Basic Calculus - Third Quarterly Post Test
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Questions and Answers

What is the relationship between $\lim_{x \to c} f(x)$ and $f(c)$ when the limit exists?

  • $\lim_{x \to c} f(x) \geq f(c)$
  • $\lim_{x \to c} f(x) \neq f(c)$
  • $\lim_{x \to c} f(x) = f(c)$ (correct)
  • $\lim_{x \to c} f(x) > f(c)$

    • Which theorem is applied first to solve the expression $\lim_{x \to 0} (2x^3 + 4x^2 - 1)$?

  • Multiplication
  • Division
  • Addition (correct)
  • Constant Multiple

    • What is the appropriate term to divide the expression $\lim_{x \to \infty} \frac{9x^2}{x^2 - 1}$?

  • $x^2$ (correct)
  • $x^4$
  • $x^3$
  • $x$

    • What is the computed value of A when evaluating $\lim_{x \to \infty} \frac{x}{x}$ for the given points?

    <p>-0.01 (A)</p> Signup and view all the answers

    What characteristic does a function exhibit if its graph can be traced without lifting the pen when approaching a specific x-value?

    <p>Continuous (D)</p> Signup and view all the answers

    What does it imply if $\lim_{x \to c} f(x) \neq f(c)$?

    <p>The function is not continuous at x = c. (A)</p> Signup and view all the answers

    Which operation must be used first to evaluate $\lim_{x \to 0} (4x^2)$?

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

    What can be inferred about the value of a function if as x approaches a large value, f(x) approaches a constant?

    <p>The function is convergent. (A)</p> Signup and view all the answers

    What is the value of 𝑓(𝑐) at 𝑥 = 0 using the function $f(x) = \frac{3x^2 + x - 2}{x - 1}$?

    <p>2 (B)</p> Signup and view all the answers

    In which interval is the function $k(x) = { x^2 + 1, \text{ if } 0 \leq x \leq 1 } \text{ and } { \frac{2}{x}, \text{ if } x > 1 }$ continuous?

    <p>(−6, 5] (B)</p> Signup and view all the answers

    In the Intermediate Value Theorem (IVT), where should the value of m fall?

    <p>(f(a), f(b)) (B)</p> Signup and view all the answers

    Which of the following is NOT a notation for the derivative of 𝑓 if 𝑦 = 𝑓(𝑥)?

    <p>[𝑓(𝑥)] (B)</p> Signup and view all the answers

    Which statement regarding continuity of a function is correct?

    <p>A function 𝑓 is continuous everywhere if it is continuous at every real number. (A)</p> Signup and view all the answers

    Which of the following represents the constant multiple rule?

    <p>𝑓 ′ (𝑥) = 𝑘ℎ′ (𝑥) (D)</p> Signup and view all the answers

    According to the Extreme Value Theorem, what must be fulfilled first?

    <p>The function should be continuous over the interval [a, b]. (D)</p> Signup and view all the answers

    What kind of rule can be used to solve the derivative of 𝑓(𝑥) = (4𝑥 − 1)⁴ easily?

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

    Which function is to be solved first in determining the derivative of 𝑗(𝑥) = sin⁴(2𝑥 + 1)?

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

    Utilizing implicit differentiation, which of the following is the derivative of 𝑦² = 4𝑥?

    <p>8y (B)</p> Signup and view all the answers

    What is the limit of the polynomial function if 𝑓(𝑐) = −12 and lim 𝑥→𝑐 𝑓(𝑥) = 𝑓(𝑐)?

    <p>−12 (B)</p> Signup and view all the answers

    What is the value of the expression lim (2𝑥³ + 4𝑥² − 1) as 𝑥 approaches 0?

    <p>−1 (C)</p> Signup and view all the answers

    What is the value of the expression lim 1/𝑥 as 𝑥 approaches infinity?

    <p>0 (B)</p> Signup and view all the answers

    What would be the correct statement about the continuity of 𝑓(𝑥) at 𝑥 = 0 after applying the third condition of continuity?

    <p>𝑓(𝑥) is continuous at 𝑥 = 0 since 𝑓(𝑐) = lim 𝑓(𝑥) as 𝑥 approaches 𝑐. (B)</p> Signup and view all the answers

    What is the slope of the tangent line of the graph of 𝑓(𝑥) = 𝑥² at point (2, 4) where 𝑥₀ = 2?

    <p>4 (C)</p> Signup and view all the answers

    What is the derivative of 𝑔(𝑥) = 2𝑥(𝑥 + 1)?

    <p>4𝑥 + 2 (D)</p> Signup and view all the answers

    What is the derivative of the function 𝑓(𝑥) = (4𝑥 − 1)⁴?

    <p>4(4𝑥 − 1)³ (A)</p> Signup and view all the answers

    Using implicit differentiation, what is the derivative of the equation 3𝑥² + 4𝑦² = 11?

    <p>−(3𝑥)/(4𝑦) (A)</p> Signup and view all the answers

    What is the objective function for making an open-top rectangular box from a 24 cm by 9 cm cardboard?

    <p>𝑓(𝑥) = 4𝑠³ − 66𝑠² + 216𝑠 (B)</p> Signup and view all the answers

    What is the volume of the open-type box created from the cardboard?

    <p>200 cm³ (D)</p> Signup and view all the answers

    What is the resulting form of the equation sin(cos(𝑥𝑦)) = 3 after integration?

