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Lesson 7: The Derivative (handout)

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Many functions in nature are described as the rate of change of another function. The concept is called the derivative. Algebraically, the process of finding the derivative involves a limit of …

Many functions in nature are described as the rate of change of another function. The concept is called the derivative. Algebraically, the process of finding the derivative involves a limit of difference quotients.

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  • 1. . V63.0121.001: Calculus I . Sec on 2.1–2.2: The Deriva ve . February 14, 2011 Notes Sec on 2.1–2.2 The Deriva ve V63.0121.001: Calculus I Professor Ma hew Leingang New York University . February 14, 2011 . NYUMathematics . Notes Announcements Quiz this week on Sec ons 1.1–1.4 No class Monday, February 21 . . Objectives Notes The Derivative Understand and state the defini on of the deriva ve of a func on at a point. Given a func on and a point in its domain, decide if the func on is differen able at the point and find the value of the deriva ve at that point. Understand and give several examples of deriva ves modeling rates of change in science. . . . 1.
  • 2. . V63.0121.001: Calculus I . Sec on 2.1–2.2: The Deriva ve . February 14, 2011 Objectives Notes The Derivative as a Function Given a func on f, use the defini on of the deriva ve to find the deriva ve func on f’. Given a func on, find its second deriva ve. Given the graph of a func on, sketch the graph of its deriva ve. . . Notes Outline Rates of Change Tangent Lines Velocity Popula on growth Marginal costs The deriva ve, defined Deriva ves of (some) power func ons What does f tell you about f′ ? How can a func on fail to be differen able? Other nota ons The second deriva ve . . The tangent problem Notes A geometric rate of change Problem Given a curve and a point on the curve, find the slope of the line tangent to the curve at that point. Solu on . . . 2.
  • 3. . V63.0121.001: Calculus I . Sec on 2.1–2.2: The Deriva ve . February 14, 2011 Notes A tangent problem Example Find the slope of the line tangent to the curve y = x2 at the point (2, 4). . . Notes Graphically and numerically y x2 − 22 x m= x−2 3 5 9 2.5 4.5 2.1 4.1 6.25 2.01 4.01 limit 4.41 4.0401 4 3.9601 3.61 1.99 3.99 2.25 1.9 3.9 1 1.5 3.5 . x 1 3 1 1.52.1 3 1.99 2.01 1.92.5 2 . . The velocity problem Notes Kinematics—Physical rates of change Problem Given the posi on func on of a moving object, find the velocity of the object at a certain instant in me. . . . 3.
  • 4. . V63.0121.001: Calculus I . Sec on 2.1–2.2: The Deriva ve . February 14, 2011 Notes A velocity problem Example Solu on Drop a ball off the roof of the Silver Center so that its height can be described by h(t) = 50 − 5t2 where t is seconds a er dropping it and h is meters above the ground. How fast is it falling one second a er we drop it? . . Notes Numerical evidence h(t) = 50 − 5t2 Fill in the table: h(t) − h(1) t vave = t−1 2 1.5 1.1 1.01 1.001 . . Notes Velocity in general Upshot y = h(t) If the height func on is given h(t0 ) by h(t), the instantaneous ∆h velocity at me t0 is given by h(t0 + ∆t) h(t) − h(t0 ) v = lim t→t0 t − t0 h(t0 + ∆t) − h(t0 ) = lim ∆t→0 ∆t . ∆t t t0 t . . . 4.
  • 5. . V63.0121.001: Calculus I . Sec on 2.1–2.2: The Deriva ve . February 14, 2011 Population growth Notes Biological Rates of Change Problem Given the popula on func on of a group of organisms, find the rate of growth of the popula on at a par cular instant. Solu on . . Notes Population growth example Example Suppose the popula on of fish in the East River is given by the func on 3et P(t) = 1 + et where t is in years since 2000 and P is in millions of fish. Is the fish popula on growing fastest in 1990, 2000, or 2010? (Es mate numerically) Answer . . Notes Derivation Solu on Let ∆t be an increment in me and ∆P the corresponding change in popula on: ∆P = P(t + ∆t) − P(t) This depends on ∆t, so ideally we would want ( ) ∆P 1 3et+∆t 3et lim = lim − ∆t→0 ∆t ∆t→0 ∆t 1 + et+∆t 1 + et But rather than compute a complicated limit analy cally, let us approximate numerically. We will try a small ∆t, for instance 0.1. . . . 5.
