Many problems in science are about rates of change. They boil down to the mathematical question of finding the slope of a line tangent to a curve. We state this quantity as a limit and give it a name: the derivative

1. Sections 2.6 and 2.7
Tangents, Velocity, and the Derivative
Math 1a
February 11, 2008
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2. The tangent problem
Problem
Given a curve and a point on the curve, find the slope of the line
tangent to the curve at that point.

3. The tangent problem
Problem
Given a curve and a point on the curve, find the slope of the line
tangent to the curve at that point.
Example
We can do this in Geogebra for y = x 2 .

4. The tangent problem
Problem
Given a curve and a point on the curve, find the slope of the line
tangent to the curve at that point.
Example
We can do this in Geogebra for y = x 2 .
Upshot
If the curve is given by y = f (x), and the point on the curve is
(a, f (a)), then the slope of the tangent line is given by
f (x) − f (a)
mtangent = lim
x→a x −a

5. Rates of Change
Problem
Given the position function of a moving object, find the velocity of
the object at a certain instant in time.
Example
Drop a ball off the roof of the science center so that is height can
be described by
h(t) = 30 − 10t 2
where t is seconds after dropping it and h is meters above the
ground. How fast is it falling one second after we drop it?

6. Rates of Change
Problem
Given the position function of a moving object, find the velocity of
the object at a certain instant in time.
Example
Drop a ball off the roof of the science center so that is height can
be described by
h(t) = 30 − 10t 2
where t is seconds after dropping it and h is meters above the
ground. How fast is it falling one second after we drop it?
Solution
The answer is
(30 − 10t 2 ) − 20
lim = −20.
t→1 t −1

7. Upshot
The instantaneous velocity is given by
h(t + ∆t) − h(t)
v = lim
∆t→0 ∆t

8. Rates of Change
Problem
Given the population function of a group of organisms, find the
rate of growth of the population at a particular instant.

9. Rates of Change
Problem
Given the population function of a group of organisms, find the
rate of growth of the population at a particular instant.
Example
Suppose the population of fish in the Charles River is given by the
function
3e t
P(t) =
1 + et
where t is in years since 2000 and P is in millions of fish. Is the
fish population growing fastest in 1990, 2000, or 2010? (Estimate
numerically)?

10. Rates of Change
Problem
Given the population function of a group of organisms, find the
rate of growth of the population at a particular instant.
Example
Suppose the population of fish in the Charles River is given by the
function
3e t
P(t) =
1 + et
where t is in years since 2000 and P is in millions of fish. Is the
fish population growing fastest in 1990, 2000, or 2010? (Estimate
numerically)?
Solution
The estimated rates of growth are 0.000136, 0.75, and 0.000136.

11. Upshot
The instantaneous population growth is given by
P(t + ∆t) − P(t)
lim
∆t→0 ∆t

12. Rates of Change
Problem
Given the production cost of a good, find the marginal cost of
production after having produced a certain quantity.

13. Rates of Change
Problem
Given the production cost of a good, find the marginal cost of
production after having produced a certain quantity.
Example
Suppose the cost of producing q tons of rice on our paddy in a
year is
C (q) = q 3 − 12q 2 + 60q
We are currently producing 5 tons a year. Should we change that?

14. Rates of Change
Problem
Given the production cost of a good, find the marginal cost of
production after having produced a certain quantity.
Example
Suppose the cost of producing q tons of rice on our paddy in a
year is
C (q) = q 3 − 12q 2 + 60q
We are currently producing 5 tons a year. Should we change that?
Example
If q = 5, then C = 125, MC = 15, while AC = 25. So we should
produce more to lower average costs.

15. Upshot
The marginal cost after producing q is given by
C (q + ∆q) − C (q)
MC = lim
∆q→0 ∆q

16. The definition
All of these rates of change are found the same way!

17. The definition
All of these rates of change are found the same way!
Definition
Let f be a function and a a point in the domain of f . If the limit
f (a + h) − f (a)
f (a) = lim
h→0 h
exists, the function is said to be differentiable at a and f (a) is
the derivative of f at a.

18. Derivative of the squaring function
Example
Suppose f (x) = x 2 . Use the definition of derivative to find f (x).

19. Derivative of the squaring function
Example
Suppose f (x) = x 2 . Use the definition of derivative to find f (x).
Answer
f (x) = 2x.

20. Worksheet