Calculus Review

Texas A&M University

Dept. of Statistics

In This Review

• Differentiation– Differentiation formulas– Examples– Trig. functions

• Integration– Indefinite integrals– Definite integrals– Trig. Functions

Also In This Review

• Calculus in statistics– Continuous distributions– Expected value and mean– Variance– Median

About Notation

As is common in statistics, we will use log to mean loge instead of log10 as is commonly used in mathematics.

Differentiation

The derivative provides us information about the rate of change of a function.

The derivative of a function is also a function.

Example: The derivative of the rate function is the

acceleration function.

Ways to Write the Derivative

Given the function f(x), we can write its derivative in the following ways:

- f`(x)

- f(x)

The derivative of x is commonly written dx.

dx

d

Differentiation Formulas

The following are common differentiation formulas:

- The derivative of a constant is 0.

- The derivative of a sum is the sum of the derivatives.

0cdu

d

)(')('))()(( ugufugufdu

d

Examples

- The derivative of a constant is 0.

- The derivative of a sum is the sum of the derivatives.

7du

d

)4(tdt

d

More Formulas

- The derivative of u to a constant power:

- The derivative of e:

- The derivative of log:

dueedu

d uu

duu

udu

d 1)log(

duunudu

d nn 1*

More Examples

- The derivative of u to a constant power:

- The derivative of e:

- The derivative of log:

yedy

d 4

)log(3 xdx

d

33xdx

d

Product and Quotient

The product rule and quotient rules are commonly used in differentiation.

- Product rule:

- Quotient rule:

)(')()(')())(*)(( ufugugufugufdu

d

2))((

)(')()(')(

)(

)(

ug

ugufufug

ug

uf

du

d

Product and Quotient Examples

- Product rule:

- Quotient rule:

))log(3( zzdz

d

ue

u

du

d3

7

Chain Rule

The chain rule allows you to combine any of the differentiation rules we have already covered.

- First, do the derivative of the outside and then do the derivative of the inside.

duugugfugfdu

d*)('*))(('))((

Chain Rule Examples)log( ve

dv

d

)log( 2 vevdv

d

vvee

dv

d

Try These

11)( zzf

yyyg 24)( 3

xexh 3)(

x

xxp

)log()(

2

3)()( zezq z

yyeys 24)(

Solutions

1)(' zf

212)(' 2 yyg

xexh 33)('

2

2 )log(2)('

x

xxp

)1()(3)(' 2 zz ezezq

yy eyeys 22 48)('

Derivatives of Trig. Functions

duuudu

d)cos()sin( duuu

du

d)(csc)cot( 2

duuudu

d)sin()cos( duuuu

du

d)tan()sec()sec(

duuudu

d)(sec)tan( 2 duuuu

du

d)cot()csc()csc(

More About Trig. Functions

The chain rule also applies to the trig. functions.

- First, do the derivative of the trig. function and then do the derivative of the inside.

Example:

)cos(22*)cos()sin( 222 xxxxxdx

d

Integration

The integral provides us with information about the area under a function.

The indefinite integral of a function is a function, but the definite integral may be a number.

Example: The integral of the rate function is the

distance function.Integrating over a specified time tells us how far we have gone.

Ways to Write the Integral

The two forms of integration, indefinite and definite, can be distinguished by the presence of limits on the integral.

- indefinite

- definite

b

adxxf )(

dxxf )(

Integration Formulas

The following are common integration formulas. *Note these are the ‘opposite’ of the derivative formulas, hence the integral is also called the antiderivative.

When doing the indefinite integral, you must add a constant to your function.

Integration Formulas

- The integral of a sum is the sum of the integrals.

- The integral of u-1:

du

uduudu

uu nn 11

cuduu

)log(1

Examples

- The integral of a sum is the sum of the integrals.

- The integral of u-1:

dxx 4

dtt2

1

More Formulas

- The integral of u to a constant power:

for

- The integral of e:

- The integral of the derivative:

cun

duu nn

1

1

1

cedue uu

1n

cufduuf )()('

More Examples

- The integral of u to a constant power:

- The integral of e:

dyy3

dve v2

Definite Integrals

We will use the formulas already given for the indefinite integral, but we will take one additional step.

