04/19/23 Perkins

AP Calculus AB

Day 9Section 5.4

ude

dx Theorem:

122. x

de

dx

u due

dx

3. lnxde x

dx

1. Find the slope of the line tangent to f (x) at x = 3.

2' 2xf x e 2xf x e 6' 3 2f e

6

2

e

2 32xe x

2xde

dx

123

2xx e

1lnx xe e x

x

1

lnxe xx

1 1ln

xx

e x

1 lnx

x x

xe

4. Find extrema and inflection points for 2 xy x e

2 2x xdyx e e x

dx

20 2 0xe x x Crit #’s:

Can’t ever work.

0,2x

2 2xe x x

2

22

2 2 2x xd ye x e x x

dx

22 2 2xe x x x

2 4 2xe x x Crit #’s:

20 4 2 0xe x x none

24 4 4 1 2

2 1x

4 8

2x

4 2 2

2x

2 2x

2 1.4x

.6,3.4x

Intervals: ,0 0,.6 .6,2

Test values: 1 12 1

f ’(test pt) f(x) dec inc inc

rel min rel max

0x 2x 0,0 2

42,e

f ’’(test pt) f(x) up up down

2,3.4 3.4,3 5

dec dec

down up

2 2xdye x x

dx

22

24 2xd y

e x xdx

2xe x x

Inf pt Inf pt

.6x 3.4x 0.586,0.191 3.414,0.384

2 xy x e

Perkins

AP Calculus AB

Day 9Section 5.4

ude

dx Theorem:

122. x

de

dx

3. lnxde x

dx

1. Find the slope of the line tangent to f (x) at x = 3.

2xf x e

4. Find extrema and inflection points for 2 xy x e