# The Fundamental Theorem of Calculus

# The Fundamental Theorem of Calculus Terms

## Get down with the lingo

### Fundamental Theorem Of Calculus

This theorem provides a relation between the derivative and the definite integral of a function.### Second Fundamental Theorem Of Calculus

This theorem provides a method of differentiating an integral.### Integration By Parts

A technique for performing integration where .### Improper Integral

An definite integral is called an improper integral when the limits of integration are infinite (*a*=-∞,

*b*=∞) or the function becomes unbounded in [

*a*,

*b*].

### Reimann Sum

The Reimann Sum of a function*f*(

*x*) over an interval [

*a*,

*b*] is defined as where

*x*and

_{i-1 }< x_{i}^{* }< x_{i},*x*for

_{i}*i*= 1,…,

*n*divides the interval [

*a*,

*b*] into

*n*subintervals.

### Left-Hand Sum

The Left-Hand Sum of a function*f*(

*x*) over an interval [

*a*,

*b*] is defined as where

*x*for

_{i}*i*= 1,…,

*n*divides the interval [

*a*,

*b*] into

*n*subintervals.

### Right-Hand Sum

The Right-Hand Sum of a function*f*(

*x*) over an interval [

*a*,

*b*] is defined as where

*x*for

_{i}*i*= 1,…,

*n*divides the interval [

*a*,

*b*] into

*n*subintervals.

### Midpoint Sum

The Midpoint Sum of a function*f*(

*x*) over an interval [

*a*,

*b*] is defined as where , and

*x*for

_{i}*i*= 1,…,

*n*divides the interval [

*a*,

*b*] into

*n*subintervals.

### Trapezoid Sum

The trapezoid sum is the average between the Left-Hand and Right-Hand Sum.### Average Value Of A Function

Average value of a continuous function f on the closed interval [*a*,

*b*] is defined as .

### Concavity

This describes whether the function is curving up, down or not curving at all.### Critical Point

The derivative of the function at the critical point is 0.### Inflection Point

The point (*x*-value) where the function changes concavity.

### Secant Line

A line joining two points on the graph of a function.### Tangent Line

A line that touches the graph of a function*f*(

*x*) at a point.

### Differentiability

If the limit exists, the function*f*(

*x*) is differentiable at

*x*=

*a*.