$\lim_{x \to 0} \frac{\sin x}{x}$

$\lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$

$1 \times 4 + 2 \times 0 + 3 \times (-1) = 1$

If $\lim_{x \to a} f(x)$ exists, then $f(x)$ approaches a unique value as $x$ approaches $a$.

Question 1

Given the function $f(x) = x^3 - 6x^2 + 9x + 2$, find the local minimum value of $f(x)$. What is the $x$-coordinate where it occurs?

Accepted Answer

$x = 2$

Question 2

Which of the following is the correct Taylor series expansion for $e^x$ centered at $x=0$?

Accepted Answer

$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$

Question 3

A chemical reaction rate is modeled by the function $R(t) = 5e^{-0.2t}$. What is the instantaneous rate of change of $R$ at $t = 0$?

Accepted Answer

$-1$

Question 4

Which of the following limits evaluates to $1$?

Accepted Answer

$\lim_{x 	o 0} \frac{\sin x}{x}$

Question 5

Given the vector $\vec{v} = (2, -1, 3)$, what is the magnitude $|\vec{v}|$?

Accepted Answer

$\sqrt{14}$

Question 6

If $f(x) = \ln(x^2 + 1)$, what is $f'(x)$?

Accepted Answer

$\frac{2x}{x^2 + 1}$

Question 7

A population of bacteria grows according to $P(t) = 100e^{0.05t}$. After how many hours will the population double?

Accepted Answer

$t = \frac{\ln 2}{0.05}$

Question 8

Which of the following is the correct definition of the derivative of a function $f(x)$ at $x = a$?

Accepted Answer

$\lim_{h 	o 0} \frac{f(a+h) - f(a)}{h}$

Question 9

If $A = \begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix}$, what is $\det(A)$?

Accepted Answer

$-2$

Question 10

A projectile is launched with height $h(t) = -4.9t^2 + 20t + 1.5$. At what time does it reach its maximum height?

Accepted Answer

$t = 2.04$

Question 11

Which of the following is the correct antiderivative of $f(x) = 3x^2$?

Accepted Answer

$x^3 + C$

Question 12

If $\lim_{x 	o 2} f(x) = 5$ and $\lim_{x 	o 2} g(x) = -3$, what is $\lim_{x 	o 2} [2f(x) - 3g(x)]$?

Accepted Answer

$19$

Question 13

Which of the following is the correct formula for the sum of a geometric series with first term $a$ and ratio $r$ ($|r| < 1$)?

Accepted Answer

$S = \frac{a}{1 - r}$

Question 14

Given $f(x) = x^2$ and $g(x) = 2x + 1$, what is $(f \circ g)(x)$?

Accepted Answer

$(2x + 1)^2$

Question 15

If $\vec{a} = (1, 2, 3)$ and $\vec{b} = (4, 0, -1)$, what is the dot product $\vec{a} \cdot \vec{b}$?

Accepted Answer

$1 	imes 4 + 2 	imes 0 + 3 	imes (-1) = 1$

Question 16

Which of the following is the correct formula for the derivative of $f(x) = \sin(x)$?

Accepted Answer

$\cos(x)$

Question 17

A medication is administered and its concentration in the bloodstream is modeled by $C(t) = 50e^{-0.1t}$. What is the concentration after 10 hours?

Accepted Answer

$50e^{-1}$

Question 18

Which of the following statements about limits is true?

Accepted Answer

If $\lim_{x 	o a} f(x)$ exists, then $f(x)$ approaches a unique value as $x$ approaches $a$.

Precalculus-Related Topics from UFF First Semester Common Disciplines

Study Guide - Practice Questions