Transcript The Derivative and the Tangent Line Problem

2.1 The Derivative and the Tangent
Line Problem
Main Ideas
• Find the slope of the tangent line to a curve at
a point.
• Use the limit definition to find the derivative
of a function.
• Understand the relationship between
differentiability and continuity.
Calculus was developed during the seventeenth
century because of four major problems that
mathematicians of the time could not solve.
1. Tangent line- to a curve
2. Velocity and acceleration
3. Minimum and Maximum
4. Area-under a curve
Each problem involves the idea of a limit.
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Who were the mathematicians?
Pierre de Fermat, Rene Descartes, Christian
Huygens, Isaac Barrow, Isaac Newton, Gottfried
Leibniz
Definition of a tangent line
A line sharing a common point with a curve or surface
and being the closest linear approximation of the
curve or surface at that point.
Tangent Line Problem
Given a function f and a point P on its graph
Find the equation of the tangent line to the graph
at point P.
How can you estimate the slope of the tangent
line?
Secant lines
Definition of Tangent line
If f is defined on an open interval containing c, and if the
limit
exists, then the line passing through (c, f(c)) with slope m
is the tangent line to the graph of f at the point (c, f(c)).
“You have now arrived at a crucial point
in the study of calculus. “ p99
Definition of the Derivative of a function
The derivative of f at x is given by
provided the limit exists. For all x for which this
limit exists, f’ is a function of x.
Other notations used to find the derivative of a
function
Note that f’ is a function of f.
This “new” function f ’ gives the slope of the
tangent line to the entire graph of f or just one
point depending on which version of the
definition you use.
When does a limit fail to exist at a given point?
 Jump at a point
 Asymptote at a point
 Sharp turn at a point (this includes a cusp)
 Vertical tangent line at a point