Solve these questions before class:
y cos x
Give the equation for the tangent line to the graph of
f x 2 3 s i n x at the point where x = 0.
a a x ln a
a u' au ln a
log a x
x ln a
log a u
u ln a
a dx ln a C
a du ln a C
Example: Find the derivatives:
y 52 x
y 2 sin x
y log 2 5 x
y log6 x 2 x
log x 3
Example: 2 dx
Example: Find the solution to this differential equation subject to the given
y' 5y y 0 6
Growth and Decay
For example, exponential Growth and Decay models are used in:
Newton’s Law of Cooling
Write and solve a differential equation for the rate of change of y with
respect to time is proportional to y, given that y > 0.
Example: Find a function that satisfies y’ = –2y and y(0) = 3
P(t) = population at time t
P t P0 e k t
P0= P(0) = initial population
k = growth rate
A t A 0 e kt
A(t) = amount at time t
A0= A(0) = initial amount
k = growth/decay rate
Continuous Compound Interest:
A t A 0 e rt
A(t) = principle at time t
A0= A(0) = initial investment
r = annual interest rate
Note that in all cases we have a quantity that changes at a rate proportional to
Example: At what rate r of continuous compounding does a sum of money
double in 10 years?
Example: In a bacteria growing experiment, a biologist observes that the
number of bacteria in a certain culture triples every 4 hours. After 12
hours, it is estimated that there are 1 million bacteria in the culture.
a. How many bacteria were present initially?
b. What is the doubling time for the bacteria population?
Example: A 100-liter tank initially full of water develops a leak at the
bottom. Given that 10% of the water leaks out in the first 5 minutes, find
the amount of water left in the tank 15 minutes after the leak develops if the
water drains off at a rate that is proportional to the amount of water present.
Next week, we will cover derivatives and integrals of inverse trig functions. Review
inverse trig functions before class!
Exercise: Suppose a culture of bacteria is growing in such a way that the
change in the number of bacteria is proportional to the number present.
The number of bacteria doubles every 200 minutes and there are currently
5000 bacteria in the culture. How many bacteria were present 2 hours ago?