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. Inspection

2. Observe graph

3. Create a table of values

4. Re-write algebraically

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

6. Squeeze theorem (rarely used!!)

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!!!

limx

xsin= ?

0x

1

lim0x

?cos1

x

x

0Zero! Zip…

What are the three main types of discontinuities?

1. Hole – at x=3 in the example

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

3. Vertical asymptote – at x=1 in the example

)3)(1(

)3(

xx

x

2xx2

for x<0

for x>0f(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 bfGiven (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 Squeeze Theorem say?

ax

If both f(x) and g(x) as L ax

Then h(x) also. L

Given f(x) > h(x) > g(x) near

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 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

The differential or df or dy or “error”

= f ‘(a) h

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

ktAey Calculus trivia: doubling time is =

k

2ln

What’s general formula for a Riemann Sum?

n

kkk xxf

1

)(

n

knab

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)([ 121021

nn xfxfxfxfxfxT

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

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 Newton’s Formula for approximating the zeros of a function?

)('

)(1

n

nnn xf

xfxx

Ain’t that ducky!!

What is the Fundamental Theorem of Calculus???

b

abFaFdxxf )()()(

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)()(

What is the general integral for computing volume by slicing?

(Assume we are revolving f(x) about the x-axis)

dxxf 2))((

What if we revolve f(x)

around y=a ?

dxaxf 2))((

What if we revolve the area between 2 functions: f(x) and

g(x) around the x-axis?

dxxgxf 22 ))(())((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)