Miscellaneous Topics

Calculus Drill!!

Miscellaneous Topics

• I’m going to ask you about various unrelated but

important calculus topics.

• It’s important to be fast as time is your enemy

on the AP Exam.

• When you think you know the answer,

(or if you give up ) click to get to the next

slide to see if you were correct.

How many different

methods are there for

evaluating limits?

Can you name

several?

1. Direct Substitution

2. Observe graph

3. Create a table of values

4. Re-write algebraically

(Simplify)

5. Use L’Hopitals Rule (only if the form is indeterminate)

How many indeterminate

forms can you name?

0

0

∞

∞

∞−∞

00

0∞

∞1

∞⋅0

Did you know all 7?

1.

2.

3.

4.

5.

6.

7.

Math Wars!!!

lim x

xsin= ?

0→x

1

lim0→x

?cos1

=−

x

x

0Zero! Zip:

What are the three main types

of discontinuities?

1. Hole – at x=3 in the example

2. Vertical asymptote – at x=1 in

the example

3. Step – usually the function’s description is split up :

)3)(1(

)3(

−−

−

xx

x

32

for x<0

for x>0 f(x)={

Under what conditions does the derivative NOT

exist at x=a

If there is a discontinuity at

x=a or if there is a sharp

corner at x=a, then the

derivative is undefined at

x=a

What is the definition of

continuity at a point?

lim=)(af )(xfax→

What is a monotone function?

A function that is either always

increasing or always decreasing.

(i.e. the derivative is always positive

or always negative.)

What is a normal line?

The line perpendicular to the

tangent line.

?)()'( 1 =− bf

Given (a,b) is on the graph of f(x)

)(

1

af ′

Did you remember that one? It’s a bit esoteric, eh?

What does the

Intermediate Value Theorem

say?

If f(x) is continuous and p is a y-value

between f(a) and f(b), then there is at

least one x-value, c, between a and b

such that f(c) = p.

What is the formula for the

slope of the secant line

through (a,f(a)) and (b,f(b))

and what does it represent?

ab

afbf

−

− )()( average rate of

change in f(x)

from x=a to x=b =

Note: This differs from the derivative which gives exact

instantaneous rate of change values at single x-value

but you can use it to the derivative value at some

values of x=c between a and b.

≈

What does the Mean Value

Theorem say?

If f(x) is continuous and differentiable,

then for some c between a and b

ab

afbfcf

−

−=′

)()()(

That is the exact rate of change equals

the average (mean) rate of change at

some point in between a and b.

What does f ‘ (a) = 0 tell you about the

graph of f(x) ?

Warning: irrelevant picture

The graph has a horizontal tangent line

at x=a.

f(a) might be a minimum or

maximum:or perhaps just a horizontal

inflection point.

What else must happen in

addition to the derivative being

zero or undefined at x=a in order

for f(a) to be an extrema?

The derivative must change signs at x=a

What is the First Derivative Test?

FIRST DERIVATIVE TEST

If f ‘(x) changes from + to – at x=a then f(a) is a local maximum.

If f ‘(x) changes from – to + at x=a then f(a) is a local minimum.

Dam that’s

a good test!! Dam, that’s

a great

test!!

What’s the Second Derivative Test?

Given f ‘(a)=0 then:

1. If f “ (a) < 0, f(a) is a relative max

2. If f “ (a) > 0, f(a) is a relative min

3. If f “ (a) = 0 the test fails

The Second Derivative Test:

Don’t be

Stumped...

Ha ha ha:

What do you know about the

graph of f(x) if f “ (a) = 0

(or does not exist)?

You know there might be an inflection

point at x = a.

(Check to see if there is also a sign change in f “ at x = a to confirm the inflection point actually occurs)

How do you determine velocity?

Velocity = the first derivative of the

position function,

or

v(a) +

(initial velocity + cumulative change in velocity)

∫b

adtta )(

How do you determine speed?

Speed = absolute value of velocity

How do you determine acceleration?

acceleration =

first derivative of velocity =

second derivative of position

Using differentials to approximate f(a+h)

with a point near (a,f(a)) on the tangent

line: what does f(a+h) ? ≈

This is

driving

me

nuts!!!!

f(a+h) f(a) + f ‘(a) h ⋅≈

Don’t make an

If f ‘(x) is negative:.

Then f(x) is decreasing:.

If f ‘(x) is positive:.

Then f(x) is increasing:.

If f “ (x) is negative then:

f(x) is concave down

If f “ (x) is positive then:

f(x) is concave up

How do you compute the

average value of

?

∫b

a______________________

b - a

dx

Note: This is also known as the

Mean (average) Value Theorem for Integrals

How do you locate and confirm vertical and

horizontal asymptotes?

Vertical – suspect them at x-values which

cause the denominator of f(x) to be zero.

Confirm that the limit as x a is infinite:. →

Horizontal – suspect rational functions

Confirm that as x , y a ±∞→ →

If = ky

What does y = ?

dt

dy

kty Ce=

Calculus trivia: doubling time is =

If you know ½ life, how do you find k?

k

2ln

ln(what it does)

time it takes to get therek =

What’s general formula for a

Riemann Sum?

∑=

∆n

k

kk xxf1

)(

∑=

−− ⋅+n

k

nab

nab kaf

1

)(

or:more specifically

Calculus trivia: as n (number of rectangles)

goes to the summation sign becomes

the integral sign and x becomes dx ∞

∆

What’s the

Trapezoidal Rule?

The Trapezoidal Rule is the formula for estimating a

definite integral with trapezoids. It is more accurate

than a Riemann Sum which uses rectangles.

)]()(2)(2)(2)([12102

1nnxfxfxfxfxfxT ++⋅⋅⋅+++∆= −

Notice that all the y-values except the first and last are doubled.

Use this only if the partitions are even, otherwise do them all by

hand separately.

Do we need to take a short

break?

Back already?

What is L’Hopital’s Rule? ^

Given that as x both f and g

)(

)(

xg

xf

a→ 0→

or both f and g then the limit of ∞→

= the limit of )('

)('

xg

xf

as x a→

L’Hopital’s Rule: ^

What is the

Fundamental

Theorem of

Calculus???

∫ −=b

aaFbFdxxf )()()(

where F ‘(x) = f(x)

Do you know the other form?

The one that is less commonly “used”?

The FUNdamental Theorem of Calculus:

∫ =x

axfdttf

dx

d)()(

YAY!! Steve’s Theorem!!

What is the general integral for

computing volume by slicing?

(Assume we are revolving f(x)

about the x-axis)

∫ dxxf2))((π

What if we revolve f(x)

around y=a ?

∫ − dxaxf2))((π

What if we revolve the area

between 2 functions: f(x) and

g(x) around the x-axis?

∫ − dxxgxf 22 ))(())((π

Be sure to square the radii

separately!!! (and put the larger function first)

1. How do you compute displacement?

(distance between starting & ending points)

2. How do you compute total distance

traveled?

displacement:

total distance:

∫kt

tdttv

0

)(

∫kt

tdttv

0

|)(|

Yea!!! That’s all

folks!

(Be sure to check out the

other calculus power point

drill and practices)