Calculus I MATH 150/151 Course Notes
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Questions and Answers

Which set of numbers includes only positive counting numbers?

  • Integers
  • Rational Numbers
  • Real Numbers
  • Natural Numbers (correct)

    • What is the definition of a function's range?

  • The total number of elements in the codomain.
  • The unique elements of set X.
  • The set of all elements in Y corresponding to an element of X. (correct)
  • The set of all possible inputs.

    • How can a function be represented numerically?

  • By a formula.
  • By a verbal description.
  • Using a graph.
  • By a table of values. (correct)

    • Which of the following statements about rational numbers is true?

    <p>They can be expressed as the ratio of two integers.</p> Signup and view all the answers

    What is the primary difference between the sets of rational and real numbers?

    <p>Real numbers include all rational numbers and irrational numbers.</p> Signup and view all the answers

    Which of the following is not a valid way to define a function?

    <p>Expressively.</p> Signup and view all the answers

    Which of the following best describes the set of integers?

    <p>All whole numbers, both positive and negative.</p> Signup and view all the answers

    Which representation of a function is described as giving a verbal description of its behavior?

    <p>Verbal representation.</p> Signup and view all the answers

    What is the velocity at time 0.6 seconds?

    <p>0.8 m/s</p> Signup and view all the answers

    Which of the following is true about the function f(x) = x^2?

    <p>f(π) = π^2</p> Signup and view all the answers

    What would be the expression for the height y as a function of x in the ladder problem?

    <p>y = √(L^2 - x^2)</p> Signup and view all the answers

    What is the domain of the function g(x) = 1/(x^2 - x)?

    <p>x ≠ 0 and x ≠ 1</p> Signup and view all the answers

    Which of the following accurately represents the range of the function h(t) = √(16 - t^2)?

    <p>[0, 4]</p> Signup and view all the answers

    What is the value of f(2/3) if f(x) = x^2?

    <p>4/9</p> Signup and view all the answers

    What is the correct way to express the velocity values as a set of ordered pairs?

    <p>{(0, 0), (0.2, 0.2), (1.0, 0.6)}</p> Signup and view all the answers

    For the function f(x) = |x|, what is the graph's property?

    <p>It is symmetric about the y-axis</p> Signup and view all the answers

    What best describes the vertical line test?

    <p>It verifies if every vertical line intersects the curve at most once.</p> Signup and view all the answers

    Which of the following is a correct piecewise function for f(x) = |x|?

    <p>f(x) = x if x &gt; 0 or f(x) = -x if x ≤ 0</p> Signup and view all the answers

    Which of the following best describes power functions?

    <p>Functions of the form f(x) = xa where a is a fixed real number.</p> Signup and view all the answers

    In which case is the function g(x) = x^2 equal to f(x) = |x|?

    <p>When x is positive.</p> Signup and view all the answers

    How is a polynomial defined?

    <p>As a function of the form f(x) = a_n x^n + a_n-1 x^(n-1) + ... + a_1 x + a_0.</p> Signup and view all the answers

    What is the shape of the graph for the function p(x) = |x| + |x + 1|?

    <p>A V-shape with a single vertex.</p> Signup and view all the answers

    When finding the equation of a line given point P(1, 3) and slope 2, what is it?

    <p>y = 2x + 3</p> Signup and view all the answers

    Which of the following is NOT a type of function mentioned?

    <p>Exponential functions</p> Signup and view all the answers

    What is the main focus of Part One in the Calculus I course notes?

    <p>Review of Functions and Models</p> Signup and view all the answers

    Which rule would be most appropriate for finding the derivative of a product of two functions?

    <p>Product Rule</p> Signup and view all the answers

    Which of the following is a key aspect of the Mean Value Theorem?

    <p>It guarantees at least one point where the derivative is zero.</p> Signup and view all the answers

    What type of functions are primarily reviewed in the section dedicated to Mathematical Models?

    <p>Essential Functions</p> Signup and view all the answers

    What is the primary subject of the section dedicated to Limits at Infinity?

    <p>Understanding horizontal asymptotes</p> Signup and view all the answers

    What does implicit differentiation allow a student to do?

    <p>Differentiate functions that are not explicitly solved for y.</p> Signup and view all the answers

    Which of the following best describes the Chain Rule in differentiation?

    <p>A technique applied when differentiating composite functions.</p> Signup and view all the answers

    What is one application of derivatives mentioned in the course notes?

    <p>Determining the tangent line at a point</p> Signup and view all the answers

    What differentiates polar coordinates from Cartesian coordinates?

