Midterm II Review
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Transcript of Midterm II Review
Review for Midterm II
Math 1a
December 2, 2007
Announcements
I Midterm II: Tues 12/4 7:00-9:00pm (SC Hall B)
I I have office hours Monday 1–2 and Tuesday 3–4 (SC 323)
I I’m aware of the missing audio on last week’s problem sessionvideo
Outline
DifferentiationThe Product RuleThe Quotient RuleThe Chain RuleImplicit DifferentiationLogarithmic Differentiation
The shape of curvesThe Mean Value TheoremThe Extreme Value Theorem
The Closed Interval MethodThe First Derivative TestThe Second Derivative Test
ApplicationsRelated RatesOptimization
MiscellaneousLinear ApproximationLimits of indeterminateforms
DifferentiationLearning Objectives
I state and use the product,quotient, and chain rules
I differentiate all“elementary” functions:
I polynomialsI rational functions:
quotients of polynomialsI root functions: rational
powersI trignometric functions:
sin/cos, tan/cot, sec/cscI inverse trigonometric
functionsI exponential and
logarithmic functionsI any composition of
functions like the above
I use implicit differentiationto find the derivative of afunction defined implicitly.
I use logarithicdifferentiation to find thederivative of a function
I given a function f and apoint a in the domain of a,compute the linearizationof f at a
I use a linear approximationto estimate the value of afunction
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)
The Quotient Rule
Theorem (The Quotient Rule)
Let u and v be differentiable at x, with v(x) 6= 0 Then(u
v
)′(x) =
u′v − uv ′
v2
The Chain Rule
Theorem (The Chain Rule)
Let f and g be functions, with g differentiable at a and fdifferentiable 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
dx=
dy
du
du
dx
Implicit Differentiation
Any time a relation is given between x and y , we may differentiatey as a function of x even though it is not explicitly defined.
Derivatives of Exponentials and Logarithms
Fact
Id
dxax = (ln a)ax
Id
dxex = ex
Id
dxln x =
1
x
Id
dxloga x =
1
(ln a)x
Logarithmic Differentiation
If f involves products, quotients, and powers, then ln f involves itto sums, differences, and multiples
Outline
DifferentiationThe Product RuleThe Quotient RuleThe Chain RuleImplicit DifferentiationLogarithmic Differentiation
The shape of curvesThe Mean Value TheoremThe Extreme Value Theorem
The Closed Interval MethodThe First Derivative TestThe Second Derivative Test
ApplicationsRelated RatesOptimization
MiscellaneousLinear ApproximationLimits of indeterminateforms
The shape of curvesLearning Objectives
I use the Closed IntervalMethod to find themaximum and minimumvalues of a differentiablefunction on a closedinterval
I state Fermat’s Theorem,the Extreme ValueTheorem, and the MeanValue Theorem
I use the First DerivativeTest and Second Derivative
Test to classify criticalpoints as relative maxima,relative minima, or neither.
I given a function, graph itcompletely, indicating
I zeroes (if they are easilyfound)
I asymptotes (ifapplicable)
I critical pointsI relative/absolute
max/minI inflection points
The Mean Value Theorem
Theorem (The Mean ValueTheorem)
Let f be continuous on [a, b]and differentiable on (a, b).Then there exists a point c in(a, b) such that
f (b)− f (a)
b − a= f ′(c). •
a
•b
•c
The Mean Value Theorem
Theorem (The Mean ValueTheorem)
Let f be continuous on [a, b]and differentiable on (a, b).Then there exists a point c in(a, b) such that
f (b)− f (a)
b − a= f ′(c). •
a
•b
•c
The Mean Value Theorem
Theorem (The Mean ValueTheorem)
Let f be continuous on [a, b]and differentiable on (a, b).Then there exists a point c in(a, b) such that
f (b)− f (a)
b − a= f ′(c). •
a
•b
•c
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 anabsolute minimum value f (d) at numbers c and d in [a, b].
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:
I Either the maximumoccurs at an endpoint ofthe interval, i.e., c = aor c = b,
I Or the maximum occursinside (a, b). In this case,c is also a localmaximum.
I Either f isdifferentiable at c , inwhich case f ′(c) = 0by Fermat’s Theorem.
I Or f is notdifferentiable at c .
This means to find themaximum value of f on [a, b],we need to check:
I a and b
I Points x where f ′(x) = 0
I Points x where f is notdifferentiable.
The latter two are both calledcritical points of f . Thistechnique is called the ClosedInterval Method.
