1a1b
Table of Trig Functions
Obtain co-functions in right column from left column by replacing and .
Two Famous Triangles
sketch 45°-45°-90° triangle, label angles and sides
Pythagorean theorem
or
sketch 30°-60°-90° triangle, label angles and sides
Pythagorean theorem
or
Derivatives of Trig Functions
where .
similarly, obtain the following table (to memorize)
to obtain the right hand column from the left hand column, replace functions by co-functions and multiply by .
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Example. Find an equation of the line tangent to the curve
at the point .
Solution.
Point-slope form of tangent line
Need slope
where
equation for tangent line
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§2.5 The Chain Rule
Composition
Example. Consider
Let ,
is the composition of and
Function machines:
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Notation for composition
is the outer function,
is the inner function
Derivatives of Compositions
Let
or , .
The Chain Rule
If is differentiable at and is differentiable at , then
The derivative of a composition of two functions is the product of their derivatives
Write another way
Leibniz notation
Let and and
then
Easy to remember: looks as though a term “du” cancels on the right hand side!
Example. Let . Find .
Write as a composition
where
apply the chain rule
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Example. Let . Find .
Write as a composition
where
apply the chain rule
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?? Class Practice
Let . Find .
Let . Find .
Let
Plausibility Argument for the Chain Rule
Recall the limit definition of derivative
Change notation
Let
then
Statement of Chain Rule
Let or with
If is differentiable at , and is differentiable at , then
or
Plausibility Argument
Let be the change in corresponding to .
The corresponding change in is
Then
as long as
product rule for limits
but as , so
The only problem is that we may have . ■
Generalized Power Rule
Combine the power rule and the chain rule
Consider
write as a composition
apply the chain rule
this is frequently written
Example. Consider
this has the form given above, with
then
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Example.
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WARNING It is a common mistake to forget the factor !
Compositions of Three Functions
Consider
write as a composition
chain rule
it looks as though ‘’ and ‘’ factors cancel
Example
write as a composition
chain rule
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Example
Find the line tangent to the graph of at .
Solution. Point slope form of a tangent line
,
slope?
gives the tangent line
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?? Class practice differentiating!
Find
1.
2.
3.
4.
5.
6.
§2.6 Implicit Differentiation
circle of radius 1
-, -, circle
defines implicitly as a function of
solve for :
two possibilities
[1a]
[1b]
derivatives
[2a][2a]
[2b]
UsingImplicit Differentiation is easier in many cases!
[3][3]
regard as a function of
or
recall the generalized power rule
write this as
must remember !
differentiate equation [3] implicitly with respect to
solve for
from [1], two possibilities
these match [2]!
Example. Find if
.[1]
Solution. Regard as a function of
Use the generalized power rule
differentiate [1] implicitly with respect to
solve for
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Example. Find the equation of the line tangent to the curve
[1]
at .
Solution. Recall .
think of as a function of
differentiate [1] implicitly
solve for
slope of curve at
point-slope form of tangent line
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?? Class practice with implicit differentiation
1. Let , where is a constant. Find .
Solution
2. Let Find .
Solution
Find an equation of the line tangent to the following curve at the given point.
at the point
Solution. Differentiate implicitly with respect to
Solve for
evaluate at and
point slope form of tangent line
in slope intercept form
show Maple image of curve and tangent line ■
§2.7 Related Rates
Example. A descending balloonist lets helium escape at a rate of 10 cubic feet / minute. Assume the balloon is spherical. How fast is the radius decreasing when the radius is 10 feet?
sketch balloon, basket, escaping gas
Given
volume of balloon
radius of balloon feet
Find when
Equation relating these quantities
[1]
differentiate with respect to time using the chain rule
generalized power rule:
consider radius a function of time
Differentiate [1] with respect to time to get
Solve for
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Problem Solving Strategy
- Read problem carefully
- Draw picture if possible
- Assign notation to given information and unknown rate
- Write down an equation relating the given information and the function whose rate is unknown
- Differentiate with respect to time
- Solve for the unknown rate
Example. The length of a rectangle is increasing at a rate of 7 cm/s and its width is increasing at a rate of 7 cm/s. When the length is 6cm and the width is 4 cm, how fast is the area of the rectangle increasing?
2. picture
3. Given information
cm, cm
cm/s, cm/s
Unknown rate is , where .
4.
5.
6.
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Example. One end of a 13 foot ladder is on the ground and the other end rests on a vertical wall. If the bottom end is drawn from the wall at 3 feet/second, how fast is the top of the ladder sliding down the wall when the bottom is 5 feet from the wall?
-, -, ladder betw., 13 ft
given information
ft./sec.
unknown rate
?
Pythagorean theorem
differentiate with respect to time
solve for
to evaluate we need
then
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Example. Pat walks feet/second towards a street light whose lamp is 20 feet above the ground. If Pat is 6 feet tall, find how rapidly Pat’s shadow changes in length.
ground, lamp, Pat, , , ,
given information
feet, 6 feet,
unknown rate
by similar triangles
differentiate wrt time
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Example. The beacon of a lighthouse 1 mile from the shore makes 5 rotations per minute. Assuming that the shoreline is straight, calculate the speed at which the spotlight sweeps along the shoreline as it lights up sand 2 miles from the lighthouse.
-, -, lighthouse at , beam to , length , angle
given information
mile 2 miles
unknown rate
from the definition of tangent
differentiate wrt time
using that is a constant
we need
then
■ STOP
§2.8 Linear Approximations and Differentials
Linear or Tangent Line Approximation
-, -, ,
Point-slope form of the tangent line
Point
Slope
Then
or
approximates near
Example. Find the linear approximation to near .
this straight line approximates near .
-, -, ,
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Example (continued). Use linear approximation to estimate .
Exact value ■
Example (continued). For what values of is the linear approximation
accurate to within ? This means
or
show Maple output
From the plot, the linear approximation is accurate to within for
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Realistic Example. Estimate the amount of paint required to paint a spherical tank of radius 20 feet with a coat 0.01 inches thick.
Exact calculation: volume of sphere
volume between two concentric spheres of radii and
cubic inches
1 gallon 231 cubic inches
Approximate calculation:
Linear approximation, usual notation
Use instead of
and
then
surface area of sphere thickness of paint
cubic inches
the error is only 0.3 cubic inches! ■
STOP Differentials
-, -, , , , ,
tangent line approximation
write
add
is approximated by
Note that . This is clever notation.
Example. Estimate the amount of paint required to paint a spherical tank of radius 20 feet with a coat of paint 0.01 inches thick.
Volume of sphere
approximate amount of paint
if inches, then
the same result as above. This is a shorthand for linear approximation. ■
Newton’s Method – An Application of Linear Approximation
-, -,
A root of is a number such that .
How to find roots geometrically?
guess add
let be the line tangent to at add
follow to the axis, get an intersection at add
repeat this process
if the initial guess was good, the sequence rapidly converges to
How to find roots algebraically?
Make an initial guess .
Find the line tangent to at .
by the linear or tangent line approximation!
intersects the axis at . Find .
Find the line tangent to at .
intersects the axis at . Find .
repeat this procedure to obtain a general formula for
If the initial guess was good, the iterates will quickly approach .
Example.
-, -,
from the boxed formula above
guess
Exact value of positive root
Transparency – How Newton’s Method can go wrong!