Calculus and Absolute Value Quiz
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Questions and Answers

Which of the following best describes the absolute value of a number?

  • The number itself.
  • The number multiplied by -1.
  • The number squared.
  • The distance of the number from the origin. (correct)

    • The absolute value of a number can be negative.

    False (B)

    What is the absolute value of -7?

    7

    The absolute value is denoted by the symbol | | which means ______.

    <p>modulus</p> Signup and view all the answers

    Match the number with its absolute value:

    <p>-5 = 5 0 = 0 3 = 3 -10 = 10</p> Signup and view all the answers

    According to the definition, if $a < 0$, then $|a|$ is equal to:

    <p>$-a$ (D)</p> Signup and view all the answers

    If $|x| = 0$, then $x$ must be equal to 0.

    <p>True (A)</p> Signup and view all the answers

    Absolute value describes the ______ of a number on the number line from the origin.

    <p>distance</p> Signup and view all the answers

    Which of the following topics are covered in the provided material?

    <p>Both A and B (D)</p> Signup and view all the answers

    The document includes information on advanced statistical analysis.

    <p>False (B)</p> Signup and view all the answers

    Who prepared the material in exam practice book?

    <p>Mr. E. Chauke</p> Signup and view all the answers

    Chapter 1 covers _________ /.

    <p>ABSOLUTE VALUES/MODULUS</p> Signup and view all the answers

    Match the chapter number with the corresponding topic:

    <p>Chapter 5 = The Limit of a Functions Chapter 9 = Implicit and Higher Order Differentiation Chapter 14 = Integration Chapter 16 = The Arc Length</p> Signup and view all the answers

    Which of these topics is included in the 'Summery (sic) and the Formulas' section?

    <p>All of the above (D)</p> Signup and view all the answers

    L'Hopital's Rule is covered in Chapter 10.

    <p>False (B)</p> Signup and view all the answers

    Name one application of calculus covered in the material.

    <p>Application of Calculus</p> Signup and view all the answers

    Chapter 2 discusses 'The Relation between _________ and _________.

    <p>Radians, Degrees</p> Signup and view all the answers

    Which chapter discusses 'Rolle’s & Mean Theorem'?

    <p>Chapter 12 (C)</p> Signup and view all the answers

    What is the first step in solving the inequality $|4x - 3| \ge 5$?

    <p>Apply the properties of inequalities and absolute value to obtain $4x - 3 \le -5$ or $4x - 3 \ge 5$. (D)</p> Signup and view all the answers

    When solving $|3x + 9| = |2x + 1|$, squaring both sides is a valid method to remove the modulus sign.

    <p>True (A)</p> Signup and view all the answers

    What are the critical values (CV) obtained when solving $|2x - 3| \ge |x + 3|$?

    <p>x = 0 or x = 6</p> Signup and view all the answers

    When solving $x + 6 > |3x + 2|$, the modulus sign can be addressed by ______ both sides.

    <p>squaring</p> Signup and view all the answers

    Solve for $x$: $e^{2x+3} = 7$

    <p>$x = \frac{ln(7) - 3}{2}$ (D)</p> Signup and view all the answers

    What operation is performed to simplify $6 < 3 < 2$?

    <p>Dividing all sides by 3 (B)</p> Signup and view all the answers

    The solution to $ln(5-2x) = -3$ is approximately $x = 2.475$.

    <p>True (A)</p> Signup and view all the answers

    The solution to $|2x + 1| < 5$ is $x < 2$ or $x > -3$.

    <p>False (B)</p> Signup and view all the answers

    After rearranging the equation $(3x + 9)^2 = (2x + 1)^2$, what is the simplified quadratic equation before factoring?

    <p>x^2 + 10x + 16 = 0</p> Signup and view all the answers

    Solve for $x$: $\frac{10}{1 + e^{-x}} = 2$

    <p>-ln 4</p> Signup and view all the answers

    To solve $4 + 3^{x+1} = 8$, after transposing, you take the natural logarithm of both sides resulting in $(x+1)$ln(3) = ln(______).

    <p>4</p> Signup and view all the answers

    Match each inequality with its corresponding solution:

    <p>$|4x - 3| \ge 5$ = $x \le -1/2$ or $x \ge 2$ $|3x + 2| &lt; 4$ = $-2 &lt; x &lt; 2/3$ $x + 6 &gt; |3x + 2|$ = $x^2 - 4 &lt; 0$</p> Signup and view all the answers

    Given $e^{2x} - 3e^x + 2 = 0$, which of the following are solutions for $x$?

    <p>$x = 0$ or $x = ln(2)$ (C)</p> Signup and view all the answers

    The first step to solving $\log_3(7x + 3) = \log_3(5x + 9)$ is to exponentiate both sides using base 10.

