AP CALCULUS PERIODIC REVIEW. 1: Limits and Continuity A function y = f(x) is continuous at x = a if: i) f(a) is defined (it exists) ii) iii) Otherwise,

AP CALCULUS PERIODIC AP CALCULUS PERIODIC REVIEWREVIEW

1: 1: Limits and ContinuityLimits and Continuity

A function A function y = f(x)y = f(x) is continuous at is continuous at x = ax = a if: if:

i) i) f(a)f(a) is defined (it exists) is defined (it exists)

ii)ii) andexists, )(lim xfax

iii)iii) )()(lim afxfax

Otherwise, Otherwise, ff is discontinuous at is discontinuous at x = ax = a..

isthat ---equal are and

exist limits sided-one ingcorrespondboth

ifonly and if exists )(limit The lim xfax

)()()( limlimlim xfLxfLxfaxaxax

2: 2: IIntermediate ntermediate VValue alue TTheoremheoremA function A function y = f(x)y = f(x) that is that is continuouscontinuous on a closed on a closed interval [a,b] takes on every value between interval [a,b] takes on every value between f(a)f(a)

and and f(b).f(b).

Note: If Note: If ff is continuous on is continuous on [a,b] [a,b] andand f(a)f(a) andand f(b) f(b) differ in sign, then the equation differ in sign, then the equation f(x) = 0 f(x) = 0 has at has at least one solution in the open interval (a,b).least one solution in the open interval (a,b).

a bf(a)

f(b)

3: 3: Limits of Rational Functions as Limits of Rational Functions as xx

)()(

0)(

)(lim

xgxf

xg

xf

x

of degree the of degree the

if

3

23

2

lim

x

xx

x

:Example


)()(

)(

)(lim

xgxf

xg

xf

x


if infinity is

8

22

3

lim

x

xx

x

:Example


)()(

)(

)(lim

xgxf

xg

xf

x


if finite is

Note: The limit will be the ratio of the leading Note: The limit will be the ratio of the leading coefficient of f(x) to g(x).coefficient of f(x) to g(x).

5

2

510

2322

2

lim

xx

xx

x

:Example

4: 4: Horizontal and Vertical Horizontal and Vertical AsymptotesAsymptotes

.)()( bxfbxff(x) y

by

xxlim or lim either if of

graph the of asymptote horizontal a is lineA

4: 4: Horizontal and Vertical Horizontal and Vertical AsymptotesAsymptotes

.)()(

xfxff(x) y

ay

aa xxlim or lim either if of

graph the of asymptote vertical a is lineA

5: 5: Average Rate vs. Instantaneous Average Rate vs. Instantaneous Rate of ChangeRate of Change

Average Rate of ChangeAverage Rate of Change:: If If (a, f(a))(a, f(a)) andand (b, f(b))(b, f(b))

are points on the graph of are points on the graph of y=f(x),y=f(x), then the then the average rate of change of average rate of change of yy with respect to with respect to xx over over the interval [the interval [a, ba, b] is:] is:

x

y

ab

afbf

)()(

a bf(a)

f(b)

5: 5: Average Rate vs. Instantaneous Average Rate vs. Instantaneous Rate of ChangeRate of Change

Instantaneous Rate of ChangeInstantaneous Rate of Change:: If If (x(x00, y, y00)) is a is a

point on the graph of point on the graph of y=f(x),y=f(x), then the then the instantaneous rate of change of instantaneous rate of change of yy with respect to with respect to xx at at xx00 is f’( is f’(xx00).).

a bf(a)

f(b)

6: 6: Limit Definition of a DerivativeLimit Definition of a Derivative

h

xfhxfxf

h

)()(lim)('

0

6: 6: Limit Definition of a DerivativeLimit Definition of a Derivative

AKA Difference QuotientAKA Difference Quotient

ax

afxfaf

ax

)()(lim)('

Geometrically, the derivative of a function at a Geometrically, the derivative of a function at a point is the slope of the tangent line to the graph point is the slope of the tangent line to the graph of the function at that point.of the function at that point.

7: 7: The Number The Number ee is actually a limit is actually a limit

en

n

n

11lim

8: 8: Rolle’s TheoremRolle’s Theorem

If f is continuous on If f is continuous on [a,b][a,b] and differentiable on and differentiable on (a, b)(a, b) such that such that f(a) = f(b),f(a) = f(b), then there is at least then there is at least one number one number cc in the open interval in the open interval (a, b)(a, b) such that such that f’(c) = 0f’(c) = 0..