The First Derivative Test
Let f be continuous on [a, b] and c in (a, b) a critical point of f .
Theorem
I If f ′(x) > 0 on (a, c) and f ′(x) < 0 on (c , b), then f (c) is alocal maximum.
I If f ′(x) < 0 on (a, c) and f ′(x) > 0 on (c , b), then f (c) is alocal minimum.
I If f ′(x) has the same sign on (a, c) and (c, b), then (c) is nota local extremum.
The Second Derivative Test
Let f , f ′, and f ′′ be continuous on [a, b] and c in (a, b) a criticalpoint of f .
Theorem
I If f ′′(c) < 0, then f (c) is a local maximum.
I If f ′′(c) > 0, then f (c) is a local minimum.
I If f ′′(c) = 0, the second derivative is inconclusive (this doesnot mean c is neither; we just don’t know yet).
Outline
DifferentiationThe Product RuleThe Quotient RuleThe Chain RuleImplicit DifferentiationLogarithmic Differentiation
The shape of curvesThe Mean Value TheoremThe Extreme Value Theorem
The Closed Interval MethodThe First Derivative TestThe Second Derivative Test
ApplicationsRelated RatesOptimization
MiscellaneousLinear ApproximationLimits of indeterminateforms
ApplicationsLearning Objectives
I model word problems with mathematical functions (this is amajor goal of the course!)
I apply the chain rule to mathematical models to relate rates ofchange
I use optimization techniques in word problems
Related RatesStrategies for Related Rates Problems
1. Read the problem.
2. Draw a diagram.
3. Introduce notation. Give symbols to all quantities that arefunctions of time (and maybe some constants)
4. Express the given information and the required rate in termsof derivatives
5. Write an equation that relates the various quantities of theproblem. If necessary, use the geometry of the situation toeliminate all but one of the variables.
6. Use the Chain Rule to differentiate both sides with respect tot.
7. Substitute the given information into the resulting equationand solve for the unknown rate.
OptimizationStrategies for Optimization Problems
1. Read the problem.
2. Draw a diagram.
3. Introduce notation. Give symbols to all quantities that arefunctions of time (and maybe some constants)
4. Write an equation that relates the various quantities of theproblem. If necessary, use the geometry of the situation toeliminate all but one of the variables.
5. Use either the Closed Interval Method, the First DerivativeTest, or the Second Derivative Test to find the maximumvalue of this function
Outline
DifferentiationThe Product RuleThe Quotient RuleThe Chain RuleImplicit DifferentiationLogarithmic Differentiation
The shape of curvesThe Mean Value TheoremThe Extreme Value Theorem
The Closed Interval MethodThe First Derivative TestThe Second Derivative Test
ApplicationsRelated RatesOptimization
MiscellaneousLinear ApproximationLimits of indeterminateforms
Linear Approximation
Let f be differentiable at a. What linear function bestapproximates 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)
Linear Approximation
Let f be differentiable at a. What linear function bestapproximates 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)
Linear Approximation
Let f be differentiable at a. What linear function bestapproximates 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)
Linear Approximation
Let f be differentiable at a. What linear function bestapproximates 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)
Theorem (L’Hopital’s Rule)
Suppose f and g are differentiable functions and g ′(x) 6= 0 near a(except possibly at a). Suppose that
limx→a
f (x) = 0 and limx→a
g(x) = 0
or
limx→a
f (x) = ±∞ and limx→a
g(x) = ±∞
Then
limx→a
f (x)
g(x)= lim
x→a
f ′(x)
g ′(x),
if the limit on the right-hand side is finite, ∞, or −∞.
L’Hopital’s Rule also applies for limits of the form∞∞
.
Theorem (L’Hopital’s Rule)
Suppose f and g are differentiable functions and g ′(x) 6= 0 near a(except possibly at a). Suppose that
limx→a
f (x) = 0 and limx→a
g(x) = 0
or
limx→a
f (x) = ±∞ and limx→a
g(x) = ±∞
Then
limx→a
f (x)
g(x)= lim
x→a
f ′(x)
g ′(x),
if the limit on the right-hand side is finite, ∞, or −∞.
L’Hopital’s Rule also applies for limits of the form∞∞
.
Cheat Sheet for L’Hopital’s Rule
Form Method
00 L’Hopital’s rule directly
∞∞ L’Hopital’s rule directly
0 · ∞ jiggle to make 00 or ∞∞ .
∞−∞ factor to make an indeterminate product
00 take ln to make an indeterminate product
∞0 ditto
1∞ ditto