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The derivative is the instantaneous rate of change of a function.

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- 1. V63.0121.041, Calculus I Section 2.1–2.2 : The Derivative September 26, 2010 Section 2.1–2.2 Notes The Derivative and Rates of Change The Derivative as a Function V63.0121.041, Calculus I New York University September 26, 2010 Announcements Quiz this week in recitation on §§1.1–1.4 Get-to-know-you/photo due Friday October 1 Announcements Notes Quiz this week in recitation on §§1.1–1.4 Get-to-know-you/photo due Friday October 1 V63.0121.041, Calculus I (NYU) Section 2.1–2.2 The Derivative September 26, 2010 2 / 46 Format of written work Notes Please: Use scratch paper and copy your ﬁnal work onto fresh paper. Use loose-leaf paper (not torn from a notebook). Write your name, lecture section, assignment number, recitation, and date at the top. Staple your homework together. See the website for more information. V63.0121.041, Calculus I (NYU) Section 2.1–2.2 The Derivative September 26, 2010 3 / 46 1
- 2. V63.0121.041, Calculus I Section 2.1–2.2 : The Derivative September 26, 2010 Objectives for Section 2.1 Notes Understand and state the deﬁnition of the derivative of a function at a point. Given a function and a point in its domain, decide if the function is diﬀerentiable at the point and ﬁnd the value of the derivative at that point. Understand and give several examples of derivatives modeling rates of change in science. V63.0121.041, Calculus I (NYU) Section 2.1–2.2 The Derivative September 26, 2010 4 / 46 Objectives for Section 2.2 Notes Given a function f , use the deﬁnition of the derivative to ﬁnd the derivative function f’. Given a function, ﬁnd its second derivative. Given the graph of a function, sketch the graph of its derivative. V63.0121.041, Calculus I (NYU) Section 2.1–2.2 The Derivative September 26, 2010 5 / 46 Outline Notes Rates of Change Tangent Lines Velocity Population growth Marginal costs The derivative, deﬁned Derivatives of (some) power functions What does f tell you about f ? How can a function fail to be diﬀerentiable? Other notations The second derivative V63.0121.041, Calculus I (NYU) Section 2.1–2.2 The Derivative September 26, 2010 6 / 46 2
- 3. V63.0121.041, Calculus I Section 2.1–2.2 : The Derivative September 26, 2010 The tangent problem Notes Problem Given a curve and a point on the curve, ﬁnd the slope of the line tangent to the curve at that point. Example Find the slope of the line tangent to the curve y = x 2 at the point (2, 4). 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 V63.0121.041, Calculus I (NYU) Section 2.1–2.2 The Derivative September 26, 2010 7 / 46 Graphically and numerically Notes y x 2 − 22 x m= x −2 3 5 9 2.5 4.5 2.1 4.1 2.01 4.01 6.25 limit 4.41 1.99 3.99 4.0401 4 3.9601 1.9 3.9 3.61 1.5 3.5 2.25 1 3 1 x 1 1.5 2.1 3 1.99 1.9 2.5 2.01 2 V63.0121.041, Calculus I (NYU) Section 2.1–2.2 The Derivative September 26, 2010 8 / 46 Velocity Notes 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 Silver Center so that its height can be described by h(t) = 50 − 5t 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 V63.0121.041, Calculus I (NYU) Section 2.1–2.2 The Derivative September 26, 2010 10 / 46 3
