Q: A function $$f(x, y)$$ has continuous second partial derivatives and $$f_{xx}(1,2) = 4$$, $$f_{yy}(1,2) = 5$$, $$f_{xy}(1,2) = 2. $$What is the value of the discriminant $$D at$$ $$(1,2)$$ used in the second derivative test?

$$D = f_{xx} f_{yy} - (f_{xy})^2$ = 4 \times 5 - 2^2$ = 20 - 4 = 16$$

Q: Which of the following is the correct formula for the directional derivative of $$f(x, y) at$$ $$(x_0$, y_0$) in $$the direction of a unit vector $$\vec{u}$$?

$$D_{\vec{u}} f(x_0$, y_0$) = \nabla f(x_0$, y_0$) \cdot \vec{u}$$

Question 1

Which of the following best describes the geometric meaning of the dot product of two vectors $$\vec{a}$$ and $$\vec{b} in$$ $$\mathbb{R}^3$$?

Accepted Answer

It gives the product of the magnitudes of $$\vec{a}$$ and $$\vec{b}$$ and the cosine of the angle between them: $$\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos 	heta.$$

Question 2

Suppose a particle moves in space along the path $$\vec{r}(t) = \langle t, t^2$, t^3$ \rangle. $$What is the velocity vector at time $$t=2$$?

Accepted Answer

$$\langle 1, 4, 12 \rangle$$

Question 3

Let $$f(x, y) = x^2$ + y^2$. Which of the following points is a critical point of f$?

Accepted Answer

$$(0, 0)$$

Question 4

A company manufactures a product at a cost given by $$C(x, y) = 4x^2$ + y^2$ - 2xy + 3$$, where $$x$$ and $$y$$ are the quantities of two resources used. What is the rate of change of cost with respect to $$x$$ when $$x=1$$ and $$y=2$$?

Accepted Answer

$$\frac{\partial C}{\partial x} = 8x - 2y$$, so at $$(1,2) it is$$ $$8 - 4 = 4$$

Question 5

Which of the following integrals represents the area of the region bounded by $$y = x^2$ and y = 4$?

Accepted Answer

$$\int_{-2}^{2} (4 - x^2$) \, dx$$

Question 6

A weather balloon rises vertically such that its height at time $$t is $$given by $$h(t) = 3t^2$ + 2t + 1. $$What is the instantaneous rate of change of the balloon's height at $$t = 5$$ seconds?

Accepted Answer

$$h'(5) = 30 + 2 = 32$$

Question 7

Which of the following statements about the gradient vector $$
abla f(x, y) is $$TRUE?

Accepted Answer

It points in the direction of the steepest increase of $$f.$$

Question 8

Evaluate the double integral $$\iint_R (x + y) \, dA$$ over the rectangle $$R$$ defined by $$0 \leq x \leq 2$$, $$0 \leq y \leq 3.$$

Accepted Answer

$$15$$

Question 9

A function $$f(x, y)$$ has continuous second partial derivatives and $$f_{xx}(1,2) = 4$$, $$f_{yy}(1,2) = 5$$, $$f_{xy}(1,2) = 2. $$What is the value of the discriminant $$D at$$ $$(1,2)$$ used in the second derivative test?

Accepted Answer

$$D = f_{xx} f_{yy} - (f_{xy})^2$ = 4 	imes 5 - 2^2$ = 20 - 4 = 16$$

Question 10

Suppose $$f(x, y) = x^2$ + y^2$ - 4x - 6y + 13. $$What are the coordinates of the minimum point of $$f$$?

Accepted Answer

$$(2, 3)$$

Question 11

A chemical reaction's rate depends on temperature $$T$$ and pressure $$P$$ according to $$R(T, P) = T^2$ P. If $$temperature increases at $$2$$ units/sec and pressure at $$3$$ units/sec, what is the rate of change of $$R at$$ $$T=1$$, $$P=2$$?

Accepted Answer

$$\frac{dR}{dt} = 2TP \frac{dT}{dt} + T^2$ \frac{dP}{dt} = 2 	imes 1 	imes 2 	imes 2 + 1^2$ 	imes 3 = 8 + 3 = 11$$

Question 12

Which of the following is the equation of the tangent plane to the surface $$z = x^2$ + y^2$ at the point (1$$, 2, 5)$$?

Accepted Answer

$$z = 2(x-1) + 4(y-2) + 5$$

Question 13

A hospital monitors the temperature at a point $$(x, y) in a $$room, given by $$T(x, y) = 100 - x^2$ - y^2$. At which point is the temperature highest?

Accepted Answer

$$(0, 0)$$

Question 14

Which of the following is the correct setup for the double integral to find the volume under $$z = x + y$$ over the triangle with vertices $$(0,0)$$, $$(1,0)$$, and $$(1,1)$$?

Accepted Answer

$$\int_{x=0}^{1} \int_{y=0}^{x} (x + y) \, dy \, dx$$

Question 15

A vector-valued function $$\vec{r}(t)$$ describes the position of a drone in space. If $$\vec{r}(t) = \langle \cos t, \sin t, t \rangle$$, what is the speed of the drone at $$t = 0$$?

Accepted Answer

$$\sqrt{2}$$

Question 16

Which of the following statements about partial derivatives is correct?

Accepted Answer

The partial derivative $$\frac{\partial f}{\partial x}$$ measures the rate of change of $$f as$$ $$x$$ changes, keeping $$y$$ constant.

Question 17

A rectangular garden is to be built along a river, with no fence needed along the river. If the garden must have area $$120 m$^2$$ and only $$3$$ sides need fencing, what is the minimum amount of fencing required?

Accepted Answer

Let $$x be $$the length parallel to the river, $$y$$ the width. Area: $$xy=120. $$Fencing: $$F=2y+x. $$Minimize $$F$$ with constraint. The minimum is at $$x=24$$, $$y=5$$, so $$F=2	imes5+24=34 m.$$

Question 18

Which of the following is the correct formula for the directional derivative of $$f(x, y) at$$ $$(x_0$, y_0$) in $$the direction of a unit vector $$\vec{u}$$?

Accepted Answer

$$D_{\vec{u}} f(x_0$, y_0$) = 
abla f(x_0$, y_0$) \cdot \vec{u}$$

Math 20C: Multivariable Calculus Syllabus and Course Structure

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