2.1 The Derivative and the Tangent Line Problem. I’m going nuts over derivatives!!!. Calculus grew out of 4 major problems that European mathematicians were working on in the seventeenth century. 1. The tangent line problem. 2. The velocity and acceleration problem.
The Derivative and the
Tangent Line Problem
I’m going nuts
European mathematicians were working on
in the seventeenth century.
1. The tangent line problem
2. The velocity and acceleration problem
3. The minimum and maximum problem
4. The area problem
(c, f(c)) is the point of tangency and
f(c+ ) – f(c)
is a second point on the graph of f.
Definition of Tangent Line with Slope m
the point (-1,2). Then, find the equation of the
Therefore, the slope
at any point (x, f(x))
is given by m = 2x
What is the slope
at the point (-1,2)?
m = -2
The equation of the tangent line is
y – 2 = -2(x + 1)
line is also used to define one of the two funda-
mental operations of calculus --- differentiation
Definition of the Derivative of a Function
f’(x) is read “f prime of x”
Other notations besides f’(x) include:
the slope of the graph of f at the points (1,1) & (4,2). What happens at the point (0,0)?
slope is ½, and at the point (4,2),
the slope is ¼.
What happens at the point (0,0)?
The slope is undefined, since it produces division
1 2 3 4
The derivative of f at x = c is given by
Example of a point that is not differentiable.
is continuous at x = 2 but let’s
look at it’s one sided limits.
, x is not differentiable at x = 2. Also, the
graph of f does not have a tangent line at the
point (2, 0).
A function is not differentiable at a point at
which its graph has a sharp turn or a vertical
tangent line(y = x1/3 or y = absolute value of x).
Differentiability can also be destroyed by
a discontinuity ( y = the greatest integer of x).