Question 1

Suppose $$z = f(x, y) is a $$differentiable function and you are given the point $$(a, b, f(a, b)). $$Which of the following is the correct equation for the tangent plane to the surface at this point?

Accepted Answer

$$z = f(a, b) + f_x(a, b)(x - a) + f_y(a, b)(y - b)$$

Question 2

Given $$f(x, y) = x^2$ + y^2$ and the point (1$$, 3)$$, what is the equation of the tangent plane to the surface z = f(x$$, y)$$ at this point?

Accepted Answer

$$z = 10 + 2(x - 1) + 6(y - 3)$$

Question 3

Which of the following best describes the geometric meaning of the tangent plane to $$z = f(x, y) at$$ $$(a, b, f(a, b))$$?

Accepted Answer

It is the plane that contains both tangent lines to the curves $$C_1$$ and $$C_2 at $$the point.

Question 4

The linear approximation $$L(x, y) of$$ $$f(x, y) at$$ $$(a, b) is $$given by which formula?

Accepted Answer

$$L(x, y) = f(a, b) + f_x(a, b)(x - a) + f_y(a, b)(y - b)$$

Question 5

Suppose $$f(x, y) = x^3$ + y^3$ - 2xy$$ and you want the linear approximation at $$(1, 2). $$What is $$L(x, y)$$?

Accepted Answer

$$L(x, y) = 5 - 1(x - 1) + 10(y - 2)$$

Question 6

If $$z = f(x, y) is $$differentiable at $$(a, b)$$, which of the following is true about the increment $$\Delta z$$ when $$x$$ changes from $$a to$$ $$a + \Delta x$$ and $$y$$ from $$b to$$ $$b + \Delta y$$?

Accepted Answer

$$\Delta z \approx f_x(a, b)\Delta x + f_y(a, b)\Delta y$$

Question 7

Which of the following is the correct expression for the differential $$dz of$$ $$z = f(x, y)$$?

Accepted Answer

$$dz = f_x(x, y)dx + f_y(x, y)dy$$

Question 8

A weather balloon's altitude is modeled by $$z = f(x, y)$$, where $$x$$ and $$y$$ are wind velocities in two perpendicular directions. If at $$(x_0$, y_0$)$$, $$f_x(x_0$, y_0$) = 2$$ and $$f_y(x_0$, y_0$) = -3$$, what is the best linear approximation for a small change $$dx = 0.1$$, $$dy = -0.2$$?

Accepted Answer

$$dz = 2(0.1) + (-3)(-0.2) = 0.2 + 0.6 = 0.8$$

Question 9

Which of the following statements is true about the linear approximation of a differentiable function $$f(x, y) at$$ $$(a, b)$$?

Accepted Answer

It provides the best linear estimate of $$f(x, y)$$ near $$(a, b).$$

Question 10

Suppose $$f(x, y) is $$differentiable and $$f_x(a, b) = 0$$, $$f_y(a, b) = 0. $$What does the tangent plane to $$z = f(x, y) at$$ $$(a, b)$$ look like?

Accepted Answer

It is horizontal and parallel to the $$xy-$$plane.

Question 11

A medical researcher models the temperature $$T on a $$skin surface as $$T(x, y)$$, where $$x$$ and $$y$$ are spatial coordinates. If $$T_x(2, 3) = 4$$ and $$T_y(2, 3) = -2$$, what is the approximate change in temperature if a probe moves from $$(2, 3) to$$ $$(2.01, 2.98)$$?

Accepted Answer

$$dT = 4(0.01) + (-2)(-0.02) = 0.04 + 0.04 = 0.08$$

Question 12

Which of the following is NOT a requirement for the existence of a tangent plane to $$z = f(x, y) at$$ $$(a, b)$$?

Accepted Answer

$$f$$ must be linear in $$x$$ and $$y.$$

Question 13

If $$f(x, y) = e$$^{xy}$$, what is the linear approximation at (0$$, 0)$$?

Accepted Answer

$$L(x, y) = 1$$

Question 14

A chemical reaction rate is modeled by $$R(x, y) = x^2$y + y^3$. If x$ increases by 0.05$ and y$ decreases by 0.02$ from the point (1$$, 2)$$, what is the approximate change in R$?

Accepted Answer

$$dR = 4(0.05) + 12(-0.02) = 0.2 - 0.24 = -0.04$$

Question 15

Which of the following best describes the role of differentials $$dx$$ and $$dy in $$the context of $$z = f(x, y)$$?

Accepted Answer

They represent independent changes in $$x$$ and $$y$$ used to approximate the change in $$z.$$

Question 16

Suppose $$f(x, y) is $$differentiable and $$f_x(a, b) = 5$$, $$f_y(a, b) = -2. If $$you move from $$(a, b) to$$ $$(a + 0.1, b + 0.2)$$, what is the approximate change in $$f$$?

Accepted Answer

$$df = 5(0.1) + (-2)(0.2) = 0.5 - 0.4 = 0.1$$

Question 17

Which of the following is the best reason to use a linear approximation for a function $$f(x, y)$$?

Accepted Answer

To estimate values of $$f(x, y)$$ near a known point using only first derivatives.

Section 4.4 - Tangent Planes and Linear Approximations in Multivariable Calculus

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