AP Calc BC: Final Formulas Flashcards
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Questions and Answers

What does the Intermediate Value Theorem state?

  • If a function is continuous, it must be increasing.
  • The derivative of a function is always positive.
  • If f(1)=-4 and f(6)=9, then there exists an x-value between 1 and 6 where f crosses the x-axis. (correct)
  • A function can cross the x-axis multiple times.

    • What is the Average Rate of Change?

    Slope of secant line between two points.

    What does the Instantaneous Rate of Change represent?

    Slope of tangent line at a point.

    What is the formal definition of the derivative?

    <p>Limit as h approaches 0 of [f(a+h)-f(a)]/h.</p> Signup and view all the answers

    What is the alternate definition of the derivative?

    <p>Limit as x approaches a of [f(x)-f(a)]/(x-a).</p> Signup and view all the answers

    When f '(x) is positive, f(x) is increasing.

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

    When f '(x) is negative, f(x) is increasing.

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

    When f '(x) changes from negative to positive, what does f(x) have?

    <p>Relative minimum.</p> Signup and view all the answers

    When f '(x) changes from positive to negative, what does f(x) have?

    <p>Relative maximum.</p> Signup and view all the answers

    When f '(x) is increasing, f(x) is concave up.

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

    When f '(x) is decreasing, f(x) is concave up.

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

    What is a point of inflection?

    <p>When f '(x) changes from increasing to decreasing or decreasing to increasing.</p> Signup and view all the answers

    What are the conditions when a function is not differentiable?

    <p>Vertical tangent</p> Signup and view all the answers

    What is the Product Rule?

    <p>uv' + vu'.</p> Signup and view all the answers

    What is the Quotient Rule?

    <p>(uv'-vu')/v².</p> Signup and view all the answers

    What is the Chain Rule?

    <p>f '(g(x)) g'(x).</p> Signup and view all the answers

    What is the derivative of y = x cos(x)?

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

    What is the derivative of y = ln(x)/x²?

    <p>Quotient Rule.</p> Signup and view all the answers

    What is the derivative of y = cos²(3x)?

    <p>Chain Rule.</p> Signup and view all the answers

    If velocity is positive, the particle is moving to the right/up.

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

    If velocity is negative, the particle is moving to the right/up.

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

    What does the absolute value of velocity represent?

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

    What is the derivative of y = sin(x)?

    <p>y' = cos(x).</p> Signup and view all the answers

    What is the derivative of y = cos(x)?

    <p>y' = -sin(x).</p> Signup and view all the answers

    What is the derivative of y = tan(x)?

    <p>y' = sec²(x).</p> Signup and view all the answers

    What is the derivative of y = csc(x)?

    <p>y' = -csc(x)cot(x).</p> Signup and view all the answers

    What is the derivative of y = sec(x)?

    <p>y' = sec(x)tan(x).</p> Signup and view all the answers

    What is the derivative of y = cot(x)?

    <p>y' = -csc²(x).</p> Signup and view all the answers

    What is the derivative of y = sin⁻¹(x)?

    <p>y' = 1/√(1 - x²).</p> Signup and view all the answers

    What is the derivative of y = cos⁻¹(x)?

    <p>y' = -1/√(1 - x²).</p> Signup and view all the answers

    What is the derivative of y = tan⁻¹(x)?

    <p>y' = 1/(1 + x²).</p> Signup and view all the answers

    What is the derivative of y = cot⁻¹(x)?

    <p>y' = -1/(1 + x²).</p> Signup and view all the answers

    What is the derivative of y = e^x?

    <p>y' = e^x.</p> Signup and view all the answers

    What is the derivative of y = a^x?

    <p>y' = a^x ln(a).</p> Signup and view all the answers

    What is the derivative of y = ln(x)?

    <p>y' = 1/x.</p> Signup and view all the answers

    What is the derivative of y = log (base a) x?

    <p>y' = 1/(x lna).</p> Signup and view all the answers

    How do you find the absolute maximum on the closed interval [a, b]?

    <p>Consider critical points and endpoints.</p> Signup and view all the answers

    What does the Mean Value Theorem state?

    <p>If f(x) is continuous and differentiable, the slope of the tangent line equals the slope of the secant line at least once in the interval (a, b).</p> Signup and view all the answers

    If f '(x) = 0 and f''(x) > 0, then f(x) has a relative minimum.

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

    If f '(x) = 0 and f''(x) < 0, then f(x) has a relative minimum.

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

    What is linearization?

    <p>Use tangent line to approximate values of the function.</p> Signup and view all the answers

    What does the term 'rate' refer to in calculus?

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

    What is the left Riemann sum?

