Historical Note Maria Gaetana Agnesi

1718-1799 - Milan, Habsburg Empire (now Italy) Oldest of 21children (3 mothers) Wealthy (and busy) father She spoke three languages Writer/Debater of philosophy and natural science Studied religious books and mathematics Admired Newton Learned Calculus from a monk - Ramiro Rampinelli He encouraged her to write a book on Calculus - She is

famous for writing that book in 1748

Increasing and Decreasing Functions

(page 290)

Closed and Open Interval Notation

(page 291)

There is not universal agreement on derivative interval notation.

The theorem below shows this author’s preference. AP Exam will accept open or closed notation.

First and Second Derivative Summary Concepts

0 is negative

0 is decrea

Graph Below x-axi

sing

0 is concave d n

s

ow

f x f x

f x f x

f x f x

0 is a root/solution/x-intercept

0 is a maximum/minimum/inflection point

0 behavior is incon

Graph at x-

clusi

a

ve

xis

f x x

f x f x

f x f x

0 is positive

0 is increasing

0 is concave

Graph Abov

up

e x-axis

f x f x

f x f x

f x f x

5.2 Extrema(page 300-305)

Relative (Local) Maximum or Minimum Values

Absolute (Global) Maximum or Minimum Value

Critical Numbers(page 302)

Critical Numbers Summary(page 300)

Definition of Critical Numbers - The numbers a which either 0 or is not differentiable are called of .critical numbers

f xf f

Points where 0 are called stationary points and occur where has a maximum, minimum or an inflection point.

f xf x

Inflection points mark the places on the curve where the rate of change of , , with respect to changes form increasing to decreasing, or visa versa.

y f xy f x x

Points where DNE occur where has a cusp, corner point, vertical tangent line or point of discontinuity.

f x f x

Critical Numbers(page 302)

Critical Numbers and Relative Extrema

(page 302)

Relative Extrema occur at critical numbers.

However, not every critical number has a relative extrema.

First Derivative Test(page 302)

If changes from increasing to decreasing at a critical number, then there is relative maximum.f x

If changes from decreasing to increasing at a critical number, then there is relative minimum.f x

If has the same sign on either side of a critical number, then there is no extreme value.f x

Second Derivative Test(page 303)

If =0 and >0, then has a minimum.f x f x f

If =0 and <0, then has a maximum.f x f x f

Homework Problem #3(page 298)

Homework Problem #4(page 296)

Homework Problem #5(page 298)

Homework Problem #6(page 298)

Homework Problem #7(page 298)

Example 5a,5b,5c(page 295)

6a xf x xe 6b sin ,

0 2

f x x

x

16c tanf x x

Example 2(page 303)

Example 3(page 303)

Example 4(page 304)