AP Calculus BC
Monday, 01 February 2016
• OBJECTIVE TSW (1) differentiate and integrate
exponential functions that have bases other than e, and
(2) use exponential functions to model compound interest
and exponential growth.
• TESTS are not graded.
• Reminder
– AMC test is tomorrow (down LGI).
AP Exam Registration
• Registration must be done on line.
www.TotalRegistration.net/AP/443381
• Regular registration: 01/05/2016 – 03/04/2016.
• Late registration: 03/05/2016 – 03/20/2016
– Additional $10 per test fee added on.
• Cost per test:
– $96/test
• Can apply for a Cy-Hope scholarship (reduction of $25/test, up to 3
tests).
• Pick up applications at Counselors’ Corner or at Mr. Hernandez’s
office or Ms. Lewis’ office.
– $11/test (free/reduced lunch program)
• Not eligible for Cy-Hope scholarship.
Bases Other Than e and
Applications
Bases Other Than e and
Applications
“There are 10 types of people in this world:
Those who understand binary . . .
. . . and those who don’t.”
Bases Other Than e and
Applications
Bases Other Than e and
Applications
Another way to think of this is
loge x
1
loga x 
ln x 
ln a
loge a
Bases Other Than e and
Applications
loga x
has the same properties as other logarithms:
1)
loga 1  0
2)
loga xy  loga x  loga y
3)
x
loga  loga x  loga y
y
4)
loga x n  n loga x
Bases Other Than e and
Applications
loga x also shares the inverse properties:
Bases Other Than e and
Applications
Find the derivative of each of the following:
1) a x a is a constant
x
ln a
loge a
x

e
a  e

  
x
 e
lna  x
d
d  lna  x 
x
a  
e

dx
dx 
 e
ln a  x
 ln a
 ln a  e
ln a  x
 ln a  a
x
2) au
d
u
u du
a    ln a  a
dx
dx
Bases Other Than e and
Applications
Find the derivative of each of the following:
ln x
3) loga x 
ln a
d
d  ln x 
loga x  

dx
dx  ln a 
4) loga u
1 du
d
loga u  

ln a  u dx
dx
d  1


 ln x 

dx  ln a

1 1
1



lna x
lna  x
Bases Other Than e and
Applications
Bases Other Than e and
Applications
Find the derivative:
x
x

y  ln2  2
3x
3x

y  ln5  5  3
a) y  2
b) y  5
y    3ln5  53 x
 sin x
c) y  log cos x y  
ln10  cos x
(What's the base?)
1
y  
tan x
ln10
Bases Other Than e and
Applications
Integral of an exponential function to a base
other than e:
 1  x
 a dx   ln a  a  C
x
Bases Other Than e and
Applications
Integrate:
1 x
 6 dx  ln 6 6  C
x
Bases Other Than e and
Applications
ROM!!!
Bases Other Than e and
Applications

Continuous Compound Interest Formula
A  Pe r t
Memorize ! ! !