Ap Calc AB Unit 5

Mean Value Theorem (MVT):

If a function is continuous on [a,b] and differentiable on (a,b), then there exists a value c in (a,b) such that f′(c) = (f(b) − f(a)) / (b − a).

Average Rate of Change:

The change in a function’s value over an interval, calculated as (f(b) − f(a)) / (b − a).

Instantaneous Rate of Change:

The rate of change of a function at a single point, given by the derivative.

Geometric Meaning of the Mean Value Theorem:

There exists at least one point where the tangent line is parallel to the secant line over the interval.

Extreme Value Theorem (EVT):

If a function is continuous on a closed interval [a,b], then it has an absolute maximum and an absolute minimum on that interval.

Absolute Maximum:

The greatest value of a function on a given interval.

Absolute Minimum:

The least value of a function on a given interval.

Local (Relative) Maximum:

A point where the function value is greater than the values of the function nearby.

Local (Relative) Minimum:

A point where the function value is less than the values of the function nearby.

Critical Point:

A value of x where f′(x) = 0 or f′(x) does not exist, provided x is in the domain of the function.

Increasing Interval:

An interval where f′(x) > 0 and the function values increase.

Decreasing Interval:

An interval where f′(x) < 0 and the function values decrease.

Sign Chart:

A diagram used to determine where a derivative is positive or negative.

First Derivative Test:

A test that uses the sign of f′(x) before and after a critical point to classify local extrema.

Positive to Negative Sign Change:

Indicates a local maximum using the First Derivative Test.

Negative to Positive Sign Change:

Indicates a local minimum using the First Derivative Test.

Candidates Test:

A method for finding absolute extrema by evaluating the function at critical points and endpoints.

Candidates:

The critical points and endpoints where absolute extrema may occur.

Concave Up:

A function is concave up on an interval where f″(x) > 0.

Concave Down:

A function is concave down on an interval where f″(x) < 0.

Inflection Point:

A point where the concavity of a function changes and f″(x) is zero or undefined.

Second Derivative Test:

A test that uses the value of f″(c) at a critical point to determine whether a local extremum exists.

Second Derivative Test:

f″(c) > 0: The function has a local minimum at c.

Second Derivative Test:

f″(c) < 0: The function has a local maximum at c.

Second Derivative Test:

f″(c) = 0: The test is inconclusive.

Horizontal Tangent:

A tangent line with slope zero that occurs where f′(x) = 0.

Relationship Between f and f′:

If f′(x) is positive then f is increasing; if f′(x) is negative then f is decreasing.

Relationship Between f′ and f″:

If f″(x) is positive then f′ is increasing; if f″(x) is negative then f′ is decreasing.

Optimization Problem:

A problem that involves finding the maximum or minimum value of a real-world quantity.

Objective Function:

The function that represents the quantity being optimized.

Constraint:

An equation that limits the possible values of the variables in an optimization problem.

Feasible Domain:

The set of all values that satisfy the constraints of an optimization problem.

Overall Behavior of a Function:

Includes increasing and decreasing intervals, extrema, concavity, inflection points, and end behavior.

End Behavior:

The behavior of a function as x approaches positive or negative infinity.

when f‘(x) is increasing

f(x) is concave up

when f‘(x) is decreasing

f(x) is concave down

when f‘(x) is positive

f(x) is increasing

when f‘(x) is negative

f(x) is decreasing

when f’’(x) is positive

f(x) is concave up

when f’’(x) is negative

f(x) is concave down