  • 6. . V63.0121.001: Calculus I . Sec on 2.1–2.2: The Deriva ve . February 14, 2011 Notes Numerical evidence Solu on (Con nued) To approximate the popula on change in year n, use the difference P(t + ∆t) − P(t) quo ent , where ∆t = 0.1 and t = n − 2000. ∆t ( ) P(−10 + 0.1) − P(−10) 1 3e−9.9 3e−10 r1990 ≈ = − 0.1 0.1 1 + e−9.9 1 + e−10 = ( ) P(0.1) − P(0) 1 3e0.1 3e0 r2000 ≈ = − 0.1 0.1 1 + e0.1 1 + e0 = . . Notes Solu on (Con nued) ( ) P(10 + 0.1) − P(10) 1 3e10.1 3e10 r2010 ≈ = 10.1 − 0.1 0.1 1+e 1 + e10 = . . Marginal costs Notes Rates of change in economics Problem Given the produc on cost of a good, find the marginal cost of produc on a er having produced a certain quan ty. Solu on The marginal cost a er producing q is given by C(q + ∆q) − C(q) MC = lim ∆q→0 ∆q . . . 6.
  • 7. . V63.0121.001: Calculus I . Sec on 2.1–2.2: The Deriva ve . February 14, 2011 Notes Marginal Cost Example Example Suppose the cost of producing q tons of rice on our paddy in a year is C(q) = q3 − 12q2 + 60q We are currently producing 5 tons a year. Should we change that? Answer . . Notes Comparisons Solu on C(q) = q3 − 12q2 + 60q Fill in the table: q C(q) AC(q) = C(q)/q ∆C = C(q + 1) − C(q) 4 5 6 . . Notes Outline Rates of Change Tangent Lines Velocity Popula on growth Marginal costs The deriva ve, defined Deriva ves of (some) power func ons What does f tell you about f′ ? How can a func on fail to be differen able? Other nota ons The second deriva ve . . . 7.
  • 8. . V63.0121.001: Calculus I . Sec on 2.1–2.2: The Deriva ve . February 14, 2011 Notes The definition All of these rates of change are found the same way! Defini on Let f be a func on and a a point in the domain of f. If the limit f(a + h) − f(a) f(x) − f(a) f′ (a) = lim = lim h→0 h x→a x−a exists, the func on is said to be differen able at a and f′ (a) is the deriva ve of f at a. . . Notes Derivative of the squaring function Example Suppose f(x) = x2 . Use the defini on of deriva ve to find f′ (a). Solu on f(a + h) − f(a) (a + h)2 − a2 f′ (a) = lim = lim h→0 h h→0 h (a2 + 2ah + h2 ) − a2 2ah + h2 = lim = lim h→0 h h→0 h = lim (2a + h) = 2a h→0 . . Notes Derivative of the reciprocal Example 1 Suppose f(x) = . Use the defini on of the deriva ve to find f′ (2). x Solu on y . x . . . 8.
  • 9. . V63.0121.001: Calculus I . Sec on 2.1–2.2: The Deriva ve . February 14, 2011 Notes What does f tell you about f′? If f is a func on, we can compute the deriva ve f′ (x) at each point x where f is differen able, and come up with another func on, the deriva ve func on. What can we say about this func on f′ ? If f is decreasing on an interval, f′ is nega ve (technically, nonposi ve) on that interval If f is increasing on an interval, f′ is posi ve (technically, nonnega ve) on that interval . . Notes What does f tell you about f′? Fact If f is decreasing on the open interval (a, b), then f′ ≤ 0 on (a, b). Picture Proof. If f is decreasing, then all secant lines point downward, hence have y nega ve slope. The deriva ve is a limit of slopes of secant lines, which are all nega ve, so the limit must be ≤ 0. . x . . Notes What does f tell you about f′? Fact If f is decreasing on on the open interval (a, b), then f′ ≤ 0 on (a, b). The Real Proof. If ∆x > 0, then f(x + ∆x) − f(x) f(x + ∆x) < f(x) =⇒ <0 ∆x If ∆x < 0, then x + ∆x < x, and f(x + ∆x) − f(x) f(x + ∆x) > f(x) =⇒ <0 . ∆x . . 9.
