Calculus 3

Calculate the gradient vector for a given function $f(x, y)$ and describe its significance in the context of a 3D graph.

What direction should you travel to increase your height on a mountain as fast as possible?

What direction should you travel to keep your height constant (i.e. travel on a contour aka a level curve)?

Using a topographical map, analyze the contours to plan a route through the mountains with minimal elevation changes. Discuss the importance of this analysis in winter sports like skiing or snowshoeing.

Compute the gradient of the function $f(x, y) = x^2 \sin(y)$.

Compute the gradient of a multivariable function by finding its partial derivatives and forming a vector.

Find the gradient of the function $f(x, y) = 3x^2 + y + 6$ at the point $(1, -1)$.

Calculate the partial derivative of a function Z with respect to X, holding Y constant.

Calculate the partial derivative of a function Z with respect to Y, holding X constant.

Find the gradient of a scalar function $z = f(x, y) = x^2 + y^2$, and evaluate it at the points (2, 1) and (-1, -1).

Find the gradient of the function $f(x, y) = \sqrt{36 - x^2 - y^2}$ and evaluate it at the point $(-3, 2)$.

For the function $f(x, y) = x^2y - xy^2$ at the point $(1, -1)$, find the direction and rate of greatest increase, greatest decrease, and a direction of no change.

For the function $f(x, y) = x^2y - xy^2$ at the point (1, -1), find the direction and rate of greatest increase, greatest decrease, and a direction of no change.

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).

Linearize the multivariable function $f(x,y) = 1 + x \ln(xy - 5)$ at the point (2, 3).

Given the function $f(x, y) = 1 + x \cdot \ln(xy - 5)$, find the linearization of the function at the point $(2, 3)$.

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.

Explain and visualize different types of multivariable functions.