9: 9: Mean Value TheoremMean Value Theorem

If f is continuous on If f is continuous on [a,b][a,b] and differentiable on and differentiable on (a, b),(a, b), then there is at least one number then there is at least one number cc in the in the open interval open interval (a, b)(a, b) such that: such that:

)(')()(

cfab

afbf

a bf(a)

f(b))(' cf

10: 10: Extreme Value TheoremExtreme Value Theorem

If If ff is continuous on a closed interval is continuous on a closed interval [a,b][a,b], then , then f(x)f(x) has both a maximum and a minimum on has both a maximum and a minimum on [a,b][a,b]..

a b

CONSIDERCONSIDER

11: 11: Max / Min of FunctionsMax / Min of FunctionsTo find maximum and minimum values of a To find maximum and minimum values of a function function y = f(x)y = f(x),, locate locate

1. the point(s) where 1. the point(s) where f’(x)f’(x) changes sign. To find changes sign. To find the candidates first find where the candidates first find where f’(x) = 0f’(x) = 0 or is or is infinite or does not exist.infinite or does not exist.

2. the end points, if any, on the domain of 2. the end points, if any, on the domain of f(xf(x))..

12: 12: Increasing and Decreasing Increasing and Decreasing IntervalsIntervals

If If f’(x) > 0f’(x) > 0 for every x in (a, b), then f is increasing for every x in (a, b), then f is increasing on [a, b].on [a, b].

If If f’(x) < 0f’(x) < 0 for every x in (a, b), then f is decreasing for every x in (a, b), then f is decreasing on [a, b].on [a, b].

a b

13: 13: Concavity and POIConcavity and POI

If If f’’(x) > 0f’’(x) > 0 for every for every xx in (a, b), then in (a, b), then ff is concave is concave up [a, b].up [a, b].

If If f’’(x) < 0f’’(x) < 0 for every for every xx in (a, b), then in (a, b), then ff is concave is concave down [a, b].down [a, b].

a b

To locate the points of inflection of To locate the points of inflection of y = f(x),y = f(x), find find the points where the points where f’’(x) = 0f’’(x) = 0 or where or where f’’(x)f’’(x) fails to fails to exist. These are the only candidates where exist. These are the only candidates where f(x)f(x) may have a POI. may have a POI.

Then test these points to make sure that Then test these points to make sure that f’’(x) < 0f’’(x) < 0 on one side and on one side and f’’(x) > 0f’’(x) > 0 on the other (changes on the other (changes sign).sign).

a b

14: 14: Differentiability and ContinuityDifferentiability and ContinuityDifferentiability implies continuity: If a function is Differentiability implies continuity: If a function is differentiable at a point differentiable at a point x = ax = a, it is continuous at , it is continuous at that point.that point.

The converse is false, that is, continuity does The converse is false, that is, continuity does NOT imply differentiability.NOT imply differentiability.

15: 15: Linear ApproximationLinear ApproximationThe linear approximation of f(x) near The linear approximation of f(x) near x = xx = xoo is is

given by given by

))((')( 000 xxxfxfy

1 1.1

16: 16: Comparing Rates of ChangeComparing Rates of Change

The exponential function The exponential function y = ey = exx grows rapidly as grows rapidly as xx while the logarithmic function while the logarithmic function y = ln xy = ln x grows grows very slowly as very slowly as xx..

ln x

x2

x3

3x

Exponential functions like Exponential functions like y = 2y = 2xx or or y = ey = exx grow grow more rapidly as more rapidly as xx than any positive power of than any positive power of x. The function x. The function y = ln xy = ln x grows slower as grows slower as xx than any nonconstant polynomial.than any nonconstant polynomial.

Another way to look at this, as Another way to look at this, as xx::

1. 1. f(x)f(x) grows grows fasterfaster than than g(x)g(x) if if

0)(

)(lim

)(

)(lim

xf

xg

xg

xfxx

if or

If If f(x)f(x) grows faster than grows faster than g(x) g(x) as as xx, then , then g(x)g(x) grows grows slowerslower than than f(x)f(x) as as xx..

3

3 limx

exxe

x

x

x since as than faster grows :Ex

x

xxxx

x lnlimln

44 since as than faster grows :Ex

Another way to look at this, as Another way to look at this, as xx::

2. 2. f(x)f(x) and and g(x)g(x) grow at the grow at the samesame rate as rate as xx if if

nonzero) and finite is LLxg

xfx

(0)(

)(lim

12

lim

2

2

2

22

x

xx

xxxx

x

since since as rate same the at grows :Ex

17: 17: Inverse FunctionsInverse Functions

1. If 1. If ff and and gg are two functions such that are two functions such that f(g(x)) = xf(g(x)) = x for every for every xx in the domain of in the domain of gg, and, , and, g(f(x)) = xg(f(x)) = x, for , for every every xx in the domain of in the domain of ff, then, , then, ff and and gg are are inverse functions of each other.inverse functions of each other.

