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

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

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

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

<p>False (B)</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 (A), Cusp (B), Corner (C), Discontinuity (D)</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 (A)</p> Signup and view all the answers

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

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

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

<p>False (B)</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

Flashcards

Intermediate Value Theorem

The Intermediate Value Theorem guarantees that if a continuous function f(x) changes sign between two points, a and b, then there must be at least one point c between a and b where f(c) = 0. This means a solution exists within the interval.

Average Rate of Change

The average rate of change of a function f(x) over an interval [a, b] is the slope of the secant line connecting the points (a, f(a)) and (b, f(b)). It measures the overall change in the function over the interval.

Instantaneous Rate of Change

The instantaneous rate of change of a function f(x) at a point x = a is the slope of the tangent line to the curve at that point. This is equivalent to the value of the derivative f'(a).

Formal Definition of Derivative

The formal definition of the derivative of a function f(x) at a point x = a is the limit as h approaches 0 of [f(a + h) - f(a)]/h. This represents the slope of the tangent line.

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Alternate Definition of Derivative

The alternate definition of the derivative of a function f(x) at a point x = a is the limit as x approaches a of [f(x) - f(a)]/(x - a). It provides another way to calculate the slope of the tangent line.

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Increasing Function

A function f(x) is increasing on an interval if its derivative f'(x) > 0 for all x in that interval. This means the function is getting larger as x increases.

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Decreasing Function

A function f(x) is decreasing on an interval if its derivative f'(x) < 0 for all x in that interval. This means the function is getting smaller as x increases.

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Relative Minimum

A relative minimum of a function f(x) occurs at a point x = c if the derivative f'(x) changes from negative to positive as x passes through c. The function reaches a local minimum at this point.

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Relative Maximum

A relative maximum of a function f(x) occurs at a point x = c if the derivative f'(x) changes from positive to negative as x passes through c. The function reaches a local maximum at this point.

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Concave Up

A function f(x) is concave up on an interval if its derivative f'(x) is increasing for all x in that interval. The graph of the function will look like a cup.

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Concave Down

A function f(x) is concave down on an interval if its derivative f'(x) is decreasing for all x in that interval. The graph of the function will look like an upside-down cup.

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Point of Inflection

A point of inflection is a point on the graph of a function f(x) where the concavity changes. This means the derivative changes from increasing to decreasing, or vice versa.

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Non-differentiable Conditions

A function f(x) is non-differentiable at a point if the following conditions occur:

  1. Corners: The graph has a sharp bend.
  2. Cusps: The graph has a sharp point.
  3. Vertical Tangents: The tangent line is vertical, meaning the slope is undefined.
  4. Discontinuities: The graph has a break or jump.
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Product Rule

The product rule states that the derivative of the product of two functions, u(x) and v(x), is given by: (uv)' = uv' + vu'.

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Quotient Rule

The quotient rule states that the derivative of the quotient of two functions, u(x) and v(x), is given by: (u/v)' = (uv' - vu')/v².

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Chain Rule

The chain rule states that the derivative of a composite function, f(g(x)), is given by: f'(g(x))g'(x). This rule helps you differentiate functions within functions.

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Derivative of sin(x)

The derivative of y = sin(x) is y' = cos(x). Remember that the derivative of a trigonometric function is its cofunction.

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Derivative of cos(x)

The derivative of y = cos(x) is y' = -sin(x). Remember that the derivative of a trigonometric function is its cofunction.

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Derivative of tan(x)

The derivative of y = tan(x) is y' = sec²(x). Remember that the derivative of a trigonometric function is its cofunction.

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Derivative of csc(x)

The derivative of y = csc(x) is y' = -csc(x)cot(x). Remember that the derivative of a trigonometric function is its cofunction.

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Derivative of sec(x)

The derivative of y = sec(x) is y' = sec(x)tan(x). Remember that the derivative of a trigonometric function is its cofunction.

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Derivative of cot(x)

The derivative of y = cot(x) is y' = -csc²(x). Remember that the derivative of a trigonometric function is its cofunction.

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Derivative of sin⁻¹(x)

The derivative of y = sin⁻¹(x) is y' = 1/√(1 - x²). This is a useful formula for differentiating inverse trigonometric functions.

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Derivative of cos⁻¹(x)

The derivative of y = cos⁻¹(x) is y' = -1/√(1 - x²). This is a useful formula for differentiating inverse trigonometric functions.

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Derivative of tan⁻¹(x)

The derivative of y = tan⁻¹(x) is y' = 1/(1 + x²). This is a useful formula for differentiating inverse trigonometric functions.

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Derivative of cot⁻¹(x)

The derivative of y = cot⁻¹(x) is y' = -1/(1 + x²). This is a useful formula for differentiating inverse trigonometric functions.

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Derivative of e^x

The derivative of y = e^x is y' = e^x. This is a unique property of the exponential function.

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Derivative of a^x

The derivative of y = a^x is y' = a^x ln(a). This is a general formula for the derivative of exponential functions.

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Derivative of ln(x)

The derivative of y = ln(x) is y' = 1/x. This is the derivative of the natural logarithm.

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Derivative of log (base a) x

The derivative of y = log (base a) x is y' = 1/(x ln a). This is the general formula for the derivative of logarithmic functions.

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Finding Absolute Maximums and Minimums

To find the absolute maximum and minimum values of a continuous function f(x) on a closed interval [a,b], you need to examine the critical points (where f'(x) = 0 or is undefined) and the endpoints a and b.

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Mean Value Theorem

The Mean Value Theorem states that for a function f(x) that is continuous and differentiable on a closed interval [a, b], there exists at least one point c in (a, b) where the instantaneous rate of change (derivative) is equal to the average rate of change over the interval.

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Linearization

Linearization is the approximation of a function using a tangent line at a specific point. It's used to estimate function values near the point of tangency.

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Rate

A rate is a measure of how a quantity changes over time or with respect to another variable. It is mathematically represented by the derivative of the function.

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Left Riemann Sum

A left Riemann sum estimates the area under a curve by using rectangles whose heights correspond to the left endpoint of each subinterval.

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Right Riemann Sum

A right Riemann sum estimates the area under a curve by using rectangles whose heights correspond to the right endpoint of each subinterval.

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Trapezoidal Rule

The trapezoidal rule uses trapezoids instead of rectangles to approximate the area under a curve. It provides a more accurate approximation than Riemann sums.

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Area of a Trapezoid

The area of a trapezoid is determined by the formula: Area = [(h1 + h2)/2] × base. This formula is used in the trapezoidal rule.

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Trapezoid

Trapezoid is a quadrilateral with two parallel sides, calculated with (h1 + h2) / 2 multiplied by the base.

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Definite Integral

A definite integral of a function f(x) over an interval [a, b], denoted as ∫[a,b]f(x)dx, represents the area under the curve of the function between the limits a and b.

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Indefinite Integral

An indefinite integral of a function f(x), denoted as ∫f(x)dx, represents the family of all antiderivatives of f(x). The result is a function F(x) plus a constant of integration, C.

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Area Under a Curve

The area under a curve of a function f(x) between two points a and b is calculated by integrating the function with respect to x from a to b: Area = ∫[a,b]f(x)dx.

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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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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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