AP CALCULUS AB Flashcards
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Questions and Answers

What does the term 'Trig Identities' refer to?

  • Theorems in geometry
  • Relationships between trigonometric functions (correct)
  • Methods of integration
  • Types of calculus derivatives

    • What is represented by dy/dx?

    The derivative of a function y with respect to x

    What is the Intermediate Value Theorem?

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

    What does the Extreme Value Theorem state?

    <p>There must be a maximum and minimum if the function is continuous on [a,b].</p> Signup and view all the answers

    What is Linear Approximation?

    <p>F(x) ≈ f(a) + f'(a)(x-a)</p> Signup and view all the answers

    What is the formula for Finding Average Velocity?

    <p>(F(x+h) - F(x)) / h</p> Signup and view all the answers

    What are Inflection Points?

    <p>Points where the concavity of the function changes.</p> Signup and view all the answers

    What does the Mean Value Theorem state?

    <p>If a function is continuous on [a,b] and differentiable on (a,b), then there exists at least one c in (a,b) such that f'(c) = (f(b) - f(a)) / (b - a).</p> Signup and view all the answers

    What is the condition for F(x) to be Increasing or Decreasing?

    <p>If f' is positive, then f(x) is increasing; if f' is negative, then f(x) is decreasing.</p> Signup and view all the answers

    What does Concavity refer to in calculus?

    <p>If f'' is positive, then f' is increasing, and the concavity of f(x) is upwards; if f'' is negative, then f' is decreasing, and the concavity of f(x) is downwards.</p> Signup and view all the answers

    Where do Absolute Maximums and Minimums occur?

    <p>Only at critical points or endpoints of a continuous function.</p> Signup and view all the answers

    What is the Product Rule in Differentiation?

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    What is the Quotient Rule in Differentiation?

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    What is the Chain Rule in Differentiation?

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    What is the significance of Inverse Functions in Differentiation?

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    What does the Power Rule in Differentiation state?

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    What is the derivative of ln|x|?

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

    What happens in Related Rates problems?

    <p>Two variables related by an equation, differentiated with respect to time.</p> Signup and view all the answers

    What is Optimization with Constraint?

    <p>A method to find the best solution under given constraints.</p> Signup and view all the answers

    What is an Indefinite Integral?

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    What is a Definite Integral?

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    What does the Fundamental Theorem of Calculus state?

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    What is the Second Fundamental Theorem of Calculus?

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    What is the Mean Value Theorem for Integrals?

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    What is the Power Rule for Integration?

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    What is the Natural Logarithm in Integration?

    <p>(squiggly line antiderivative)d/du = ln|u| + C</p> Signup and view all the answers

    What is Integration of Trigonometric Functions?

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    What does Area Between Curves represent?

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    What are Volumes of Solids of Revolution?

    <p>Rotating the region between two curves about a line to find the volume created.</p> Signup and view all the answers

    What is Disk Volume?

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    What is Washer Volume along the x-axis?

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    What is Washer Volume along the y-axis?

    <p>Same as along the x-axis simply replace x with y.</p> Signup and view all the answers

    What is the formula for Cylindrical Shell Volume?

    <p>h = R - r</p> Signup and view all the answers

    What is a Riemann Sum?

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    What is a Riemann Sum: Trapezoidal?

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    Study Notes

    Trigonometric Identities

    • Fundamental building blocks for simplifying and solving trigonometric equations.

    Derivative Notation (dy/dx)

    • Represents the rate of change of y with respect to x in calculus.

    Intermediate Value Theorem

    • States that a continuous function on an interval [a, b] takes every value between f(a) and f(b).

    Extreme Value Theorem

    • A continuous function on a closed interval [a,b] must achieve both a maximum and minimum value.

    Linear Approximation

    • f(x) can be approximated as f(a) + f'(a)(x - a), useful for estimating values near a point.

    Finding Average Velocity

    • Average velocity is calculated as the total change in position over the total time.

    Numerical Differentiation

    • Provides an estimate of the derivative using the formula (f(x+h) - f(x)) / h.

    Inflection Points

    • Occur where the second derivative changes sign; involves finding where f''(x) = 0 and analyzing f(x) values.

