1. Unit 2 - Derivatives

2. Calculus overview
3 main concepts
Limits
Derivatives
Integrals

3. Describe the graph in words. Include
Distance
Average speed
Constant speed
Stops or not moving

4. Describe the graph in words. Include
Distance
Average speed
Constant speed
Stops or not moving

5. Slopes: list as many ways as you can, to describe slope:

6. Derivative Can also be written This will give a numeric value
Formula to calculate slope of a tangent line at any point on the curve.
Can also be written
This will give a numeric value

7. What is the derivative?
The derivative is a “formula” to calculate slopes (ie. Tangents) at any point on a curve.
Therefore if you have a graph, you can estimate the derivative by looking at the slope of the graph at that point

8. Example 1
Estimate the derivative at the points marked on the graph below.

9. Slope of secant line vs. tangent line
As h shrinks to zero, slope of secant line approaches slope of tangent line
nt2-1.html
Slope of tangent line =

10. Slope of secant line vs. tangent line
As h shrinks to zero, slope of secant line approaches slope of tangent line
Slope of tangent line =

11. Slope of secant line vs. tangent line

12. Secant vs. tangent Secant Line Tangent Line
Uses 2 points (a, f(a)) and (b,f(b))
Uses 1 point (c, f(c))
The following are equivalent:
Slope of secant line between [a,b]
Slope of tangent line at x = c
Average rate of change on [a,b]
Instantaneous rate of change at
x=c
f’(c)
Derivative at x = c
dy/dx

13. Example 2 a
For f(x) = x2 , use the definition of the derivative to find the derivative function f’(x), then find the equation of the tangent line to f(x) at x=2.

14. Example 2 b
For f(x) = x2 , use the definition of the derivative to find the derivative function f’(x), then find the equation of the tangent line to f(x) at x=2.

15. Example 3
For g(t) = t/(t+1), use the definition of the derivative to find the derivative function g’(t), then find the equation of the normal line to g(t) at t=1.

16. Example 4
Find the derivative using the alternative from of the definition of f(x)= x2 – 3x at x=1

17. Differentiability
A function f(x) is differentiable at x=c if and only if
L is a finite value
This basically says that for a function to be differentiable at a point, the graph must be continuous but also must connect in a way that the slopes merge into each other smoothly.

18. Example 5
Discuss the differentiability of
at x = 2

19. Example 6
Discuss the differentiability of
at x = 0

20. Differentiability Places a function is not differentiable
Anywhere there is a discontinuity
Corner or cusp
Vertical tangent

21. Local Linearity
A function that is differentiable closely resembles its own tangent line when viewed very closely
In other words, when zoomed in on a few times, a curve will look like a straight line if it is differentiable.

22. Example 7

23. Continuity vs. Differentiability
If f(x) is differentiable at x = c, then f is continuous at x = c.
Be careful! The converse is not true! Being continuous does not imply differentiability (think about the absolute value function)

24. Piecewise function
You must always be careful about piecewise functions
Example 8
For the piecewise function find a piecewise derivative function f’(x) by using the definition of the derivative. Then evaluate the following
What can you say about the differentiability based on your results?