- 4. V63.0121.041, Calculus I Section 2.1–2.2 : The Derivative September 26, 2010 Numerical evidence Notes h(t) = 50 − 5t 2 Fill in the table: h(t) − h(1) t vave = t −1 2 1.5 1.1 1.01 1.001 V63.0121.041, Calculus I (NYU) Section 2.1–2.2 The Derivative September 26, 2010 11 / 46 Velocity in general Notes y = h(t) Upshot h(t0 ) If the height function is given by h(t), the instantaneous velocity ∆h at time 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 V63.0121.041, Calculus I (NYU) Section 2.1–2.2 The Derivative September 26, 2010 13 / 46 Population growth Notes 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 East 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 V63.0121.041, Calculus I (NYU) Section 2.1–2.2 The Derivative September 26, 2010 14 / 46 4
- 5. V63.0121.041, Calculus I Section 2.1–2.2 : The Derivative September 26, 2010 Derivation Notes Let ∆t be an increment in time and ∆P the corresponding change in population: ∆P = P(t + ∆t) − P(t) This depends on ∆t, so ideally we would want ∆P 1 3e t+∆t 3e t lim = lim − ∆t→0 ∆t ∆t→0 ∆t 1 + e t+∆t 1 + et But rather than compute a complicated limit analytically, let us approximate numerically. We will try a small ∆t, for instance 0.1. V63.0121.041, Calculus I (NYU) Section 2.1–2.2 The Derivative September 26, 2010 15 / 46 Numerical evidence Notes To approximate the population change in year n, use the diﬀerence P(t + ∆t) − P(t) quotient , where ∆t = 0.1 and t = n − 2000. ∆t P(−10 + 0.1) − P(−10) 1 3e −9.9 3e −10 r1990 ≈ = −9.9 − 0.1 0.1 1+e 1 + e −10 = P(0.1) − P(0) 1 3e 0.1 3e 0 r2000 ≈ = − 0.1 0.1 1 + e 0.1 1 + e 0 = P(10 + 0.1) − P(10) 1 3e 10.1 3e 10 r2010 ≈ = 10.1 − 0.1 0.1 1+e 1 + e 10 = V63.0121.041, Calculus I (NYU) Section 2.1–2.2 The Derivative September 26, 2010 16 / 46 Population growth in general Notes Upshot The instantaneous population growth is given by P(t + ∆t) − P(t) lim ∆t→0 ∆t V63.0121.041, Calculus I (NYU) Section 2.1–2.2 The Derivative September 26, 2010 18 / 46 5
- 6. V63.0121.041, Calculus I Section 2.1–2.2 : The Derivative September 26, 2010 Marginal costs Notes 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? Answer V63.0121.041, Calculus I (NYU) Section 2.1–2.2 The Derivative September 26, 2010 19 / 46 Comparisons Notes Solution C (q) = q 3 − 12q 2 + 60q Fill in the table: q C (q) AC (q) = C (q)/q ∆C = C (q + 1) − C (q) 4 5 6 V63.0121.041, Calculus I (NYU) Section 2.1–2.2 The Derivative September 26, 2010 20 / 46 Marginal Cost in General Notes Upshot The incremental cost ∆C = C (q + 1) − C (q) is useful, but is still only an average rate of change. The marginal cost after producing q given by C (q + ∆q) − C (q) MC = lim ∆q→0 ∆q is more useful since it’s an instantaneous rate of change. V63.0121.041, Calculus I (NYU) Section 2.1–2.2 The Derivative September 26, 2010 22 / 46 6
- 7. V63.0121.041, Calculus I Section 2.1–2.2 : The Derivative September 26, 2010 Outline Notes Rates of Change Tangent Lines Velocity Population growth Marginal costs The derivative, deﬁned Derivatives of (some) power functions What does f tell you about f ? How can a function fail to be diﬀerentiable? Other notations The second derivative V63.0121.041, Calculus I (NYU) Section 2.1–2.2 The Derivative September 26, 2010 23 / 46 The deﬁnition Notes 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 (x) − f (a) f (a) = lim = lim h→0 h x→a x −a exists, the function is said to be diﬀerentiable at a and f (a) is the derivative of f at a. V63.0121.041, Calculus I (NYU) Section 2.1–2.2 The Derivative September 26, 2010 24 / 46 Derivative of the squaring function Notes Example Suppose f (x) = x 2 . Use the deﬁnition of derivative to ﬁnd f (a). Solution 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 V63.0121.041, Calculus I (NYU) Section 2.1–2.2 The Derivative September 26, 2010 25 / 46 7
