1. 4.6 Related rates

2. Useful formulae a^2 +b^2 = c^2 Cube V= s^3 Sphere V= 4/3 pi r^3
SA=4 pi r^2
Cone V= 1/3 pi r^2 h
Lateral SA= pi r (r^2 + h^2)^(1/2)
Right circular cylinder V=pi r^2 h
Lateral SA= 2 pi r h
Circle A= pi r^2
C= 2 pi r

3. triples
3,4,5
5,12,13
6,8,10
7,24,25
8,15,17
9,12,15

4. Implicit differentiation
Change wrt time
Each changing quantity is differentiated wrt time.

5. Example the radius of a circle is increasing at 0. 03 cm/sec
Example the radius of a circle is increasing at 0.03 cm/sec. What is the rate of change of the area at the second the radius is 20 cm?

6. Example
A circle has area increasing at 1.5 pi cm^2/min. what is the rate of change of the radius when the radius is 5 cm?

7. Example Circle Area decreasing 4.8 pi ft^2/sec
Radius decreasing 0.3 ft/sec
Find radius

8. Example
What is the radius of a circle at the moment when the rate of change of its area is numerically twice as large as the rate of change of its radius?

9. Example
The length of a rectangle is decreasing at 5 cm/sec. And the width is increasing at 2 cm/sec. What is the rate of change of the area when l=6 and w=5?

10. Same rectangle Find rate of change of perimeter
Find rate of change of diagonal

11. Example
The edges of a cube are expanding at 3 cm/sec. How fast is the volume changing when:
e= 1 cm
e=10 cm

12. Example
V= l w h
dV/dt=

13. Example
A 25 ft ladder is leaning against a house. The bottom is being pulled out from the house at 2 ft/sec.

14. Part a
How fast is the top of the ladder moving down the wall when the base is 7 ft. from the end of the ladder?

15. Part b
Find the rate at which the area of the triangle formed is changing when the bottom is 7 ft. from the house.

16. Part c
Find the rate at which the angle between the top of the ladder and the house changes.

17. Spherical soap bubble r= 10 cm air added at 10 cm^2/sec.
Find rate at which radius is changing.

18. Rectangular prism Length increasing 4 cm/sec
Height decreasing 3 cm/sec
Width constant
When l=4.w=5,h=6
Find rate of change of SA

19. Cylindrical tank with circular base
Drained at 3 l/sec
Radius=5
How fast is the water level dropping?

20. Cone-shaped cup Being filled with water at 3 cm^3/sec H=10, r=5
How fast is water level rising when level is 4 cm.

21. Cone, r=7,h=12 Draining at 15 m^3/sec When r=3
How fast is the radius changing?

22. Cone, r=10, h=7 Filled at 2 m^3/sec H=5m
How fast is the radius changing?

23. Water drains from cone at the rate of 21 ft^3/min
Water drains from cone at the rate of 21 ft^3/min. how fast is the water level dropping when the height is 5 ft?
Cone, r=3, h=8

24. A hot-air balloon rises straight up from a level field.
It is tracked by a range-finder 500 ft from lift-off. When the range-finder’s angle of elevation is pi/4, the angle increases at 0.14 rad/min. How fast is the balloon rising?

25. P 329 19 20

26. How fast is the tip of her shadow moving?
A 5 ft girl is walking toward a 20 ft lamppost at the rate of 6 ft/sec.
How fast is the tip of her shadow moving?

27. A 6 ft man is moving away from the base of a streetlight that is 15 ft high.
If he moves at the rate of 18 ft/sec., how fast is the length of his shadow changing?

28. A balloon rises at 3 m/sec
A balloon rises at 3 m/sec. from a point on the ground 30 m from an observer.
Find rate of change of the angle of elevation of the balloon from the observer when the balloon is 30 m above ground.

29. P 326 30 32 31

30. Sure you remember!!! f’ ( c ) = f(b)-f(a) b-a
4.7 Mean Value Theorem
Sure you remember!!!
f’ ( c ) = f(b)-f(a)
b-a

31. 4.7 Mean Value Theorem Sure you remember!!! And
Corollary 1 is the first derivative test for increasing and decreasing.

32. Corollary 2
If f’(x)=0 for all x in (a,b) then there is a constant ,c, such that
f (x) = c,
for all x in (a,b).

33. This is the converse of : the derivative of a constant is zero.
Corollary 2
This is the converse of :
the derivative of a constant is zero.

34. Corollary 3
If F’(x)=G’(x) at each x in (a,b), then there is a constant,c, such that
F(x)=G(x)+c for all x in (a,b).

35. Definitions
Antiderivative
General antiderivative
Arbitrary constant

36. At every point of the interval.
antiderivative
A function F is an anti-derivative of a function f over an interval I if
F’(x)=f(x)
At every point of the interval.

37. General antiderivative
If F is an antiderivative of f, then the family of functions F(x)+C (C any real no.) is the general antiderivative of f over the interval I.

38. The constant C is called the arbitrary constant.

39. 4.7 Initial value problems
Uses general antiderivatives
With “initial values”
To find the specific function of the family