1. 2.4 The Derivative

2. The tangent problem The slope of a line is given by:
The slope of the tangent to f(x)=x2 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), i.e. (3,9):
Even better would be the point (2,4):

3. The tangent problem 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?

4. The tangent problem slope slope at
The slope of the curve at the point is:
Note: This is the slope of the tangent line to the curve at the point.

5. The velocity problem
Consider a graph of displacement (distance traveled) vs. time.
Average velocity can be found by taking:
time (hours)
distance
(miles) B A
The speedometer in your car does not measure average velocity, but instantaneous velocity.
(The velocity at one moment in time.)

6. Derivatives
Definition: The derivative of a function at a number a, denoted by f ′(a), is
if this limit exists.
Example: Find f ′(a) for f(x)=x2+3.

7. Equation of the tangent line
The tangent line to y=f(x) at (a,f(a)) is the line through (a,f(a)) whose slope is equal to f ′(a).
Then the equation of the tangent line to the curve y=f(x) at the point (a,f(a)):
Example: Find an equation of the tangent line to f(x)=x2+3 at (1,4).
From previous slide: f ′(1)=21=2.
Thus, the equation is y-f(1)= f ′(1)(x-1)
y-4=2(x-1) or y=2x+2

8. Rates of Change:
Average rate of change =
Instantaneous rate of change =
These definitions are true for any function.
( x does not have to represent time. )

9. The derivative as a function
In Section 2.1 we considered the derivative at a fixed number a.
Now let number a vary. If we replace number a by a variable x, then the derivative can be interpreted as a function of x :
Alternative notations for the derivative:
D and d / dx are called differentiation operators.
dy / dx should not be regarded as a ratio.

10. The derivative is the slope of the original function.

12. Differentiable functions
A function f is differentiable at a if f ′(a) exists. It is differentiable on an open interval (a,b) [ or (a,) or (- , a) or (- , ) ] if it is differentiable at every number in the interval.
Theorem: If f is differentiable at a, then f is continuous at a.
Note: The converse is false: there are functions that are continuous but not differentiable.
Example: f(x) = | x |

13. To be differentiable, a function must be continuous and smooth.
Derivatives will fail to exist at:
corner
cusp
discontinuity
vertical tangent

14. Higher Order Derivatives:
is the first derivative of y with respect to x.
is the second derivative.
(y double prime)
is the third derivative.
We will learn later what these higher order derivatives are used for.
is the fourth derivative.