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Kathryn
Bollinger, March 12, 2014
Concepts to Know #2
• 4.3 - The Chain Rule
Math 142
4.1-4.5, 5.1-5.6
Know how to find the composite of two functions:
(f ◦ g)(x) = f (g(x))
• 4.1 - Derivatives of Powers, Exponents and
Sums
Be familiar with the different notations for the
derivative:
df
d
dy
=
=
f (x)
f ′ (x) = y ′ =
dx
dx
dx
Know the following derivative rules:
f (x) = k → f ′ (x) = 0 (k constant)
f (x) = xn → f ′ (x) = nxn−1
f (x) = ex → f ′ (x) = ex
1
f (x) = ln x → f ′ (x) =
x
f (x) = kg(x) → f ′ (x) = kg ′ (x) (k constant)
h(x) = f (x) ± g(x) → h′ (x) = f ′ (x) ± g ′ (x)
Know how to find equations of lines tangent to a
curve.
Know how to locate places where horizontal
tangent lines exist.
Know the word marginal refers to an IROC and
how to use marginal business functions to
make approximations.
• 4.2 - Derivatives of Products and Quotients
Product Rule:
h(x) = f (x) · g(x), where f and g are differentiable functions, then
h′ (x) = f ′ (x) · g(x) + f (x) · g ′ (x)
(Deriv of
1st )(2nd )
+
(1st )(Deriv
of
2nd )
Quotient Rule:
f (x)
If h(x) =
, where f and g are differeng(x)
tiable functions with g(x) 6= 0, then
h′ (x) =
g(x) · f ′ (x) − f (x) · g ′ (x)
[g(x)]2
HOdHI − HIdHO
HOHO
Know how to find the derivative of a composite
function by using the chain rule.
Know the chain rule for the power rule:
f (x) = [g(x)]n → f ′ (x) = n[g(x)]n−1 · g′ (x)
• 4.4 - Derivatives of Exponential and
Logarithmic Functions
Know the following derivative rules:
f (x) = bx → f ′ (x) = bx (ln b)
1
1
′
f (x) = logb x → f (x) =
ln b
x
Know how to use the chain rule for exp. and log.
functions:
h(x) = ef (x) → h′ (x) = ef (x) · (f ′ (x))
h(x) = bf (x) → h′ (x) = bf (x) · (f ′ (x)) · (ln b)
h(x) = ln [g(x)] →
g′ (x)
1
· g ′ (x) =
h′ (x) =
g(x)
g(x)
h(x) = logb[g(x)]→
1
1
g′ (x)
h′ (x) =
· g ′ (x) =
ln b
g(x)
g(x)(ln b)
Recall log properties that help simplify functions
before taking derivatives
• 4.5 - Elasticity of Demand
Know how to find the relative rate of change of a
function.
Know how to find and use the elasticity of demand
function:
E(p) =
−p · d′ (p)
d(p)
Know the difference in elastic, inelastic, and unit
elastic.
Know how changes in price affect revenue with different elasticity classifications.
Know how to use E(p) to maximize revenue.
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Kathryn
Bollinger, March 12, 2014
• 5.1 - The First Derivative
Second Derivative Test
(Use of f ′′ to find rel. ext. of f )
If f ′ (x) > 0 on an interval, then
f (x) is increasing on that interval.
If f ′ (x) < 0 on an interval, then
f (x) is decreasing on that interval.
If f ′ (x) = 0 on an interval, then
f (x) is constant on that interval.
Critical Number: A point in the domain of f where
f ′ (x) = 0 or f ′ (x) DNE.
Know the First Derivative Test
(Use sign chart of f ′ to find inc/dec and rel.
ext. of f )
1. Plot all critical numbers and points not in
the domain of f on a number line.
2. Test x-values in f ′ on subsequent
intervals.
3. Based on sign of f ′ ,
determine where f is inc/dec.
4. Relative ext. occur at points in the domain of f where f ′ changes sign.
If f has a rel. max or rel. min at x = c, then x = c
is a critical number of f .
Know when relative extrema occur based on the
graph of the function or information about its
derivative.
• 5.2 - The Second Derivative
Understand the notation of higher-order derivatives.
If f ′′ (x) > 0 on an interval, then
f ′ (x) is increasing on that interval, and
f (x) is concave up (⌣) on that interval.
If
Know when inflection points of f occur (when
f concavity changes, where f ′ has rel. ext.,
where f ′′ changes sign).
• 5.3 - Limits at Infinity
Know how to determine end behavior of polynomials (degree and leading coefficient).
Know how to evaluate limits if x → ±∞.
Horizontal Asymptotes
Leveling off end behavior of a graph
(as x → ±∞)
Rational Functions: Compare highest powers
of x in the numerator and denominator, if
a HA exists. If no HA exist, know how to
determine, using limits, if the end → ±∞.
Be able to use limits to find HA of a function.
Be able to find both VA and HA from a
graph.
• 5.4 - Additional Curve Sketching
f ′′ (x)
< 0 on an interval, then
is decreasing on that interval, and
f (x) is concave down (⌢) on that interval.
f ′ (x)
Know the Test for Concavity (Use sign chart of
to find concavity and inflection pts of f .)
1. Find critical numbers of f where
f ′ (c) = 0.
2. Test these types of critical numbers with
f ′′ :
If f ′′ (c) > 0, then f is concave up and a
rel. min. occurs at x = c.
If f ′′ (c) < 0, then f is concave down and
a rel. max. occurs at x = c.
If f ′′ (c) = 0 or f ′′ (c) DNE, then this test
fails. You must use the first derivative
test to determine rel. ext.
f ′′
1. Plot all points not in the domain of f and
all pts where f ′′ = 0 or f ′′ DNE on a
number line.
2. Test x-values in f ′′ on subsequent intervals.
3. Based on sign of f ′′ ,
determine where f is concave up/concave
down.
4. Inflection points occur at points in the domain of f where f ′′ changes sign.
Know how to Graph Functions Using Calculus.
1. Find info from f : Domain, intercepts, VA,
holes, HA
2. Find info from f ′ : inc/dec, rel. ext.
3. Find info from f ′′ : concave up/concave
down, infl. pts.
4. Plot important POINTS (intercepts, rel.
ext, infl. pts.) and asymptotes
5. Use info from the derivatives to sketch the
correctly shaped curve passing through
the known points.
6. Check your graph with the graph from
your calculator if given the actual function.
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Kathryn
Bollinger, March 12, 2014
• 5.5 - Absolute Extrema
Absolute Maximum: the largest value a function
obtains on its domain
Absolute Minimum: the smallest value a function
obtains on its domain
=⇒ Absolute Extrema are function values
(y-values)
Absolute Extrema on a Closed Interval
(Closed Interval Method)
If f is continuous on the interval, then
abs. ext. are guaranteed.
1. Determine critical numbers of f .
2. Evaluate f at each endpoint and at
each critical number IN the interval.
(t-chart)
3. The largest function value is the absolute maximum; the smallest is the
absolute minimum.
If f is NOT continuous on the interval, then
check the graph before proceeding.
Know how and when to use the Second Derivative
Test to determine Absolute Extrema
• 5.6 - Optimization and Modeling
Be able to solve applied absolute extrema problems using calculus.
Procedure:
1. Draw a picture, if possible.
2. Select variables, where needed.
3. Write an equation to be maximized or
minimized (only in terms of one variable).
4. Find an interval over which you are optimizing your function.
5. Apply the correct procedure for finding
absolute extrema, based on your interval
type.
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