Mastering the Fundamental Theorem of Calculus (Unit 6)

Fundamental Theorem of Calculus (FTC)

The theorem that connects differential calculus and integral calculus.

Accumulation Function

A function defined by a definite integral with a variable upper limit.

Derivative of the Accumulation Function

The derivative of g(x) = ∫_a^x f(t) dt is f(x).

Chain Rule

A rule used to differentiate composite functions.

Definite Integral

An integral that has specified limits of integration.

Net Signed Area

The area calculated by considering the sign of the function relative to the axis.

Increasing Function (g(x))

g(x) is increasing when f(t) is positive.

Decreasing Function (g(x))

g(x) is decreasing when f(t) is negative.

Local Maximum of g(x)

Occurs when f(t) changes from positive to negative.

Local Minimum of g(x)

Occurs when f(t) changes from negative to positive.

Concave Up

g(x) is concave up when f(t) is increasing.

Concave Down

g(x) is concave down when f(t) is decreasing.

Point of Inflection

Occurs when f(t) changes from increasing to decreasing or vice versa.

Zero Width Interval

The integral of a function over the same limits is zero: ∫_a^a f(x) dx = 0.

Reversing Limits of Integration

Switching the bounds of an integral changes its sign: ∫b^a f(x) dx = -∫a^b f(x) dx.

Constant Multiple Property

The integral of a constant times a function is the constant multiplied by the integral of the function.

Additivity Property

The integral can be split over an interval: ∫a^b f(x) dx = ∫a^c f(x) dx + ∫_c^b f(x) dx.

Evaluation Theorem

For a continuous function f on [a, b], the integral can be evaluated as F(b) - F(a) where F is an antiderivative of f.

Evaluation Notation

A shorthand notation used in integrals: ∫_a^b f(x) dx = F(x) | _a^b = F(b) - F(a).

Common Mistake - Chain Rule

Forgetting to apply the chain rule when differentiating an integral with a variable upper limit.

Common Mistake - Increasing vs. Decreasing

Confusing the slope of f(t) to determine if g(x) is increasing; g(x) increases when f(t) is positive.

Order of Subtraction in FTC Part 2

The correct subtraction is always F(Upper) - F(Lower).

Area below the x-axis

In definite integrals, area below the axis is considered negative.

Total Area vs. Value of the Integral

Total area refers to the absolute value of areas below the axis; value of the integral includes negatives.

Worked Example: Additivity

Use additivity property to solve for missing integral values based on known integrals.