1.
Review for Midterm II
Math 1a
December 2, 2007
Announcements
Midterm II: Tues 12/4 7:00-9:00pm (SC Hall B)
I have oﬃce hours Monday 1–2 and Tuesday 3–4 (SC 323)
I’m aware of the missing audio on last week’s problem session
video
2.
Outline
The Closed Interval Method
Diﬀerentiation The First Derivative Test
The Product Rule The Second Derivative Test
The Quotient Rule
Applications
The Chain Rule
Related Rates
Implicit Diﬀerentiation
Optimization
Logarithmic Diﬀerentiation
Miscellaneous
The shape of curves Linear Approximation
The Mean Value Theorem Limits of indeterminate
The Extreme Value Theorem forms
3.
Diﬀerentiation
Learning Objectives
state and use the product, use implicit diﬀerentiation
quotient, and chain rules to ﬁnd the derivative of a
diﬀerentiate all function deﬁned implicitly.
“elementary” functions:
use logarithic
polynomials
diﬀerentiation to ﬁnd the
rational functions:
quotients of polynomials derivative of a function
root functions: rational
powers given a function f and a
trignometric functions: point a in the domain of a,
sin/cos, tan/cot, sec/csc
compute the linearization
inverse trigonometric
of f at a
functions
exponential and
use a linear approximation
logarithmic functions
to estimate the value of a
any composition of
function
functions like the above
4.
The Product Rule
Theorem (The Product Rule)
Let u and v be diﬀerentiable at x. Then
(uv ) (x) = u(x)v (x) + u (x)v (x)
5.
The Quotient Rule
Theorem (The Quotient Rule)
Let u and v be diﬀerentiable at x, with v (x) = 0 Then
u v − uv
u
(x) =
v2
v
6.
The Chain Rule
Theorem (The Chain Rule)
Let f and g be functions, with g diﬀerentiable at a and f
diﬀerentiable at g (a). Then f ◦ g is diﬀerentiable at a and
(f g ) (a) = f (g (a))g (a)
◦
In Leibnizian notation, let y = f (u) and u = g (x). Then
dy dy du
=
dx du dx
7.
Implicit Diﬀerentiation
Any time a relation is given between x and y , we may diﬀerentiate
y as a function of x even though it is not explicitly deﬁned.
8.
Derivatives of Exponentials and Logarithms
Fact
dx
a = (ln a)ax
dx
dx
e = ex
dx
d 1
ln x =
dx x
d 1
loga x =
dx (ln a)x
9.
Logarithmic Diﬀerentiation
If f involves products, quotients, and powers, then ln f involves it
to sums, diﬀerences, and multiples
10.
Outline
The Closed Interval Method
Diﬀerentiation The First Derivative Test
The Product Rule The Second Derivative Test
The Quotient Rule
Applications
The Chain Rule
Related Rates
Implicit Diﬀerentiation
Optimization
Logarithmic Diﬀerentiation
Miscellaneous
The shape of curves Linear Approximation
The Mean Value Theorem Limits of indeterminate
The Extreme Value Theorem forms
11.
The shape of curves
Learning Objectives
use the Closed Interval Test to classify critical
Method to ﬁnd the points as relative maxima,
maximum and minimum relative minima, or neither.
values of a diﬀerentiable given a function, graph it
function on a closed completely, indicating
interval
zeroes (if they are easily
state Fermat’s Theorem, found)
asymptotes (if
the Extreme Value
applicable)
Theorem, and the Mean
critical points
Value Theorem
relative/absolute
use the First Derivative max/min
inﬂection points
Test and Second Derivative
12.
The Mean Value Theorem
Theorem (The Mean Value
Theorem)
Let f be continuous on [a, b]
and diﬀerentiable on (a, b).
Then there exists a point c in
(a, b) such that •
b
f (b) − f (a) •
= f (c). a
b−a
13.
The Mean Value Theorem
Theorem (The Mean Value
Theorem)
Let f be continuous on [a, b]
and diﬀerentiable on (a, b).
Then there exists a point c in
(a, b) such that •
b
f (b) − f (a) •
= f (c). a
b−a
14.
The Mean Value Theorem
Theorem (The Mean Value
c
Theorem) •
Let f be continuous on [a, b]
and diﬀerentiable on (a, b).
Then there exists a point c in
(a, b) such that •
b
f (b) − f (a) •
= f (c). a
b−a
15.
The Extreme Value Theorem
Theorem (The Extreme Value Theorem)
Let f be a function which is continuous on the closed interval
[a, b]. Then f attains an absolute maximum value f (c) and an
absolute minimum value f (d) at numbers c and d in [a, b].
16.
The Closed Interval Method
Let f be a continuous function deﬁned on a closed interval [a, b].
We are in search of its global maximum, call it c. Then:
This means to ﬁnd the
Either the maximum
maximum value of f on [a, b],
occurs at an endpoint of
we need to check:
the interval, i.e., c = a
a and b
or c = b,
Points x where f (x) = 0
Or the maximum occurs
inside (a, b). In this case, Points x where f is not
c is also a local diﬀerentiable.
maximum.
