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3.6 Chain rule. When gear A makes x turns, gear B makes u turns and gear C makes y turns. , . u turns 3 times as fast as x. So y turns 3/2 as fast as x. y turns ½ as fast as u. Rates are multiplied. The Chain Rule for composite functions.

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3 6 chain rule l.jpg
3.6 Chain rule

When gear A makes x turns, gear B makes u turns and gear C makes y turns.,

u turns 3 times as fast as x

So y turns 3/2 as fast as x

y turns ½ as fast as u

Rates are multiplied


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The Chain Rule for composite functions

If y = f(u) and u = g(x) then y = f(g(x)) and

multiply rates

multiply rates





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Outside/Inside method of chain rule

outside

inside

derivative of inside

derivative of outside wrt inside

think of g(x) = u


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Outside/Inside method of chain rule example

outside

derivative of inside

inside

derivative of outside wrt inside


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Outside/Inside method of chain rule

outside

inside

derivative of inside

derivative of outside wrt inside


Outside inside method of chain rule9 l.jpg
Outside/Inside method of chain rule

outside

inside

derivative of inside

derivative of outside wrt inside


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More derivatives with the chain rule

product

Simplify terms

Combine with

common denominator



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Radians Versus Degrees

The formulas for derivatives assume x is in radian measure.

sin (x°) oscillates only /180 times as oftenas sin (x)

oscillates. Its maximum slope is /180.

d/dx[sin (x)] = cos (x)

d/dx [sin (x°) ] = /180 cos (x°)


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3.7 Implicit Differentiation

Although we can not solve explicitly for y, we can assume that y is some function of x and use implicit differentiation to find the slope of the curve at a given point

y=f (x)


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If y is a function of x then its derivative is

y2 is a function of y, which in turn is a function of x.

using the chain rule:

Find the following derivatives wrt x

Use product rule


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Implicit Differentiation

  • Differentiate both sides of the equation with respect to x, treating y as a function of x. This requires the chain rule.

2. Collect terms with dy/dx on one side of the equation.

3. Factor dy/dx

4. Solve for dy/dx


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Use Implicit Differentiation

Find equations for the

tangent and normal to the curve at (2, 4).

find the slope of the tangent at (2,4)

find the slope of the normal at (2,4)


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Solution

  • Differentiate both sides of the equation with respect to x, treating y as a function of x. This requires the chain rule.

  • Collect terms with dy/dx on one side of the equation.

  • Factor dy/dx

  • 4. Solve for dy/dx


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Find dy/dx

1. Write the equation of the tangent line at (0,1)

2. Write the equation of the normal line at (0,1)


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Solution

  • Differentiate both sides of the equation with respect to x, treating y as a function of x. This requires the chain rule.

  • Collect terms with dy/dx on one side of the equation.

  • Factor dy/dx

  • 4. Solve for dy/dx


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Find dy/dx

1. Write the equation of the tangent line at (0,1)

2. Write the equation of the normal line at (0,1)


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3.8 Higher Derivatives

The derivative of a function f(x) is a function itself f ´(x). It has a derivative, called the second derivative f ´´(x)

If the function f(t) is a position function, the first derivative f ´(t)is a velocity function and the second derivative f ´´(t) is acceleration.

The second derivative has a derivative (the third derivative) and the third derivative has a derivative etc.


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Find the second derivative for

Find the third derivative for


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In algebra we study relationships among variables

  • The volume of a sphere is related to its radius

  • The sides of a right triangle are related by Pythagorean Theorem

  • The angles in a right triangle are related to the sides.

In calculus we study relationships between the rates of change of variables.

How is the rate of change of the radius of a sphere related to the rate of change of the volume of that sphere?


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3.9

Examples of rates-assume all variables are implicit functions of t = time

Rate of change in radius of a sphere

Rate of change in volume of a sphere

Rate of change in length labeled x

Rate of change in area of a triangle

Rate of change in angle,


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Solving Related Rates equations

  • Read the problem at least three times.

  • Identify all the given quantities and the quantities to be found (these are usually rates.)

  • Draw a sketch and label, using unknowns when necessary.

  • Write an equation (formula) that relates the variables.

  • ***Assume all variables are functions of time and differentiate wrt time using the chain rule. The result is called the related rates equation.

  • Substitute the known values into the related rates equation and solve for the unknown rate.


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Related Rates

Figure 2.43: The balloon in Example 3.

A hot-air balloon rising straight up from a level field is tracked by a range finder 500 ft from the liftoff point. The angle of elevation is increasing at the rate of 0.14 rad/min. How fast is the balloon rising when the angle of elevation is is /4?

Given:

Find:


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Related Rates

Figure 2.43: The balloon in Example 3.

A hot-air balloon rising straight up from a level field is tracked by a range finder 500 ft from the liftoff point. At the moment the range finder’s elevation angle is /4, the angle is increasing at the rate of 0.14 rad/min. How fast is the balloon rising at that moment?


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Related Rates

Figure 2.44: Figure for Example 4.

A police cruiser, approaching a right angled intersection from the north is chasing a speeding car that has turned the corner and is now moving straight east. The cruiser is moving at 60 mph and the police determine with radar that the distance between them is increasing at 20 mph. When the cruiser is .6 mi. north of the intersection and the car is .8 mi to the east, what is the speed of the car?

Given:

Find:


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Figure 2.44: Figure for Example 4.

Given:

Find:

then s = 1


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Related Rates

Figure 2.45: The conical tank in Example 5.

Water runs into a conical tank at the rate of 9 ft3/min. The tank stands point down and has a height of 10 ft and a base of radius 5 ft. How fast is the water level rising when the water is 6 ft. deep?

Given:

Find:


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Water runs into a conical tank at the rate of 9 ft3/min. The tank stands point down and has a height of 10 ft and a base of radius 5 ft. How fast is the water level rising when the water is 6 ft. deep?

Figure 2.45: The conical tank in Example 5.

Given:

Find:

x=3


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3. .10 The more we magnify the graph of a functionnear a point where the function is differentiable, the flatter the graph becomes and the more it resembles its tangent.

Differentiability



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Approximating the change in the function f by

the change in the tangent line of f.

Linearization



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