2. 2 Differentiation 2.1 TANGENT LINES AND VELOCITY 2.2 THE DERIVATIVE
2.3
COMPUTATION OF DERIVATIVES: THE POWER RULE
2.4
THE PRODUCT AND QUOTIENT RULES
2.5
THE CHAIN RULE
2.6
DERIVATIVES OF TRIGONOMETRIC FUNCTIONS
2.7
IMPLICIT DIFFERENTIATION
2.8
THE MEAN VALUE THEOREM
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Slide 2

3. TANGENT LINES AND VELOCITY
2.1
TANGENT LINES AND VELOCITY
Secant Lines and Tangent Lines
Consider the curve y = x2 + 1.
A secant line is a line between a pair of points on the curve.
The secant line shown here has a slope
And the secant line has the equation
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Slide 3

4. TANGENT LINES AND VELOCITY
2.1
TANGENT LINES AND VELOCITY
Secant Lines and Tangent Lines
Choose the second point to be
much closer to (1, 2), say,
(1.05, ).
It is reasonable to ask, “What is
the slope and the equation of the line as the second point approaches the first one?”
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Slide 4

5. TANGENT LINES AND VELOCITY
2.1
TANGENT LINES AND VELOCITY
Secant Lines and Tangent Lines
Continue this process by computing the slope of the secant line joining (1, 2) and the unspecified point
(1 + h, f (1 + h)), for some value of h close to 0 (but h = 0).
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Slide 5

6. TANGENT LINES AND VELOCITY
2.1
TANGENT LINES AND VELOCITY
Secant Lines and Tangent Lines
Notice that as h approaches 0, the slope of the secant line approaches 2, which we define to be the slope of the tangent line, or the line tangent to the curve at the point (1,2).
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Slide 6

7. TANGENT LINES AND VELOCITY
2.1
TANGENT LINES AND VELOCITY
2.1
Tangent lines may intersect a curve at more than one point.
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Slide 7

8. TANGENT LINES AND VELOCITY
2.1
TANGENT LINES AND VELOCITY
The General Case
The difference quotient.
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Slide 8

9. TANGENT LINES AND VELOCITY
2.1
TANGENT LINES AND VELOCITY
1.1
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Slide 9

10. TANGENT LINES AND VELOCITY
2.1
TANGENT LINES AND VELOCITY
1.1
Finding the Equation of a Tangent Line
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Slide 10

11. TANGENT LINES AND VELOCITY
2.1
TANGENT LINES AND VELOCITY
1.1
Finding the Equation of a Tangent Line
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Slide 11

12. TANGENT LINES AND VELOCITY
2.1
TANGENT LINES AND VELOCITY
1.1
Finding the Equation of a Tangent Line
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Slide 12

13. TANGENT LINES AND VELOCITY
2.1
TANGENT LINES AND VELOCITY
1.2
Tangent Line to the Graph of a Rational Function
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Slide 13

14. TANGENT LINES AND VELOCITY
2.1
TANGENT LINES AND VELOCITY
1.2
Tangent Line to the Graph of a Rational Function
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Slide 14

15. TANGENT LINES AND VELOCITY
2.1
TANGENT LINES AND VELOCITY
1.2
Tangent Line to the Graph of a Rational Function
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Slide 15

16. TANGENT LINES AND VELOCITY
2.1
TANGENT LINES AND VELOCITY
1.3
Graphical and Numerical Approximation
of Tangent Lines
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Slide 16

17. TANGENT LINES AND VELOCITY
2.1
TANGENT LINES AND VELOCITY
1.3
Graphical and Numerical Approximation
of Tangent Lines
A reasonable estimate of the slope of the tangent line at the point (0, −1) is then 2.
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Slide 17

18. TANGENT LINES AND VELOCITY
2.1
TANGENT LINES AND VELOCITY
1.3
Graphical and Numerical Approximation
of Tangent Lines
A reasonable estimate of the slope of the tangent line at the point (0, −1) is then 2.
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Slide 18

19. TANGENT LINES AND VELOCITY
2.1
TANGENT LINES AND VELOCITY
Velocity
Suppose that the function s(t) gives the position at time t of an object moving along a straight line.
That is, s(t) gives the displacement (signed distance) from a
fixed reference point, so that s(t) < 0 means that the object is located |s(t)| away from the reference point, but in the negative direction.
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Slide 19