    <p>Polar coordinates depend on angles and distances from a fixed point.</p> Signup and view all the answers

    In the context of calculus, what are related rates?

    <p>Derivatives that involve two or more quantities that change over time.</p> Signup and view all the answers

    What method can be used to solve optimization problems in calculus?

    <p>Finding critical points using derivatives.</p> Signup and view all the answers

    Which of the following derivatives specifically relate to logarithmic functions?

    <p>The property of logarithm differentiation involving the base.</p> Signup and view all the answers

    What is emphasized in the summary of curve sketching?

    <p>Understanding the influence of derivatives on graph shapes.</p> Signup and view all the answers

    The use of Newton’s Method is primarily for what purpose in calculus?

    <p>Estimating the roots of functions.</p> Signup and view all the answers

    What is the value of $ an(30^{ ext{°}})$?

    <p>$ rac{ ext{√3}}{3}$</p> Signup and view all the answers

    How does the graph of $y = f(x) + c$ relate to $y = f(x)$?

    <p>It is shifted vertically upward.</p> Signup and view all the answers

    What transformation does the equation $y = f(x - c)$ represent?

    <p>Shift to the right by $c$ units.</p> Signup and view all the answers

    What is the value of $ heta$ for which $ an( heta) = 1$?

    <p>$ rac{ ext{π}}{4}$</p> Signup and view all the answers

    Which of the following equations represents a vertical shift downward?

    <p>$y = f(x) - c$</p> Signup and view all the answers

    For the function $y = 3x^2 - 6x + 1$, what is the effect of the $-6x$ term?

    <p>It determines the position of the vertex.</p> Signup and view all the answers

    What is the value of $ ext{sin}(90^{ ext{°}})$?

    <p>1</p> Signup and view all the answers

    What effect does the factor of 2 in the equation $y = ext{sin}(2x)$ have on the graph of $y = ext{sin}(x)$?

    <p>It compresses the graph horizontally.</p> Signup and view all the answers

    Study Notes

    Overview of Calculus I (MATH 150/151)

    • Course focuses on fundamental concepts of calculus, including functions, differentiation, and applications.
    • Offers comprehensive coverage of essential mathematical models and theories.

    Functions and Models

    • Functions relate each element of a domain to a unique element of a codomain.
    • Four representations of functions:
      • Verbally (e.g., descriptions)
      • Algebraically (e.g., formulas)
      • Numerically (e.g., tables or sets of ordered pairs)
      • Visually (e.g., graphs)

    Basic Sets of Numbers

    • Natural Numbers: N = {1, 2, 3, ...}
    • Integers: Z = {..., -2, -1, 0, 1, 2, ...}
    • Rational Numbers: Q = {a/b: a, b ∈ Z, b ≠ 0}
    • Real Numbers: Includes all rational numbers and irrational numbers that fill gaps within the number line.

    Defining Functions

    • Example: For a function f such that f(x) = x², identify input-values and compute output values.
    • Domain represents possible input values; the codomain represents allowable output values.
    • Range is the actual output values corresponding to the domain.

    Mathematical Models

    • Linear Functions: Defined by slope and y-intercept. Can deduce the equation from these parameters.
    • Power Functions: Functions of the form f(x) = x^a, where a is a real number.
    • Polynomials: Functions expressed as f(x) = a_n * x^n + ... + a_1 * x + a_0, with integer n and coefficients a_i being real numbers.

    Graphing Functions

    • Vertical line test determines whether a graph represents a function; if any vertical line intersects at most once, it is a function.
    • Common function shapes include linear, parabolic, and piecewise functions.

    Transformation of Functions

    • Transformations of graphs include:
      • Shifting vertically (up/down)
      • Shifting horizontally (right/left)
      • Reflecting across axes
    • Example transformations:
      • y = f(x) + c shifts upward by c.
      • y = f(x) - c shifts downward by c.

    Differentiation and Applications

    • Concepts of limits and derivatives underpin calculus.
    • Derivatives measure rates of change and are essential for understanding curve behavior.
    • Applications include optimization problems, curve sketching, and understanding motion.

    Exam Preparation

    • Practice exercises covering all the course material are necessary.
    • Review notes and key concepts regularly.
    • Prepare sample problems and check against solutions for mastery.

    Appendix and Additional Resources

    • Solutions to exercises provided to enhance understanding.
    • Bibliography includes articles, books, and web resources for further reading and study.

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    Explore the foundational concepts of Calculus I through these comprehensive course notes from SFU. This resource provides essential information for MATH 150/151, making complex ideas more accessible. Ideal for students needing a thorough review or supplementary material.

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