    <p>False (B)</p> Signup and view all the answers

    Given $\log_2(5x + 7) = 5$, after exponentiating both sides with base 2 and simplifying, $x$ = ______

    <p>5</p> Signup and view all the answers

    Match the logarithmic equation with the first step in solving for x:

    <p>log 4 x + log 4 (x – 12) = 3 = Combine log terms using product rule and change form to 4^3= x(x-12) log(x – 3) + log(x) = log 18 = Rewrite the left side using the product rule and cancel logarithms from both sides log 4 (2x + 1) = log 4 (x + 2) – log 4 3 = Use quotient rule to combine right side and cancel logarithms</p> Signup and view all the answers

    The general solution for $\tan \theta = m$ is given by $\theta = RA + 2\pi k$, where $k \in \mathbb{Z}$ and $RA$ is the reference angle.

    <p>False (B)</p> Signup and view all the answers

    Which of the following should be added to the solution of $\theta$ when solving for $\sin \theta$?

    <p>$2\pi k$ (B)</p> Signup and view all the answers

    When $\cos \theta = 0 $, the general solution can be written as $\theta = \frac{\pi}{2} + 2 \pi k$.

    <p>False (B)</p> Signup and view all the answers

    For $\tan \theta$, including $\tan \theta = 0$, add ________ to the solution.

    <p>$\pi k$</p> Signup and view all the answers

    If $\sin \theta = -m$, which of the following is a possible general solution for $\theta$?

    <p>$\theta = (\pi + RA) + 2\pi k$, where RA is the reference angle. (A)</p> Signup and view all the answers

    Solve for $x$: $\sqrt{3} \tan x = 1$

    <p>$\frac{\pi}{6}$</p> Signup and view all the answers

    For what values of $\sin \theta$ does the equation have NO solution?

    <p>Both A and B (A)</p> Signup and view all the answers

    If $\cos \theta = -m$, then $\theta = \cos^{-1}(-m) + 2\pi k = \pm RA + $ ______

    <p>$2\pi k$</p> Signup and view all the answers

    Match the trigonometric function with what should be added to the solutions:

    <p>sin θ = $2πk$ cos θ = $2πk$ tan θ = $πk$</p> Signup and view all the answers

    If $\tan \theta = -m$, then one possible solution can be $\theta = (\pi - RA) + \pi k$

    <p>True (A)</p> Signup and view all the answers

    Given the equation $\sin(3x + \frac{18}{5\pi}) = -\sin(\frac{9}{5\pi} - 2x)$, which of the following is a general solution for $x$?

    <p>Both A and B (C)</p> Signup and view all the answers

    The solutions $x = \frac{23\pi}{90}$, $x = \frac{59\pi}{90}$, $x = -\frac{13\pi}{90}$, $x = -\frac{49\pi}{90}$, $x = -\frac{17\pi}{18}$, and $x = -\frac{5\pi}{6}$ are valid solutions for the equation given.

    <p>True (A)</p> Signup and view all the answers

    What trigonometric identity is used to simplify the equation $4\cos^2 x + \sin 2x - 1 = 0$?

    <p>$\cos^2 x + \sin^2 x = 1$</p> Signup and view all the answers

    The equation $4\cos^2 x + \sin 2x - 1 = 0$ can be factorized into $(3\cos x - \sin x)(\sin x + \cos x) = $ ______.

    <p>0</p> Signup and view all the answers

    What are the two general solutions obtained after simplification of $\sin(3x + \frac{18}{5\pi}) = \sin(2x - \frac{9}{5\pi})$?

    <p>$3x + \frac{18}{5\pi} = 2x - \frac{9}{5\pi} + 2\pi k$ and $3x + \frac{18}{5\pi} = \pi - (2x - \frac{9}{5\pi}) + 2\pi k$ (D)</p> Signup and view all the answers

    Match the trigonometric expression with its equivalent form or result:

    <p>$\sin 2x$ = $2 \sin x \cos x$ $\cos^2 x + \sin^2 x$ = 1 $\frac{18}{5\pi}$ = Constant $4\cos^2 x + \sin 2x - 1 = 0$ = $(3 \cos x - \sin x)(\sin x + \cos x) = 0$</p> Signup and view all the answers

    The only solutions to the equation $4\cos^2 x + \sin 2x - 1 = 0$ in the interval $0 \le x \le 2\pi$ can be found directly without any factorization or simplification.

    <p>False (B)</p> Signup and view all the answers

    If $x = -6 + 2\pi k$ is a solution to a trigonometric equation, what does $k$ represent?

    <p>an integer</p> Signup and view all the answers

    Flashcards

    Logarithmic Functions

    The inverse operation to exponentiation, answering the question 'to what exponent?'.

    Absolute Value

    The distance of a number from the origin on the number line.

    Notation for Absolute Value

    Denoted by |a|, representing the absolute value of a real number a.

    Definition of Absolute Value when a ≥ 0

    If a is non-negative, |a| = a.

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    Definition of Absolute Value when a < 0

    If a is negative, |a| = -a (makes it positive).

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    Absolute Value Properties

    |x| = 0 if and only if x = 0.

