Q: Which of the following statements about the dot product of two vectors $$\vec{a}$$ and $$\vec{b} is $$true?

$$\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta$$

Q: Which of the following is a necessary condition for a series $$\sum a_n to $$converge?

$$\lim\limits_{n \to \infty} a_n = 0$$

Q: A function $$f(x, y) is $$continuous at $$(a, b) if$$:

$$\lim\limits_{(x, y) \to (a, b)} f(x, y) = f(a, b)$$

Q: A real-world application: The temperature $$T at a $$point $$(x, y) on a $$metal plate is given by $$T(x, y) = 100 - x^2$ - y^2$. In which direction from the point (2$$, 1)$$ does the temperature increase most rapidly?

In the direction of the gradient $$\nabla T at$$ $$(2, 1)$$

Question 1

Which of the following series converges?

Accepted Answer

$$\sum\limits_{n=1}^{\infty} \frac{1}{n^2$}$$

Question 2

A medical researcher is modeling the concentration of a drug in the bloodstream over time using the function $$C(t) = Ce$$^{-kt}$$. What is the rate of change of the drug concentration at time t$?

Accepted Answer

$$-kCe^{-kt}$$

Question 3

Which of the following is the correct formula for the Taylor series of $$f(x)$$ centered at $$x=a$$?

Accepted Answer

$$\sum\limits_{n=0}^{\infty} \frac{f$$^{(n)}$$(a)}{n!}(x-a)^n$$

Question 4

A projectile is launched with initial velocity $$\vec{v}_0 = \langle 3, 4 \rangle m/s. $$What is the magnitude of the initial velocity?

Accepted Answer

$$5 m/s$$

Question 5

Which of the following tests can be used to determine the convergence of the series $$\sum\limits_{n=1}^{\infty} \frac{(-1$$)^{n}$$}{n}$$?

Accepted Answer

Alternating Series Test

Question 6

Given the vector-valued function $$\vec{r}(t) = \langle t^2$, \sin t \rangle$$, what is $$\vec{r}'(t)$$?

Accepted Answer

$$\langle 2t, \cos t \rangle$$

Question 7

A chemical reaction's rate is modeled by $$R(x, y) = x^2$y + y^3$. What is the partial derivative $$\frac{\partial R}{\partial y}$$?

Accepted Answer

$$x^2$$ + $$3y^2$$

Question 8

Which of the following statements about the dot product of two vectors $$\vec{a}$$ and $$\vec{b} is $$true?

Accepted Answer

$$\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos 	heta$$

Question 9

Suppose $$f(x, y) = x^2$y + y^2$. What is the value of f_x(1$$, 2)$$?

Accepted Answer

$$4$$

Question 10

A population of bacteria grows according to $$P(t) = P_0e$$^{kt}$$. If the population doubles in 3 hours, what is the value of k$?

Accepted Answer

$$\frac{\ln 2}{3}$$

Question 11

Which of the following is the correct formula for the arc length of a curve $$y = f(x)$$ from $$x = a to$$ $$x = b$$?

Accepted Answer

$$\int_a^b \sqrt{1 + (f'(x))^2$} \, dx$$

Question 12

A curve is given in parametric form by $$x = t^2$, y$$ = $$t^3. $$What is $$\frac{dy}{dx} at$$ $$t = 2$$?

Accepted Answer

$$3$$

Question 13

Which of the following is a necessary condition for a series $$\sum a_n to $$converge?

Accepted Answer

$$\lim\limits_{n 	o \infty} a_n = 0$$

Question 14

A vector-valued function $$\vec{r}(t)$$ describes the position of a particle in space. What does $$|\vec{r}'(t)|$$ represent?

Accepted Answer

The speed of the particle at time $$t$$

Question 15

Which of the following is the correct formula for the cross product of $$\vec{a} = \langle a_1$, a_2$, a_3$ \rangle$$ and $$\vec{b} = \langle b_1$, b_2$, b_3$ \rangle$$?

Accepted Answer

$$\langle a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1 \rangle$$

Question 16

A function $$f(x, y) is $$continuous at $$(a, b) if$$:

Accepted Answer

$$\lim\limits_{(x, y) 	o (a, b)} f(x, y) = f(a, b)$$

Question 17

Which of the following is the correct statement of the Chain Rule for $$z = f(x, y)$$, where $$x = g(t)$$ and $$y = h(t)$$?

Accepted Answer

$$\frac{dz}{dt} = \frac{\partial f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\partial y} \frac{dy}{dt}$$

Question 18

A curve is given in polar coordinates by $$r = 2\cos 	heta. $$What is the value of $$r$$ when $$	heta = \frac{\pi}{3}$$?

Accepted Answer

$$1$$

Question 19

A real-world application: The temperature $$T at a $$point $$(x, y) on a $$metal plate is given by $$T(x, y) = 100 - x^2$ - y^2$. In which direction from the point (2$$, 1)$$ does the temperature increase most rapidly?

Accepted Answer

In the direction of the gradient $$
abla T at$$ $$(2, 1)$$

Calculus III Syllabus and Study Guide: Infinite Sequences, Series, and Vector Calculus

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