Download AP Calculus AB - Van Buren Public Schools

AP Calculus AB
5.5 Exponential Applications
Objectives: Define exponential functions that
have a base other than e, differentiate and
integrate exponential functions, and use the
functions to model compound interest and
exponential growth.
Car Depreciation

After t years, the value of a car purchased
for $20,000 is 𝑣 𝑡 = 20000
3 𝑡
.
4
Find the rate of change of the
depreciation when 𝑡 = 1, 𝑡 = 4.
 What do we need to find first?
 Find the first derivative

◦ Rate of change at a point (Instantaneous)
Car Depreciation

After t years, the value of a car purchased for $20,000 is
𝑣 𝑡 = 20000

3 𝑡
.
4
Find the rate of change of the depreciation when 𝑡 =
1, 𝑡 = 4.
𝑣 𝑡 = 20000𝑎𝑡
𝑡
3 3
′
𝑣 𝑡 = 20000 ln
4 4
𝑣 ′ 1 = −$4315.23
𝑣 ′ 4 = −$1820.49
The older the car the less it depreciates
Compound Interest
A = Balance after t years
 P = Amount Deposited (Principal)
 r = Interest Rate (Percent needs to be
written as a decimal)
 n = Number Compounded per Year
 t = Time in Years
 Compounded n times per year
𝑟 𝑛𝑡
𝐴=𝑃 1+
𝑛
 Compounded Continuously
𝐴 = 𝑃𝑒 𝑟𝑡

Interest Example
A deposit of $2500 in an account pays 5%
per year. Find the balance after 5 years
 Compounded

A.
B.
C.
D.
Quarterly
Monthly
Daily
Continously
Interest Example
A deposit of $2500 in an account pays 5%
per year. Find the balance after 5 years
 Compounded

A. Quarterly
 P = $2500, n = 4, r = .05, t = 5
 𝐴 =𝑃 1+
𝑟 𝑛𝑡
𝑛
.05
𝐴 = 2500 1 +
4
4(5)
≈ $3205.09
Interest Example
A deposit of $2500 in an account pays 5%
per year. Find the balance after 5 years
 Compounded

B. Monthly
 P = $2500, n = 12, r = .05, t = 5
 𝐴 =𝑃 1+
𝑟 𝑛𝑡
𝑛
.05
𝐴 = 2500 1 +
12
12(5)
≈ $3208.40
Interest Example
A deposit of $2500 in an account pays 5%
per year. Find the balance after 5 years
 Compounded

C. Daily
 P = $2500, n = 365, r = .05, t = 5
 𝐴 =𝑃 1+
𝑟 𝑛𝑡
𝑛
.05
𝐴 = 2500 1 +
365
365(5)
≈ $3210.01
Interest Example
A deposit of $2500 in an account pays 5%
per year. Find the balance after 5 years
 Compounded

D. Continuously
 P = $2500, n = continuously, r = .05, t = 5
 𝐴 = 𝑃𝑒 𝑟𝑡
𝐴 = 2500𝑒 .05(5) ≈ $3210.06
The more it is compounded, the more you earn
Pg. 368 (81, 83, 89-93)

#90

a) 182.21
b) 271.83
c) 332.01
#92
a) 86%
b) 𝑃′ 3 ≈ 0.047
c) 𝑃′(10) ≈ 0.015
The answers in the back on Pg. A68 are out of order
#81 is #83, #83 is #85, #85 is #87, #87 is #81
Formative Assessment