    <p>Use rectangles with left-endpoints to evaluate integrals.</p> Signup and view all the answers

    What is the right Riemann sum?

    <p>Use rectangles with right-endpoints to evaluate integrals.</p> Signup and view all the answers

    What is the trapezoidal rule?

    <p>Use trapezoids to evaluate integrals.</p> Signup and view all the answers

    What is the formula for the area of a trapezoid?

    <p>[(h1 - h2)/2]*base.</p> Signup and view all the answers

    What is a definite integral?

    <p>Has limits a &amp; b, find antiderivative, F(b) - F(a).</p> Signup and view all the answers

    What is an indefinite integral?

    <p>No limits, find antiderivative + C.</p> Signup and view all the answers

    What does the area under a curve represent mathematically?

    <p>∫ f(x) dx integrate over interval a to b.</p> Signup and view all the answers

    Study Notes

    Fundamental Theorems and Concepts

    • Intermediate Value Theorem: Guarantees a solution exists between two points where a function changes signs.
    • Average Rate of Change: Represents the slope of the secant line connecting two function points, approximating instantaneous rate of change.
    • Instantaneous Rate of Change: Defined by the slope of the tangent line at a point, equating to the derivative's value.

    Derivatives Definitions

    • Formal Definition of Derivative: Derived from the limit as h approaches 0 of [f(a+h) - f(a)]/h.
    • Alternate Definition of Derivative: Based on the limit as x approaches a of [f(x) - f(a)]/(x - a).

    Behavior of Functions

    • Increasing Function: Occurs when the derivative f'(x) > 0.
    • Decreasing Function: Takes place when the derivative f'(x) < 0.
    • Relative Minimum: Observed when the derivative changes from negative to positive.
    • Relative Maximum: Happens when the derivative changes from positive to negative.

    Concavity and Inflection Points

    • Concave Up: When the derivative f'(x) is increasing.
    • Concave Down: When the derivative f'(x) is decreasing.
    • Point of Inflection: Occurs when the derivative changes concavity.

    Non-differentiability

    • Non-differentiable Conditions: Includes corners, cusps, vertical tangents, and discontinuities.

    Differentiation Rules

    • Product Rule: To differentiate a product of two functions: uv' + vu'.
    • Quotient Rule: For the division of two functions: (uv' - vu')/v².
    • Chain Rule: For composite functions, represented as f'(g(x))g'(x).

    Derivatives of Common Functions

    • Trigonometric Functions:

      • y = sin(x), y' = cos(x)
      • y = cos(x), y' = -sin(x)
      • y = tan(x), y' = sec²(x)
      • y = csc(x), y' = -csc(x)cot(x)
      • y = sec(x), y' = sec(x)tan(x)
      • y = cot(x), y' = -csc²(x)

    • Inverse Trigonometric Functions:

      • y = sin⁻¹(x), y' = 1/√(1 - x²)
      • y = cos⁻¹(x), y' = -1/√(1 - x²)
      • y = tan⁻¹(x), y' = 1/(1 + x²)
      • y = cot⁻¹(x), y' = -1/(1 + x²)

    • Exponential and Logarithmic Functions:

      • y = e^x, y' = e^x
      • y = a^x, y' = a^x ln(a)
      • y = ln(x), y' = 1/x
      • y = log (base a) x, y' = 1/(x lna)

    Optimization

    • Finding Absolute Maximums: Analyze critical points and endpoints on a closed interval [a, b].
    • Mean Value Theorem: States if a function is continuous and differentiable over an interval, there exists at least one point where the instantaneous slope equals the average slope.

    Linearization and Rates

    • Linearization: Using a tangent line for approximating function values at a specific point.
    • Rate: Represented mathematically by the derivative of the function.

    Riemann Sums and Integrals

    • Left Riemann Sum: Area estimation using rectangles with left endpoints.
    • Right Riemann Sum: Area estimation using rectangles with right endpoints.
    • Trapezoidal Rule: Uses trapezoids for area estimation under a curve.
    • Area of a Trapezoid: Calculated with formula [(h1 - h2)/2] × base.

    Types of Integrals

    • Definite Integral: Represents the area under a curve with limits given by a and b, expressed as F(b) - F(a).
    • Indefinite Integral: Represents family of antiderivatives, expressed as F(x) + C where C is a constant determined by initial conditions.
    • Area Under a Curve: Calculated by integrating ∫ f(x) dx over a specified interval [a, b].

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    Description

    This quiz features key formulas and concepts from AP Calculus BC, designed to help students review essential mathematical principles. Each flashcard provides definitions and explanations for important terms such as the Intermediate Value Theorem and Rate of Change. Perfect for exam preparation or quick revision.

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