ln x

exf(x) = ex

g(x) = ln x

f(g(x)) = eln x = x

17: 17: Inverse FunctionsInverse Functions

4. If 4. If ff is differentiable at every point on an interval is differentiable at every point on an interval II, and , and f’(x) f’(x) 0 0 on on II, then , then g = fg = f-1-1(x)(x) is differentiable is differentiable at every point of the interior of the interval at every point of the interior of the interval f(I)f(I) and and

))(('

1)('

xgfxg

f(x) = ex

g(x) = ln x

f(g(x)) = eln x = x

f’(g(x)) g’(x) = 1

f’(g(x)) g’(x) = 1/(f’(g(x)))

18: 18: Properties of Properties of eexx

1. The exponential function 1. The exponential function y = ey = exx is the inverse is the inverse function of function of yy = ln = ln xx. .

2. The domain of 2. The domain of y = ey = exx is the set of all real is the set of all real numbers and the range is the set of all positive numbers and the range is the set of all positive numbers, numbers, y>0y>0..


4.4. y = e y = exx is continuous, increasing, and concave is continuous, increasing, and concave up for all up for all xx..

xx eedx

d)(3.3.


0limlim.5

x

x

x

xee and

xxexxe xx all for for )ln(;0,.6 ln

19: 19: Properties of Properties of ln xln x

numbers.

real all ofset theis ln of range The .2 xy

1.1. The domain of y = ln x is the set of all positive The domain of y = ln x is the set of all positive numbers, x > x.numbers, x > x.


baab lnln)ln( .4

3.3. y = ln x is continuous and increasing y = ln x is continuous and increasing everywhere on its domain.everywhere on its domain.

bab

alnlnln .5

ara r lnln .6


xxxx

lnlim and lnlim .80

a

xxa ln

lnln .9

.1 if 0ln

and 10 if 0ln .7

xx

xxy

20: 20: Trapezoidal RuleTrapezoidal Rule

)]()(2...

)(2)(2)([2

)(

1

210

nn

b

a

xfxf

xfxfxfn

abdxxf

If a function, f, is continuous on the closed If a function, f, is continuous on the closed interval [a, b] where [a, b] has been partitioned interval [a, b] where [a, b] has been partitioned into n subintervals of equal length, each into n subintervals of equal length, each length (b – a) / n, then:length (b – a) / n, then:

.])()([2

1)( 32232112 etcbbwbbwdxxf

b

a

21: 21: Properties of the Definite Properties of the Definite IntegralIntegral

b

a

b

adxxfcdxxfc )()(

If f(x) and g(x) are continuous on [a, b]:If f(x) and g(x) are continuous on [a, b]:

0)( a

adxxf


a

b

b

adxxfdxxf )()(




b

c

c

a

b

adxxfdxxfdxxf )()()(

c] b, [a, continuous is f as long as


If f(x) is an even function, thenIf f(x) is an even function, then

aa

adxxfdxxf

0)(2)(


If f(x) is an odd function, thenIf f(x) is an odd function, then

0)( a

adxxf


If f(x) If f(x) 0 on [a, b], then 0 on [a, b], then

0)( b

adxxf


If g(x) If g(x) f(x) on [a, b], then f(x) on [a, b], then

b

a

b

adxxfdxxg )()(

22: 22: Definition of a Definite Integral Definition of a Definite Integral as the Limit of a Sumas the Limit of a Sum

Suppose that a function f(x) is continuous on the Suppose that a function f(x) is continuous on the closed interval [a, b]. Divide the interval into n closed interval [a, b]. Divide the interval into n equal subintervals, of lengthequal subintervals, of length

n

abx

22: 22: Definition of a Definite Integral Definition of a Definite Integral as the Limit of a Sumas the Limit of a Sum

Choose one number in each subinterval, i.e., xChoose one number in each subinterval, i.e., x11 in in

the first, xthe first, x22 in the second, …., x in the second, …., xii in the i in the ithth ,…., and ,…., and

xxnn in the n in the nthth. Then:. Then:

b

a

n

ii

ndxxfxxf .)()(lim

1

23: 23: First Fundamental Theorem of First Fundamental Theorem of CalculusCalculus

b

a

aFbFdxxf ),()()(

)()(' where xfxF

23: 23: Second Fundamental Theorem Second Fundamental Theorem of Calculusof Calculus

x

a

xfdttfdx

d and )()(

)(

)('))(()(xg

a

xgxgfdttfdx

d

24: 24: PVAPVA

dttvtsposition )()(

dttatvtsvelocity )()()('

)()(')('' tatvtsonaccelerati

24: 24: PVAPVA

The The speedspeed of an object is the absolute value of of an object is the absolute value of the velocity, the velocity, v(t)v(t). It tells how fast it is going . It tells how fast it is going disregarding its direction.disregarding its direction.