    <p>sin 𝑢 = 3 (B)</p> Signup and view all the answers

    Which of the following is the derivative of 𝑒^(4𝑦) = 5𝑥³ − 9𝑥?

    <p>(5𝑥² + 3)𝑒^(4𝑦) (B)</p> Signup and view all the answers

    Which of the following expresses the relationship between the area of a ripple and the radius of the ripple?

    <p>d𝐴/d𝑟 = 2𝜋𝑟 (D)</p> Signup and view all the answers

    Flashcards

    Limit Definition

    The limit of f(x) as x approaches c is the value f(c) that f(x) approaches.

    Limit Relationship

    If lim as x→c f(x) = f(c), the function is continuous at c.

    Limit Theorems

    Basic theorems include Addition, Constant Multiple, and others for evaluating limits.

    Dividing Limits

    To divide terms in a limit expression, choose a term that simplifies it, usually the highest degree.

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    Limit at Infinity

    The limit of a function as x approaches infinity assesses long-term behavior of the function.

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    Continuous Function

    A function is continuous if its graph can be drawn without lifting your pen.

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    Graph Approaching Values

    When evaluating limits, check both sides' values as they approach a point.

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    Limit Values Table

    A table can show limit behaviors by comparing function values as x approaches a point.

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    Continuity at a point

    A function f is continuous at c if lim x→c f(x) = f(c).

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    Interval of continuity

    An interval where a function is continuous for all x within it.

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    Intermediate Value Theorem (IVT)

    If f is continuous on [a, b], then for any m between f(a) and f(b), there exists c in (a, b) such that f(c) = m.

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    Derivative notation

    The derivative of f(x) can be expressed as f'(x), dy/dx, or D[f(x)].

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    Extreme Value Theorem

    A continuous function on [a, b] must attain a maximum and minimum value.

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    Constant multiple rule

    If k is a constant, then the derivative of kf(x) is kf'(x).

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    Conditions of continuity

    For f to be continuous at c: f(c) exists, lim x→c f(x) exists, and they are equal.

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    Function continuity everywhere

    A function is continuous everywhere if it meets continuity conditions at every point x.

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    Chain Rule

    A rule for finding the derivative of composite functions.

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    Trigonometric Function

    A function like sin, cos, or tan; determines first for derivatives.

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    Implicit Differentiation

    A method to find derivatives when y is defined implicitly.

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    Limit of a Polynomial

    The limit of a polynomial function at c equals f(c).

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    Limit as x approaches infinity

    The behavior of a function as x gets very large.

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    Third Condition of Continuity

    For continuity, f(c) must equal the limit as x approaches c.

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    Slope of Tangent Line

    The rate of change or the derivative at a given point.

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    Derivative at a Point

    The value of the derivative evaluated at a specific x-value.

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    Derivative of g(x)

    The derivative of g(x) = 2x(x + 1) is g'(x) = 4x + 2.

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    Derivative of f(x)

    The derivative of f(x) = (4x - 1)⁴ is f'(x) = 16(4x - 1)³.

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    Implicit differentiation result

    The derivative of 3x² + 4y² = 11 gives dy/dx = -3x/4y.

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    Objective function for box

    The objective function for an open rectangular box is related to volume maximization.

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    Volume of the box

    The volume of the open-type box can be expressed as V = base area x height.

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    Area growth of ripples

    The area of a ripple increases at the rate of dA/dt = 2π cm²/s.

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    Derivative of e^(4y)

    The derivative of e^(4y) = 5x³ - 9x involves y's rates.

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    Rate of area change

    The rate of change of area A with respect to radius r is dA/dr = 2πr.

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    Study Notes

    Basic Calculus - Third Quarterly Post Test

    • Multiple Choice Questions: The test covers various calculus concepts.
    • Relationship between limit and function value: The limit of a function as x approaches c equals the function value at c (lim f(x) = f(c)).
    • Theorems for solving limits: Various theorems such as constant multiple, addition, division, and multiplication properties are used to evaluate limits.
    • Limits and expressions: Questions involve finding limits of algebraic expressions.
    • Continuity: The test questions included a problem involving the first condition of continuity, concerning a function's value at a point. The function must be defined there, and the limit from the left and right must be equal to the function value to ensure continuity.
    • Continuity of Functions: The test assesses the understanding of function continuity. One example involves identifying the intervals where a piecewise-defined function is continuous.
    • Derivative Notation: Different notations for the derivative of a function are considered (e.g., f'(x), dy/dx).
    • Rules for differentiation: Understanding various differentiation techniques will help in answering questions.
    • Extreme Value Theorem: The Extreme Value Theorem states that a continuous function on a closed interval must attain both a maximum and a minimum value on that interval.
    • Implicit Differentiation: Some questions include implicit differentiation, where relationships among variables are defined implicitly, and the derivatives are found by differentiating implicitly.
    • Trigonometric Functions and their derivatives: Questions about the derivative of trigonometric functions, or functions composed of trigonometric functions, show up in this test.
    • Area, Radius, and Time: Questions concerning the rate of change of area and radius of circles involve calculating the derivative of the area formula (Area = πr²) with regards to time.

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    Basic Calculus Post Test Q3 2025 PDF

    Description

    This post test evaluates your understanding of key calculus concepts, including limits, continuity, and derivative functions. Multiple choice questions challenge your ability to apply theorems and assess function behavior at specific points. Prepare to demonstrate your grasp of these fundamental topics in calculus.

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