MATH1013 Mastery Exam 1 Sample

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Questions and Answers

State the Squeeze Theorem for functions defined around a point $a \in \mathbb{R}$.

If $g(x) \leq f(x) \leq h(x)$ for all $x$ in an open interval containing $a$ (except possibly at $a$) and $\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L$, then $\lim_{x \to a} f(x) = L$.

Evaluate $\lim_{x \to -4} \frac{(x + 4)^2}{x^2 - 16}$.

1/8
$\infty$
DNE
0 (correct)

Evaluate $\lim_{x \to \infty} \frac{\sqrt{x^6 + x + x^2}}{5x^3 + 4x^2}$.

0
$\infty$
DNE
1/5 (correct)

Evaluate $\lim_{x \to 1^+} \frac{e^{x-1}}{x - 1}$.

$\infty$ (A) Signup and view all the answers

Let $k$ be a real number and $f(x) = \begin{cases} \frac{x^2 - 1}{x - 1} & \text{if } x \neq 1 \ k & \text{otherwise} \end{cases}$. For which value(s) of $k$, if any, is the function $f$ continuous?

$k = 2$ Signup and view all the answers

Let $g(x) = (x + 2)^2$. Use the formal definition of the derivative to find $g'(x)$.

$g'(x) = 2x + 4$ Signup and view all the answers

Find $h'(x)$ when $h(x) = \cos((x + 7)^2)$.

$h'(x) = -2(x+7)\sin((x+7)^2)$ Signup and view all the answers

Find $j'(x)$ when $j(x) = x^2 \sin(x)$.

$j'(x) = 2x\sin(x) + x^2\cos(x)$ Signup and view all the answers

Find an equation of the tangent line to $y = \sin^2(x)$ at the point $(\frac{\pi}{2}, 1)$.

$y = 1$ Signup and view all the answers

Let $k$ be a real number and let $f(x) = \begin{cases} \frac{\ln(1 - k^2x)}{2x^2 - 4x} & \text{if } x < 0 \ k & \text{if } x \geq 0 \end{cases}$. For which values of $k$, if any, is the function $f$ differentiable?

k = 0 Signup and view all the answers

Perform row operations to reduce the matrix $\begin{bmatrix} 1 & 2 & -1 & 2 \ 1 & 2 & -1 & 4 \ 2 & 4 & -2 & 5 \end{bmatrix}$ to reduced row echelon form. Indicate which row operations you use.

$\begin{bmatrix} 1 & 2 & -1 & 0 \ 0 & 0 & 0 & 1 \ 0 & 0 & 0 & 0 \end{bmatrix}$; R2 -> R2 - R1, R3 -> R3 - 2R1, R2 -> -R2/2, R1 -> R1-2R2 Signup and view all the answers

Study the following linear systems, represented by their augmented matrices in echelon form. Which of these systems has a single unique solution?

$\begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}$ (C) Signup and view all the answers

The system represented by the augmented matrix $\begin{bmatrix} 1 & 1 & b \ 0 & k & b \end{bmatrix}$ has infinitely many solutions only when $k = 0$ and $b = 0$.

False (B) Signup and view all the answers

Write out the general solution to the linear system in vector parametric form, given the augmented matrix in reduced row echelon form: $\begin{bmatrix} 1 & 1 & 1 & -2 & 0 & -3 \ 0 & 0 & 0 & 1 & 2 \ 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix}$.

$\begin{bmatrix} x_1 \ x_2 \ x_3 \ x_4 \end{bmatrix} = \begin{bmatrix} -3 \ 0 \end{bmatrix} + s \begin{bmatrix} -1 \ 1 \ 0 \ 0 \end{bmatrix} + t \begin{bmatrix} 0\ 0 \1 \0 \end{bmatrix} + u \begin{bmatrix} 2 \ 0\ 0 \-2\end{bmatrix}$ Signup and view all the answers

For what value(s) of $k$ is the vector $\begin{bmatrix} -2 \ 2 \ k \end{bmatrix}$ in the span of the vectors $\begin{bmatrix} 1 \ 0 \ 1 \end{bmatrix}$ and $\begin{bmatrix} -1 \ 1 \ 2 \end{bmatrix}$?

k=4 Signup and view all the answers

Calculate the vector given by the matrix-vector product $\begin{bmatrix} 1 & 1 & 2 \ 1 & 1 & 0 \ -1 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 \ 0 \ 2 \end{bmatrix}$.

$\begin{bmatrix} 5 \ 1 \ 1 \end{bmatrix}$ Signup and view all the answers

Let $A$ be an $m \times n$ matrix, with its columns, in order, being the vectors $a_1, a_2, ..., a_n$ in $\mathbb{R}^m$. For a vector $x$ in $\mathbb{R}^n$, write down the definition of the matrix-vector product $Ax$.

$Ax = x_1a_1 + x_2a_2 + ... + x_na_n$ Signup and view all the answers

The set ${\begin{bmatrix} 1 \ 0 \ 2 \end{bmatrix}, \begin{bmatrix} 1 \ 0 \ 2 \end{bmatrix}, \begin{bmatrix} 1 \ 1 \ 1 \end{bmatrix}}$ is linearly independent.

False (B) Signup and view all the answers

Let $A$ be an $m \times n$ matrix (with $m$ rows and $n$ columns). Which of the following would guarantee that the columns of $A$ are linearly dependent?

$m < n$ (A) Signup and view all the answers

To pass the hurdle requirement for the exam, a score of at least ______ out of 20 is required.

16 Signup and view all the answers

Flashcards

Squeeze Theorem

The Squeeze Theorem states that if g(x) ≤ f(x) ≤ h(x) for all x in an interval containing a (except possibly at a), and lim x→a g(x) = lim x→a h(x) = L, then lim x→a f(x) = L.

Indeterminate Form 0/0

The limit of a quotient where both numerator and denominator approach zero. Requires techniques like factoring or L'Hôpital's Rule.

Continuity of a Function

A function f is continuous at a point c if f(c) is defined, lim x→c f(x) exists, and lim x→c f(x) = f(c).

Formal Definition of Derivative

The derivative of a function f(x) is f'(x) = lim h→0 (f(x+h) - f(x)) / h, provided the limit exists.