The The velocityvelocity of an object tells how fast it is going of an object tells how fast it is going andand in which direction. in which direction. VelocityVelocity is an is an instantaneous rate of change.instantaneous rate of change.

24: 24: PVAPVA

The The accelerationacceleration is the instantaneous rate of is the instantaneous rate of change of velocity—it is the derivative of the change of velocity—it is the derivative of the velocity—that is, a(t) = v’(t). velocity—that is, a(t) = v’(t).

NegativeNegative acceleration (deceleration) means that acceleration (deceleration) means that the velocity is decreasing.the velocity is decreasing.

24: 24: PVAPVA

The The averageaverage velocity of a particle over the time velocity of a particle over the time interval tinterval t0 0 to another time t, is:to another time t, is:

0

0 )()(

timeoflength

positionin changeVelocity Avg

tt

tsts

Where s(t) is the position of the particle at time t.Where s(t) is the position of the particle at time t.

25: 25: Average ValueAverage ValueThe average value of f(x) on [a, b] is The average value of f(x) on [a, b] is

dxxfab

b

a

)(1

26: 26: Area Between CurvesArea Between CurvesIf f and g are continuous functions such that If f and g are continuous functions such that g(x) g(x) f(x) on [a, b], then the area between the f(x) on [a, b], then the area between the curves is curves is

dxxfxgb

a )()(

27: 27: Volume of Solids of RevolutionVolume of Solids of RevolutionFor volumes of solids rotated around the x (or y) For volumes of solids rotated around the x (or y) axis, volume =axis, volume =

b

a

dxxfV 2)]([

a b

27: 27: Volume of Solids of RevolutionVolume of Solids of RevolutionFor washer method, volume =For washer method, volume =

b

a

dxxgxfV 22 )]([)]([

where f(x) is the large radius, and g(x) is the where f(x) is the large radius, and g(x) is the small radius.small radius.

a b

27: 27: Volume of Solids of RevolutionVolume of Solids of RevolutionFor cylinder method, volume =For cylinder method, volume =

b

a

dxheightradiusV (2

28: 28: Volume of Solids with Known Volume of Solids with Known Cross SectionsCross Sections

1. For cross sections of area A(x), taken 1. For cross sections of area A(x), taken perpendicular to the x-axis, volume =perpendicular to the x-axis, volume =

b

a

dxxA )(

2. For cross sections of area A(x), taken 2. For cross sections of area A(x), taken perpendicular to the y-axis, volume =perpendicular to the y-axis, volume =

d

c

dyyA )(

28: 28: Volume of Solids with Known Volume of Solids with Known Cross SectionsCross Sections

Some examples of these volumes are shown in Some examples of these volumes are shown in the next four slides:the next four slides:

b

a

dxxA )(

29: 29: Solving Differential Equations: Solving Differential Equations: Graphically and Numerically (Slope Graphically and Numerically (Slope

Fields)Fields)At every point (x, y) a differential equation of the At every point (x, y) a differential equation of the form dy/dx = f(x, y), gives the slope of the form dy/dx = f(x, y), gives the slope of the member of the family of solutions that contains member of the family of solutions that contains that point.that point.

At each point in the plane, a short segment is At each point in the plane, a short segment is drawn whose slope is equal to the value of the drawn whose slope is equal to the value of the derivative at that point. These segments are derivative at that point. These segments are tangent to the solution’s graph at the point.tangent to the solution’s graph at the point.

29: 29: Solving Differential Equations: Solving Differential Equations: Graphically and Numerically (Slope Graphically and Numerically (Slope

Fields)Fields)x

dx

dy2

x

y

O

You may be given an initial You may be given an initial condition:condition:

2)0( f

This tells you exactly which This tells you exactly which of the possible solutions is of the possible solutions is the answer.the answer.

30: 30: Solving Differential Equations Solving Differential Equations by by Separating the VariablesSeparating the Variables

Example of a differential equation:Example of a differential equation:

y

x

dx

dy 2

1. Rewrite the equation as an equivalent 1. Rewrite the equation as an equivalent equation with all the equation with all the xx and the and the dxdx terms on one terms on one side and all the side and all the yy and and dydy terms on the other. terms on the other.2. Antidifferentiate both sides to obtain an 2. Antidifferentiate both sides to obtain an equation withoutequation without dxdx or or dydy, , but with one constant but with one constant of integration.of integration.

3. Use the initial condition (given) to evaluate this 3. Use the initial condition (given) to evaluate this constant.constant.

f

f ’

f ’’

Documents

AP CALCULUS PERIODIC REVIEW. 1: Limits and Continuity A function y = f(x) is continuous at x = a if: i) f(a) is defined (it exists) ii) iii) Otherwise,