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    Role of Absolute Value in Inequalities

    Absolutely values can simplify the understanding of inequalities.

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    Common Misconception about Absolute Value

    Absolute value is not a function; it represents distance, not direction.

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    Applications of Absolute Value

    Used in calculus and various fields to express magnitudes without direction.

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    Inequality Notation

    A way to express the relationship between two expressions using symbols like <, >, ≤, or ≥.

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    Critical Values

    The points where a function changes direction or touches a line, often found in inequalities.

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    Squaring Both Sides

    A method used to eliminate absolute values by raising both expressions to the power of two.

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    Simplifying Inequalities

    The process of reducing complex inequalities to their simplest form.

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    Rearranging the Equation

    The process of moving terms around in an equation to isolate a particular variable.

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    Modulus Sign Removal

    The process of eliminating modulus signs often by squaring the term.

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    Inequality Solutions

    The set of values that satisfy an inequality.

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    Natural Log Property

    ln(e^x) = x, simplifies logarithmic expressions.

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    Exponential Function

    e^x is a mathematical function where e is the base of the natural logarithm.

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    Solving for x

    Find the value of x in equations involving logs or exponentials.

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    Cross Multiplication

    A method to eliminate fractions by multiplying diagonal terms.

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    Logarithmic Identity

    log_b(a) = c means b^c = a; helps convert between forms.

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    Factoring Quadratics

    Expressing a quadratic in the form (k-a)(k-b) = 0 to find roots.

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    Properties of Logarithms

    Rules that govern how to manipulate logarithmic expressions, like log(a*b) = log(a) + log(b).

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    Inverse Function

    A function that reverses the effect of the original function, like ln and e.

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    Trigonometric Equation

    An equation involving trigonometric functions that seeks values of the variable that satisfy it.

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    sin(3x + 18) = -sin(9 - 2x)

    This equation can be transformed using the sine function properties.

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    Sine Relationship

    sin(a) = sin(b) leads to multiple possible angles depending on k.

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    Factorization of Trigonometric Expressions

    Expressing a trigonometric equation in multiplicative form to find solutions.

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    Quadratic Equation in Cosine and Sine

    An equation combining cos²x and sin²x terms, like 4cos²x + sin²x - 1 = 0.

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    cos²x + sin²x = 1

    Fundamental identity in trigonometry stating the sum of squares of sine and cosine is one.

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    k ∈ ℤ

    k represents any integer, indicating solutions are infinite and periodic in trigonometric contexts.

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    0 ≤ x ≤ 2π

    A common restriction for trigonometric equations specifying the range of the variable x.

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    General Solution for Tan

    tan 𝜃 = 𝑚 leads to 𝜃 = R.A + 𝜋k or 𝜃 = (𝜋 − R.A) + 2𝜋k.

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    Sin θ = 0

    When sin 𝜃 = 0, the solutions are θ = πk (not 2π).

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    Cos θ = 0

    When cos 𝜃 = 0, the solutions are θ = π/2 + πk (not 2π).

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    No Solution Condition

    For sin and cos, values outside [-1, 1] have no solutions.

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    Negative Sin Solution

    If sin 𝜃 = -𝑚, then θ = sin⁻¹(m) + 2πk or θ = (π + R.A) + 2πk.

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    Negative Cos Solution

    If cos 𝜃 = -𝑚, then θ = cos⁻¹(-m) + 2πk or θ = (π + R.A) + 2πk.

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    Negative Tan Solution

    If tan 𝜃 = -𝑚, then θ = tan⁻¹(m) + πk.

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    Solving √3 tan x = 1

    From √3 tan x = 1, deduce x = tan⁻¹(1/√3) + πk.

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    Sin Equation Simplification

    For √2 sin(x - π/2) = 1, solve by isolating sin.

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    Quadratic Sin Equation

    For 2sin²x + 3sinx - 2 = 0, use factoring or the quadratic formula.

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    Study Notes

    Differential and Integral Calculus Study Notes

    • Books: A calculus study book, specifically "Maths Made Easy" by Mr. E. Chauke for MMTH011/MAH101M, is the source for this information.
    • Copyright: Any copying of pages from this book is strictly prohibited without permission of the copyright holder.
    • Course Content: The book covers topics including an Algebra refresher, acknowledgements, and the purpose and guidance for students, Absolute Values/Modulus, The Relationship Between Radians and Degrees, Trigonometric and Logarithmic Equations, Functions, The Limit of a Function, Derivatives of Ordinary functions, Derivatives of Trigonometric functions, Derivatives of Exponential and Logarithmic functions, Implicit and Higher-Order Differentiation Derivatives of the Inverse Trigonometric Functions, and The Area Between Curves.
    • Additional Topics: Includes a trigonometry tool box, formulas, properties, fundamental identities, co-functions, identities concerning angles, and area and sine rules.
    • Overview: A table of contents is included with page numbers for each chapter

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