2.1 The Derivative and the Tangent Line Problem

Devil’s Tower, Wyoming

Calculus grew out of 4 major problems that European mathematicians were working onin the seventeenth century.

1.The tangent line problem

2. The velocity and acceleration problem

3. The minimum and maximum problem

4. The area problem

The slope of a line is given by:y

mx

x

y

The slope at (1,1) can be approximated by the slope of the secant through (4,16).

y

x

16 1

4 1

15

3 5

We could get a better approximation if we move the point closer to (1,1). ie: (3,9)

y

x

9 1

3 1

8

2 4

Even better would be the point (2,4).

y

x

4 1

2 1

3

1 3

2f x x

The slope of a line is given by:y

mx

x

y

If we got really close to (1,1), say (1.1,1.21), the approximation would get better still

y

x

1.21 1

1.1 1

.21

.1 2.1

How far can we go?

2f x x

The tangent line problem

(c, f(c))

secant line

(c, f(c)) is the point of tangency and is a second point on the graph of f.

The slope between these two points is

Definition of Tangent Line with Slope m

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)

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)

f(x) = x2 + 1

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

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:

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)?

Therefore, at the point (1,1), the 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 by zero.

1 2 3 4

Find the derivative with respect to t for the function

Alternate Form of the DerivativeThe derivative of f at x = c is given by

(c, f(c))

c x

(x, f(x))

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

The 1-sided limits are not equal.

, x is not differentiable at x = 2. Also, thegraph of f does not have a tangent line at the point (2, 0).

A function is not differentiable at a point atwhich its graph has a sharp turn or a verticaltangent line(y = x1/3 or y = absolute value of x). Differentiability can also be destroyed by a discontinuity ( y = the greatest integer of x).