20. TANGENT LINES AND VELOCITY
2.1
TANGENT LINES AND VELOCITY
Velocity
Then, for two times a and b (where a < b), s(b) − s(a) gives the signed distance between positions s(a) and s(b). The average velocity vavg is then given by
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Slide 20

21. TANGENT LINES AND VELOCITY
2.1
TANGENT LINES AND VELOCITY
1.4
Finding Average Velocity
The position of a car after t minutes driving in a straight line is given by
where s is measured in miles and t is measured in minutes. Approximate the velocity at time t = 2.
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Slide 21

22. TANGENT LINES AND VELOCITY
2.1
TANGENT LINES AND VELOCITY
1.4
Finding Average Velocity
Averaging over the 2 minutes from t = 2 to t = 4, we get
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Slide 22

23. TANGENT LINES AND VELOCITY
2.1
TANGENT LINES AND VELOCITY
1.4
Finding Average Velocity
We get an improved approximation by averaging over
just one minute:
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Slide 23

24. TANGENT LINES AND VELOCITY
2.1
TANGENT LINES AND VELOCITY
1.4
Finding Average Velocity
It stands to reason that, if we compute the average velocity over the time interval [2, 2 + h] (where h > 0) and then let h → 0, the resulting average velocities should be getting closer and closer to the velocity at the
instant t = 2.
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Slide 24

25. TANGENT LINES AND VELOCITY
2.1
TANGENT LINES AND VELOCITY
1.4
Finding Average Velocity
It appears that the average
velocity is approaching 1 mile/minute (60 mph), as h → 0.
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Slide 25

26. TANGENT LINES AND VELOCITY
2.1
TANGENT LINES AND VELOCITY
1.2
If s(t) represents the position of an object relative to some fixed location at time t as the object moves along a straight line, then the instantaneous velocity at time
t = a is given by
provided the limit exists. The speed is the absolute value of the velocity.
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Slide 26

27. TANGENT LINES AND VELOCITY
2.1
TANGENT LINES AND VELOCITY
1.5
Finding Average and Instantaneous Velocity
Suppose that the height of a falling object t seconds after being dropped from a height of 64 feet is given by
s(t) = 64 − 16t2 feet.
Find the instantaneous velocity at time t = 2.
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Slide 27

28. TANGENT LINES AND VELOCITY
2.1
TANGENT LINES AND VELOCITY
1.5
Finding Average and Instantaneous Velocity
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Slide 28

29. TANGENT LINES AND VELOCITY
2.1
TANGENT LINES AND VELOCITY
1.5
Finding Average and Instantaneous Velocity
The negative velocity indicates that the object is moving in the negative (or downward) direction.
The speed of the object is 64 ft/s.
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Slide 29

30. TANGENT LINES AND VELOCITY
2.1
TANGENT LINES AND VELOCITY
Velocity
Note that the instantaneous velocity at time t = 2 corresponds to the slope of the tangent line at t = 2 on the graph of s(t) vs. t.
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Slide 30

31. TANGENT LINES AND VELOCITY
2.1
TANGENT LINES AND VELOCITY
Rates of Change
Velocity is a rate (more precisely, the instantaneous rate of change of position with respect to time). In general, the average rate of change of a function f between x = a and
x = b (a ≠ b) is given by
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Slide 31

32. TANGENT LINES AND VELOCITY
2.1
TANGENT LINES AND VELOCITY
Rates of Change
The instantaneous rate of change of f at x = a is given by
provided the limit exists. The units of the instantaneous rate of change are the units of f divided by (or “per”) the units of x.
Recognize this limit as the slope of the tangent line to
y = f (x) at x = a.
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Slide 32

33. TANGENT LINES AND VELOCITY
2.1
TANGENT LINES AND VELOCITY
1.7
A Graph with No Tangent Line at a Point
Determine whether there is a tangent line to the graph of y = |x| at x = 0.
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Slide 33

34. TANGENT LINES AND VELOCITY
2.1
TANGENT LINES AND VELOCITY
1.7
A Graph with No Tangent Line at a Point
The tangent line does not exit.
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Slide 34