Evaluate the integral at the upper bound and subtract the value of the integral at the lower bound.

b

aafbfcafcbfdxxf )()())(()()('

Definite Integral Examples

1

02xdx

1

1dtet

Try These

dyyy )36( 5

dxex

x21

4

0

2dxx

Solutions

cyydyyy 265

2

3)36(

cexdxex

xx

22

2

1)log(

1

3

644

0

2 dxx

Integrals of Trig. Functions

cuduu )cos()sin( cuduu )cot()(csc2

cuduu )sin()cos( cuduuu )sec()tan()sec(

cuduu )tan()(sec2 cuduuu )csc()cot()csc(

Definite Trig. Integral

dxx2/

0)cos(

2/0)sin( x

)0sin(2/sin

01

1

Calculus in Statistics

There are many calculus applications in statistics including: - continuous density functions - finding the expected value (mean) of a distribution - finding the variance (standard deviation) of a distribution - finding the median of a distribution

Probability Function

The probability function of a variable tells us the probability of a certain event.

Example:The probability of getting a jack whendrawing from a normal deck of cards is4/52 = 1/13.

Continuous Density Function

The probability density function (pdf) of a variable tells us the probability of a certain event when a continuum of events is possible.

Example:The probability of getting the exact valueof π from the set of all real numbers.

The pdf of x is usually noted by the lowercase f, i.e. f(x).

Cumulative Density Function

The cumulative density function (cdf) of a variable tells us the probability for all events in a range.

The cdf of x is usually noted by the capital F, i.e. F(x).

Example:For a standard normal distribution,F(-0.22)=P(z≤-0.22)=0.4129.

Notes About the pdf

The pdf of a variable is said to have a value of 0 for any specific event.

Example:From our example earlier, we can neverget the exact value of π.

For all x, f(x)≥0.

The integral of the pdf over all events is 1.

Notes About the cdf

The cdf has values between 0 and 1. The limit as x approaches -∞ is 0 and the limit as x approaches ∞ is 1.

The cdf is an increasing function. The cdf can be ‘flat’ for a range of values, so some people prefer to think of it as non-decreasing.

Calculus and pdf & cdf

The cdf measures the area under the pdf.

When given the pdf, we can do an indefinite integral to find the cdf. If given the cdf, we can differentiate to find the pdf.

We will also use calculus to find information about the pdf such as the mean, median, and variance.

Try This

Show that the pdf f(x)=1/6 integrates to 1 when x=(0,6). Find F(x).

Answer

Show that the pdf f(x)=1/6 integrates to 1 when x=(0,6). Find F(x).

xxF

xdxdxxf

6

1)(

10*6

16*

6

1

6

1

6

1)(

6

0

6

0

6

0

Try This One Too

Given the cdf F(t)=1-e-t when t=(0,∞), find the pdf and show that it integrates to 1.

Another Answer

Given the cdf F(t)=1-e-t when t=(0,∞), find the pdf and show that it integrates to 1.

1)1(0)()(

)(

0

000

eeedtedttf

etf

tt

t

Expected Value

The expected value is the mean of a distribution.

Example:The expected value for the standard

normal is 0.

The expected value is commonly written as E(X).

To find the expected value, we use the following formula.

The expected value of f(x)=1/6 where x=(0,6) is

Expected Value Formula

dxxfx )(*E(X)

30312

1

6

1*)(*E(X)

6

0

26

0

6

0 xdxxdxxfx

Median

The median of a distribution is the number in the middle. 50% of the area under the distribution is above the value and 50% is below it.

The median, like the expected value (mean), is a measure of the center of a distribution.

The median of the distribution function f(x) is commonly written .X~

To find the median, we use the following formula. In this formula, M is the median.

To find the median of f(x)=1/6 where x=(0,6), we solve the equation above for M.

So M= =3.

Median Formula

Mdxxf 5.0)(

66

1

6

1)(5.0

000

MxdxdxxfM

MM

X~

The variance of a distribution function tells us about how spread out the possible events are. A large variance means the events are more spread out.

The standard deviation is the square root of the variance.

The variance is often written as V(X) or Var(X).

Variance

To find the variance, we use the following formula.

E(X), the expected value, is the same as before and E(X2) is computed similarly.

Variance Formula

22 E(X))()E(XVar(X)

dxxfx )(*)E(X 22

To find the variance of f(x)=1/6 where x=(0,6), first find E(X2).

Now use the variance formula.

Variance Example

1201218

1

6

1*)(*)E(X

6

0

36

0

26

0

22 xdxxdxxfx

3)3(12E(X))()E(XVar(X) 222

Try This

Find the expected value, median, and variance for the distribution f(y)=2y for y in (0,1).

Answer

Find the expected value, median, and variance for the distribution f(y)=2y for y in (0,1).

1/18Var(X)

0.5X~

2/3E(X)