4.6 Related rates

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

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

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

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?

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?

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

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?

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?

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

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

Example V= l w h dV/dt=

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

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?

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

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

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

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

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

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.

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

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

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

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?

P 329 19 20

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?

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?

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.

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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

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

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).

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.

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).

Definitions Antiderivative General antiderivative Arbitrary constant

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.

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.

The constant C is called the arbitrary constant.

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