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