  • 10. . V63.0121.001: Calculus I . Sec on 2.1–2.2: The Deriva ve . February 14, 2011 Notes What does f tell you about f′? Fact If f is decreasing on on the open interval (a, b), then f′ ≤ 0 on (a, b). The Real Proof. f(x + ∆x) − f(x) Either way, < 0, so ∆x f(x + ∆x) − f(x) f′ (x) = lim ≤0 ∆x→0 ∆x . . Notes Going the Other Way? Ques on If a func on has a nega ve deriva ve on an interval, must it be decreasing on that interval? Answer Maybe. . . Notes Outline Rates of Change Tangent Lines Velocity Popula on growth Marginal costs The deriva ve, defined Deriva ves of (some) power func ons What does f tell you about f′ ? How can a func on fail to be differen able? Other nota ons The second deriva ve . . . 10.
  • 11. . V63.0121.001: Calculus I . Sec on 2.1–2.2: The Deriva ve . February 14, 2011 Notes Differentiability is super-continuity Theorem If f is differen able at a, then f is con nuous at a. Proof. We have f(x) − f(a) lim (f(x) − f(a)) = lim · (x − a) x→a x→a x−a f(x) − f(a) = lim · lim (x − a) x→a x−a x→a ′ = f (a) · 0 = 0 . . Differentiability FAIL Notes Kinks Example Let f have the graph on the le -hand side below. Sketch the graph of the deriva ve f′ . f(x) f′ (x) . x . x . . Differentiability FAIL Notes Cusps Example Let f have the graph on the le -hand side below. Sketch the graph of the deriva ve f′ . f(x) f′ (x) . x . x . . . 11.
  • 12. . V63.0121.001: Calculus I . Sec on 2.1–2.2: The Deriva ve . February 14, 2011 Differentiability FAIL Notes Vertical Tangents Example Let f have the graph on the le -hand side below. Sketch the graph of the deriva ve f′ . f(x) f′ (x) . x . x . . Differentiability FAIL Notes Weird, Wild, Stuff Example f(x) f′ (x) . x . x This func on is differen able But the deriva ve is not at 0. con nuous at 0! . . Notes Outline Rates of Change Tangent Lines Velocity Popula on growth Marginal costs The deriva ve, defined Deriva ves of (some) power func ons What does f tell you about f′ ? How can a func on fail to be differen able? Other nota ons The second deriva ve . . . 12.
  • 13. . V63.0121.001: Calculus I . Sec on 2.1–2.2: The Deriva ve . February 14, 2011 Notes Notation Newtonian nota on f′ (x) y′ (x) y′ Leibnizian nota on dy d df f(x) dx dx dx These all mean the same thing. . . Meet the Mathematician Notes Isaac Newton English, 1643–1727 Professor at Cambridge (England) Philosophiae Naturalis Principia Mathema ca published 1687 . . Meet the Mathematician Notes Gottfried Leibniz German, 1646–1716 Eminent philosopher as well as mathema cian Contemporarily disgraced by the calculus priority dispute . . . 13.
  • 14. . V63.0121.001: Calculus I . Sec on 2.1–2.2: The Deriva ve . February 14, 2011 Notes Outline Rates of Change Tangent Lines Velocity Popula on growth Marginal costs The deriva ve, defined Deriva ves of (some) power func ons What does f tell you about f′ ? How can a func on fail to be differen able? Other nota ons The second deriva ve . . Notes The second derivative If f is a func on, so is f′ , and we can seek its deriva ve. f′′ = (f′ )′ It measures the rate of change of the rate of change! Leibnizian nota on: d2 y d2 d2 f 2 2 f(x) dx dx dx2 . . Notes Function, derivative, second derivative y f(x) = x2 f′ (x) = 2x f′′ (x) = 2 . x . . . 14.
  • 15. . V63.0121.001: Calculus I . Sec on 2.1–2.2: The Deriva ve . February 14, 2011 Summary Notes What have we learned today? The deriva ve measures instantaneous rate of change The deriva ve has many interpreta ons: slope of the tangent line, velocity, marginal quan es, etc. The deriva ve reflects the monotonicity (increasing-ness or decreasing-ness) of the graph . . Notes . . Notes . . . 15.