Fundamental theorem of calculus Derivative and integral Limits Commentary

James Gregory, Isaac Barrow mathematics

Fundamental theorem of calculus

• Early Greek and Chinese mathematicians
• used the “method of exhaustion,”
• determining a value by disproving a series of approximations.
• This method foreshadowed the concept of a limit.
• A limit is the value
• that the result of a function approaches
• as its input approaches a given value.
• Determining limits gave European mathematicians
• the ability to prove the continuity of functions,
• and to determine their differentials and integrals.
• James Gregory and Isaac Barrow showed
• that the derivative and the integral
• are inverses of each other.
• The derivative of velocity is acceleration.
• The integral of acceleration is the velocity.

Derivative and integral

• The derivative of a continuous function
• is the slope of the tangent at a point
• which is the same as the rate of change.
• For a function giving velocity over time,
• the derivative is acceleration at an instant.
• You can derive the derivative
• by considering the slope of a line between
• a target point and a nearby point on the curve
• and finding its slope when the distance
• between the points approaches zero.
• The integral of a continuous function
• is the area under the curve.
• For a function giving the velocity over time,
• because distance is velocity times time,
• the integral is the distance covered.
• You can derive the integral
• by considering adjacent rectangular slices
• whose sum approximates the total area under the curve
• and seeing what you get when the widths
• of the rectangles approaches zero.

Limits

• Nearly everything has a limit
• and it’s generally not infinity.
• There are infinite points between zero and one,
• but you can measure their span with a ruler.
• Here I try not to be pessimistic,
• but there’s a limit to optimism.
• There’s a limit to how much I can afford.
• There’s a limit to how much I can stand.
• You live only once. Knowledge is power,
• but there’s a limit to it. Even limits have limits.
• A man’s reach should exceed his grasp,
• but there ought to be a limit to stupidity.