Find an equation of the tangent plane to the given surface...

Key Concepts

Tangent Plane

The tangent plane is a flat surface that best approximates a curved surface at a given point. It shares the same first-order behavior as the surface at that point, meaning its slope in any direction matches that of the surface. This concept is key in multivariable calculus for understanding local linear approximation.

Partial Derivatives

Partial derivatives measure the rate at which a function changes with respect to one variable while holding the others constant. In the context of finding a tangent plane, they provide the slopes of the surface in the x and y directions, which are used directly in the tangent plane equation.

Evaluation at a Specific Point

Evaluating partial derivatives at the specific point of interest ensures the tangent plane accurately reflects the surface's behavior at that location. It involves substituting the coordinates of the point into the partial derivatives to determine the plane's precise slope.

Transcript

00:01 We'll find an equation of the tangent plane to the given surface, f of x of y, with a specified point, which we have negative 1 .110.

00:08 So we need to find the derivative of f with respect of x.

00:13 So we've got to use the product rule.

00:15 So we'll say f of x equals x, f prime of x equals 1, g of x equals sine of x plus y, g prime of x equals and for this one we'll have to use a chain rule so cosine of x plus y multiplied by the derivative of x plus y with respective x in this case is just one when you click that in f prime is one times g is sine of x plus y plus g plus g prime is cosine of x plus y plus g prime is cosine of x plus y plus g prime is cosine of x plus y times f which is just x now we're going to find the derivative with respect of y because x is considered a constant we can just pull that out and we don't have to do a product rule so um the derivative of sign is cosine of x plus y then we'll have to multiply it by the derivative of with respect of y of x plus y in this case it's just one so this is our our derivative with respect of y.

01:32 So now we can start plugging in to our red z minus zero z -z -s -0 equation.

01:40 So z z z -s of zero equals f with respect of x where we plug in x and y.

01:48 So sign of x plus y, x is negative one, y is positive one, plus cosine of x plus y, plus positive 1 multiplied by x which is negative 1 so this is all this term here we're going to multiply that by x minus x of 0 which is 1 so x plus 1 we're going to add f of uh with respect of y where we plug in negative 1 and 1 so x is negative 1 times cosine of negative 1 plus 1 and we're going to multiply i move that down here.

02:34 Y minus y of 1, which is 1...

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