The latter two are both called
Either f is
critical points of f . This
diﬀerentiable at c, in
technique is called the Closed
which case f (c) = 0
Interval Method.
by Fermat’s Theorem.
Or f is not
diﬀerentiable at c.
17.
The First Derivative Test
Let f be continuous on [a, b] and c in (a, b) a critical point of f .
Theorem
If f (x) > 0 on (a, c) and f (x) < 0 on (c, b), then f (c) is a
local maximum.
If f (x) < 0 on (a, c) and f (x) > 0 on (c, b), then f (c) is a
local minimum.
If f (x) has the same sign on (a, c) and (c, b), then (c) is not
a local extremum.
18.
The Second Derivative Test
Let f , f , and f be continuous on [a, b] and c in (a, b) a critical
point of f .
Theorem
If f (c) < 0, then f (c) is a local maximum.
If f (c) > 0, then f (c) is a local minimum.
If f (c) = 0, the second derivative is inconclusive (this does
not mean c is neither; we just don’t know yet).
19.
Outline
The Closed Interval Method
Diﬀerentiation The First Derivative Test
The Product Rule The Second Derivative Test
The Quotient Rule
Applications
The Chain Rule
Related Rates
Implicit Diﬀerentiation
Optimization
Logarithmic Diﬀerentiation
Miscellaneous
The shape of curves Linear Approximation
The Mean Value Theorem Limits of indeterminate
The Extreme Value Theorem forms
20.
Applications
Learning Objectives
model word problems with mathematical functions (this is a
major goal of the course!)
apply the chain rule to mathematical models to relate rates of
change
use optimization techniques in word problems
21.
Related Rates
Strategies for Related Rates Problems
1. Read the problem.
2. Draw a diagram.
3. Introduce notation. Give symbols to all quantities that are
functions of time (and maybe some constants)
4. Express the given information and the required rate in terms
of derivatives
5. Write an equation that relates the various quantities of the
problem. If necessary, use the geometry of the situation to
eliminate all but one of the variables.
6. Use the Chain Rule to diﬀerentiate both sides with respect to
t.
7. Substitute the given information into the resulting equation
and solve for the unknown rate.
22.
Optimization
Strategies for Optimization Problems
1. Read the problem.
2. Draw a diagram.
3. Introduce notation. Give symbols to all quantities that are
functions of time (and maybe some constants)
4. Write an equation that relates the various quantities of the
problem. If necessary, use the geometry of the situation to
eliminate all but one of the variables.
5. Use either the Closed Interval Method, the First Derivative
Test, or the Second Derivative Test to ﬁnd the maximum
value of this function
23.
Outline
The Closed Interval Method
Diﬀerentiation The First Derivative Test
The Product Rule The Second Derivative Test
The Quotient Rule
Applications
The Chain Rule
Related Rates
Implicit Diﬀerentiation
Optimization
Logarithmic Diﬀerentiation
Miscellaneous
The shape of curves Linear Approximation
The Mean Value Theorem Limits of indeterminate
The Extreme Value Theorem forms
24.
Linear Approximation
Let f be diﬀerentiable at a. What linear function best
approximates f near a?
25.
Linear Approximation
Let f be diﬀerentiable at a. What linear function best
approximates f near a? The tangent line, of course!
26.
Linear Approximation
Let f be diﬀerentiable at a. What linear function best
approximates f near a? The tangent line, of course!
What is the equation for the line tangent to y = f (x) at (a, f (a))?
27.
Linear Approximation
Let f be diﬀerentiable at a. What linear function best
approximates f near a? The tangent line, of course!
What is the equation for the line tangent to y = f (x) at (a, f (a))?
L(x) = f (a) + f (a)(x − a)
28.
Theorem (L’Hˆpital’s Rule)
o
Suppose f and g are diﬀerentiable functions and g (x) = 0 near a
(except possibly at a). Suppose that
lim f (x) = 0 and lim g (x) = 0
x→a x→a
or
lim f (x) = ±∞ lim g (x) = ±∞
and
x→a x→a
Then
f (x) f (x)
lim = lim ,
x→a g (x) x→a g (x)
if the limit on the right-hand side is ﬁnite, ∞, or −∞.
29.
Theorem (L’Hˆpital’s Rule)
o
Suppose f and g are diﬀerentiable functions and g (x) = 0 near a
(except possibly at a). Suppose that
lim f (x) = 0 and lim g (x) = 0
x→a x→a
or
lim f (x) = ±∞ lim g (x) = ±∞
and
x→a x→a
Then
f (x) f (x)
lim = lim ,
x→a g (x) x→a g (x)
if the limit on the right-hand side is ﬁnite, ∞, or −∞.
∞
L’Hˆpital’s Rule also applies for limits of the form
o .
∞
30.
Cheat Sheet for L’Hˆpital’s Rule
o
Form Method
0
L’Hˆpital’s rule directly
o
0
∞
L’Hˆpital’s rule directly
o
∞
∞
0
0·∞ jiggle to make or ∞.
0
∞−∞ factor to make an indeterminate product
00 take ln to make an indeterminate product
∞0 ditto
1∞ ditto
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