Q: For $$f(x, y, z) = x^2$ + y^2$ + z^2$ at P(1$$, 2, 2)$$, in which direction does f$ increase most rapidly?

In the direction of the gradient $$\nabla f$$

Question 1

Given $$f(x, y, z) = \sqrt{49 - x^2$ - y^2$ - z^2$}$$, which of the following best describes the domain of $$f$$?

Accepted Answer

$$x^2$$ + $$y^2$$ + $$z^2$$ \leq 49$$

Question 2

For $$f(x, y, z) = \sqrt{49 - x^2$ - y^2$ - z^2$}$$, what is the equation of the level surface for $$f(x, y, z) = 1$$?

Accepted Answer

$$x^2$$ + $$y^2$$ + $$z^2 = 48$

Question 3

Given $$g(x, y) = \sqrt{x^2$ + 3y^2$}$$, what is the equation of the level curve passing through $$(2, 1)$$?

Accepted Answer

$$x^2$$ + $$3y^2 = 7$

Question 4

For $$g(x, y) = \sqrt{x^2$ + 3y^2$}$$, what is the equation of the tangent plane to the level surface at $$(2, 1)$$?

Accepted Answer

$$2(x - 2) + 6(y - 1) = 0$$

Question 5

Given $$w = x^2$ e$$^{y^2}$$, what is $$\frac{\partial w}{\partial x}$$?

Accepted Answer

$$2x e$$^{y^2}$$

Question 6

Given $$w = x^2$ e$$^{y^2}$$, what is $$\frac{\partial w}{\partial y}$$?

Accepted Answer

$$2x^2 y$$ $$e^{y^2}$$

Question 7

Let $$z = e$$^{xy}$$, x$$ = \cos t$$, y$$ = \sin t$$. What is $$\frac{dz}{dt}$$ at t$$ = \frac{\pi}{2}$$?

Accepted Answer

$$0$$

Question 8

Given $$z = e$$^{xy}$$, x$$ = \cos t$$, y$$ = \sin t$$, which of the following correctly expresses $$\frac{dz}{dt}$$?

Accepted Answer

$$e^{xy}(y$$ \frac{dx}{dt} + x \frac{dy}{dt})$$

Question 9

Suppose $$w = f(x, y, z)$$ where $$x = g(s, t)$$, $$y = h(s, t)$$, $$z = k(s, t). $$Which formula gives $$\frac{\partial w}{\partial s}$$?

Accepted Answer

$$\frac{\partial w}{\partial x} \frac{\partial x}{\partial s} + \frac{\partial w}{\partial y} \frac{\partial y}{\partial s} + \frac{\partial w}{\partial z} \frac{\partial z}{\partial s}$$

Question 10

Given $$f(x, y, z) = x^2$ + y^2$ + z^2$ and point P(1$$, 2, 2)$$, what is the gradient $$
abla f$$ at P$?

Accepted Answer

$$(2, 4, 4)$$

Question 11

For $$f(x, y, z) = x^2$ + y^2$ + z^2$ at P(1$$, 2, 2)$$, what is the directional derivative in the direction of $$\vec{v} = (2, 2, 1)$$?

Accepted Answer

$$\frac{18}{3} = 6$$

Question 12

For $$f(x, y, z) = x^2$ + y^2$ + z^2$ at P(1$$, 2, 2)$$, in which direction does f$ increase most rapidly?

Accepted Answer

In the direction of the gradient $$
abla f$$

Question 13

Suppose $$f(x, y, z) = x^2$ + y^2$ + z^2$. What is the equation of the tangent plane to the surface x^2$ + y^2$ + z^2$ = 9 at $$the point $$(2, 1, 2)$$?

Accepted Answer

$$2(x - 2) + 1(y - 1) + 2(z - 2) = 0$$

Question 14

If $$f(x, y, z) = x^2$ + y^2$ + z^2$, what is the value of f(2$$, 1, 2)$$?

Accepted Answer

$$9$$

Guided Study: Calculus III – Tangent Planes, Level Surfaces, and Partial Derivatives

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