1. 2.1 The Derivative and the Tangent Line Problem
Devil’s Tower, Wyoming
2.1 The Derivative and the Tangent Line Problem

2. 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
3. The minimum and maximum problem
4. The area problem

3. The slope of a line is given by:
The slope at (1,1) can be approximated by the slope of the secant through (4,16).
We could get a better approximation if we move the point closer to (1,1). ie: (3,9)
Even better would be the point (2,4).

4. The slope of a line is given by:
If we got really close to (1,1), say (1.1,1.21), the approximation would get better still
How far can we go?

5. The tangent line problem
secant line
(c, f(c))
(c, f(c)) is the point of tangency and
is a second point on the graph of f.

6. The slope between these two points is
Definition of Tangent Line with Slope m

7. Find the slope of the graph of f(x) = x2 +1 at the point (-1,2)
Find the slope of the graph of f(x) = x2 +1 at the point (-1,2). Then, find the equation of the
tangent line.
(-1,2)

8. f(x) = x2 + 1
Therefore, the slope at any point (x, f(x)) is given by m = 2x
What is the slope at the point (-1,2)?
The equation of the tangent line is
y – 2 = -2(x + 1)
m = -2

9. Find the slope of the tangent line of at the point (1, 2).

10. The limit used to define the slope of a tangent line is also used to define one of the two
fundamental 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:

11. Find f’(x) for f(x) = and use the result to find the slope of the graph of f at the points (1,1) & (4,2). What happens at the point (0,0)?

12. Therefore, at the point (1,1), the slope is ½ , and at the
What happens at the point (0,0)?
The slope is undefined, since it produces division by zero.

13. Find the derivative with respect to t for the function

14. Alternate Form of the Derivative
The derivative of f at x = c is given by
(x, f(x))
(c, f(c)) c x

15. Derivative from the left and from the right.
Example of a point that is not differentiable.
is continuous at x = 2 but let’s look at it’s one sided limits. -1 1

16. The 1-sided limits are not equal.
, 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).