Calculus 3: Tangent planes and linear approximations

Find the linear approximation to this multivariable function at the point $(2, 3)$ using the tangent plane, and then use the linear approximation to estimate the value of the function at $(2.1, 2.99)$.

Find the equation of the tangent plane to the graph of the function f(x, y) = 2 - x^2 - y^2 at the point $\left(\frac{1}{2}, \frac{1}{2}\right)$.

Find the linearization $L(x, y, z)$ of a function of three variables at the point (2, 1, 0).

Estimate the temperature of a pizza at the point $(2.9, 4.2)$ where the temperature function is given by $f(x, y) = 5x^2 \sqrt{x^2 + y^2} - 100$ using the tangent plane at a nearby point.

Find the equation for the tangent plane to the function $Z = f(x, y)$ at a given point $(a, b)$.

Find the equation for the tangent plane to the implicit function $F(x, y, z) = 0$ at a point $(a, b, c)$.

Using the linear approximation, approximate the value of the function at a given point near $(a, b)$.

Create the equation of a tangent plane to a given surface at a given point.

Make a tangent plane to a surface at a specified point by calculating the partial derivatives with respect to x and y and then substituting into the tangent plane equation, similar to the previously discussed method.

Find the equation of a tangent plane to the circular paraboloid $z = x^2 + y^2$ at the point (-1, -1, 2).

Find the equation of the tangent plane to the function $f(x, y) = \sqrt{36 - x^2 - y^2}$ at the point $(2, 4, 4)$.

Find a plane tangent to the ellipsoid $\frac{x^2}{1} + \frac{y^2}{25} + \frac{z^2}{9} = 1$ at the point $(-3/5, 4, 0)$.

Find the equation of the tangent plane to the surface $z = 4x^2 - y^2 + 2y$ at $(-1, 2, 4)$.

Estimate the value of $z = \ln(x-2y)$ at the point (3.1, 0.9) using a tangent plane approximation.