    Mean Value Theorem

    • Guarantees that there exists at least one point c in (a, b) where f'(c) equals the average rate of change from a to b.

    Increasing or Decreasing Functions

    • If f'(x) > 0, the function is increasing; if f'(x) < 0, it is decreasing.

    Concavity of Functions

    • If f''(x) > 0, the function is concave up; if f''(x) < 0, it is concave down.

    Absolute Maxima and Minima

    • Can only occur at critical points or endpoints of a continuous function, ensured by the Extreme Value Theorem.

    Differentiation: Product Rule

    • The derivative of a product u*v is given by u'v + uv'.

    Differentiation: Quotient Rule

    • The derivative of a quotient u/v is (u'v - uv') / v^2.

    Differentiation: Chain Rule

    • Used when differentiating composite functions; if y = f(g(x)), then dy/dx = f'(g(x)) * g'(x).

    Differentiation: Inverse Functions

    • Allows the calculation of derivatives for inverse functions using the derivative of the original function.

    Differentiation: Implicit Functions

    • Involves finding derivatives of y that are defined implicitly by an equation involving x and y.

    Differentiation: Power Rule

    • States that d/dx of x^n = nx^(n-1) for any real number n.

    Differentiation: Logarithms

    • Key rules include:
      • d/dx of ln|x| = 1/x
      • d/dx of log_a|x| = 1/(ln(a)x)
      • d/dx of ln|f(x)| = f'(x)/f(x)

    Differentiation: Trig Functions

    • Derivatives of basic trigonometric functions:
      • d/dx(sin(x)) = cos(x)
      • d/dx(cos(x)) = -sin(x)
    • Involves differentiating relationships between variables that change with time to find unknown rates.

    Optimization with Constraints

    • A methodical process to identify optimal solutions subject to certain restrictions using constraints.

    Indefinite Integral

    • Represents a family of functions whose derivative is the integrand, includes a constant of integration, C.

    Definite Integral

    • Measures the area under a curve between two specified points, represented as ∫[a,b] f(x) dx.

    Fundamental Theorem of Calculus

    • Connects differentiation and integration, establishing that the integral of a function can be evaluated using its antiderivative.

    Second Fundamental Theorem of Calculus

    • Confirms that if F is an antiderivative of f, then ∫[a,b] f(x) dx = F(b) - F(a).

    Mean Value Theorem for Integrals (MVT)

    • States that for a continuous function on [a, b], there exists at least one c in [a, b] such that f(c) equals the average value of f over the interval.

    Integration: Power Rule

    • ∫x^n dx = (x^(n+1))/(n+1) + C, valid for n ≠ -1.

    Integration: Natural Logarithm

    • The antiderivative of u is given by ∫(1/u) du = ln|u| + C.

    Integration: Exponential Functions

    • The integral of e^x is e^x + C, defining the behavior of the exponential function.

    Integration: Trig Functions

    • Trigonometric integrals have specific results; refer to specialized integral tables for detailed calculations.

    Area Between Curves

    • computed by ∫[a,b] (upper function - lower function) dx, represents the region between two curves over an interval.

    Volumes of Solids of Revolution

    • Volume calculated by rotating a region around an axis, typically using the disk or washer method.

    Disk Volume

    • V = π∫[a,b] [f(x)]^2 dx, used for solids obtained by rotating a region around the x-axis.

    Washer Volume along x-axis

    • V = π∫[a,b] ([outer radius]^2 - [inner radius]^2) dx, accounts for hollow regions when rotating.

    Washer Volume along y-axis

    • Similar to the x-axis, but involves swapping variables to cover rotations around the y-axis.

    Cylindrical Shell Volume

    • V = 2π∫[a,b] (radius)(height) dx, useful when rotating around vertical lines.

    Riemann Sum

    • An approximation of the area under a curve using rectangles; involves subdividing the interval into n parts.

    Riemann Sum: Trapezoidal

    • A more accurate method to estimate area by using trapezoids instead of rectangles, enhancing approximation.

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    Test your knowledge of key concepts in AP Calculus AB with these flashcards. Learn important terms such as Trig Identities, the Intermediate Value Theorem, and more. Perfect for quick study sessions and mastering calculus fundamentals.

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