Inverse Functions Notes 3.8

I. Inverse Functions A.) B.) Ex. –

C.) D.) Symmetric to the line y = x. E.) Notation – F.) Existence: A function has an inverse iff for any two x values Horizontal Line Test for Inverses

II. Inverse Theorems A.) B.) C.) Ex. – Given Does it have an inverse, and if so, what is it? Always positive, therefore always increasing! Cannot solve for y!

D.) Derivatives of Inverse Functions: and

E.) Ex- Given inverse functions and Notice

F.) Ex- Given inverse functions and

G.) Ex- Given 1.) Does it have an inverse? 2.) If it does, find it and then find its derivative. 3.) Verify the inverse derivative theorem on

Always positive, therefore always increasing, and it has an inverse

You verify (f (5), 5)!!

H.) Notice, f (3) = 9. The most confusing aspect of the inverse derivative theorem is that you are asked to find the derivative at a value of x. You are really being asked to find the derivative of the inverse function at the value that corresponds to f (x) = 9.

I.)

J.)

Derivatives of Inverse Trig Functions Notes 3.8 Part II

I. y = sin -1 x

II. y = cos -1 x

III. y = tan -1 x

IV. y = cot -1 x

V. y = sec -1 x

VI. y = csc -1 x

Find y’ in each of the following: VII. Examples