the quantity y minus zero minus zero...