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# Lesson 5: Tangents, Velocity, the Derivative

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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

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### Lesson 5: Tangents, Velocity, the Derivative

1. 1. Sections 2.6 and 2.7 Tangents, Velocity, and the Derivative Math 1a February 11, 2008 Announcements All HW on website Oﬃce Hours Tuesday, Wednesday 2–4pm (SC 323)
2. 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. 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. 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. 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. 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. 7. Upshot The instantaneous velocity is given by h(t + ∆t) − h(t) v = lim ∆t→0 ∆t
8. 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. 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. 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. 11. Upshot The instantaneous population growth is given by P(t + ∆t) − P(t) lim ∆t→0 ∆t
12. 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. 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. 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. 15. Upshot The marginal cost after producing q is given by C (q + ∆q) − C (q) MC = lim ∆q→0 ∆q
16. 16. The deﬁnition All of these rates of change are found the same way!
17. 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. 18. Derivative of the squaring function Example Suppose f (x) = x 2 . Use the deﬁnition of derivative to ﬁnd f (x).
19. 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. 20. Worksheet