- 8. V63.0121.041, Calculus I Section 2.1–2.2 : The Derivative September 26, 2010 Derivative of the reciprocal function Notes Example 1 Suppose f (x) = . Use the x deﬁnition of the derivative to ﬁnd f (2). x Solution 1/x − 1/2 f (2) = lim x→2 x −2 2−x x = lim x→2 2x(x − 2) −1 1 = lim =− x→2 2x 4 V63.0121.041, Calculus I (NYU) Section 2.1–2.2 The Derivative September 26, 2010 26 / 46 What does f tell you about f ? Notes If f is a function, we can compute the derivative f (x) at each point x where f is diﬀerentiable, and come up with another function, the derivative function. What can we say about this function f ? If f is decreasing on an interval, f is negative (technically, nonpositive) on that interval If f is increasing on an interval, f is positive (technically, nonnegative) on that interval V63.0121.041, Calculus I (NYU) Section 2.1–2.2 The Derivative September 26, 2010 28 / 46 What does f tell you about f ? Notes Fact If f is decreasing on (a, b), then f ≤ 0 on (a, b). Proof. If f is decreasing on (a, b), and ∆x > 0, then f (x + ∆x) − f (x) f (x + ∆x) < f (x) =⇒ <0 ∆x But if ∆x < 0, then x + ∆x < x, and f (x + ∆x) − f (x) f (x + ∆x) > f (x) =⇒ <0 ∆x f (x + ∆x) − f (x) still! Either way, < 0, so ∆x f (x + ∆x) − f (x) f (x) = lim ≤0 ∆x→0 ∆x V63.0121.041, Calculus I (NYU) Section 2.1–2.2 The Derivative September 26, 2010 32 / 46 8
- 9. V63.0121.041, Calculus I Section 2.1–2.2 : The Derivative September 26, 2010 Outline Notes Rates of Change Tangent Lines Velocity Population growth Marginal costs The derivative, deﬁned Derivatives of (some) power functions What does f tell you about f ? How can a function fail to be diﬀerentiable? Other notations The second derivative V63.0121.041, Calculus I (NYU) Section 2.1–2.2 The Derivative September 26, 2010 33 / 46 Diﬀerentiability is super-continuity Notes Theorem If f is diﬀerentiable at a, then f is continuous 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 Note the proper use of the limit law: if the factors each have a limit at a, the limit of the product is the product of the limits. V63.0121.041, Calculus I (NYU) Section 2.1–2.2 The Derivative September 26, 2010 34 / 46 Diﬀerentiability FAIL Kinks Notes f (x) f (x) x x V63.0121.041, Calculus I (NYU) Section 2.1–2.2 The Derivative September 26, 2010 35 / 46 9
- 10. V63.0121.041, Calculus I Section 2.1–2.2 : The Derivative September 26, 2010 Diﬀerentiability FAIL Cusps Notes f (x) f (x) x x V63.0121.041, Calculus I (NYU) Section 2.1–2.2 The Derivative September 26, 2010 36 / 46 Diﬀerentiability FAIL Vertical Tangents Notes f (x) f (x) x x V63.0121.041, Calculus I (NYU) Section 2.1–2.2 The Derivative September 26, 2010 37 / 46 Diﬀerentiability FAIL Weird, Wild, Stuﬀ Notes f (x) f (x) x x This function is diﬀerentiable at But the derivative is not 0. continuous at 0! V63.0121.041, Calculus I (NYU) Section 2.1–2.2 The Derivative September 26, 2010 38 / 46 10
- 11. V63.0121.041, Calculus I Section 2.1–2.2 : The Derivative September 26, 2010 Outline Notes Rates of Change Tangent Lines Velocity Population growth Marginal costs The derivative, deﬁned Derivatives of (some) power functions What does f tell you about f ? How can a function fail to be diﬀerentiable? Other notations The second derivative V63.0121.041, Calculus I (NYU) Section 2.1–2.2 The Derivative September 26, 2010 39 / 46 Notation Notes Newtonian notation f (x) y (x) y Leibnizian notation dy d df f (x) dx dx dx These all mean the same thing. V63.0121.041, Calculus I (NYU) Section 2.1–2.2 The Derivative September 26, 2010 40 / 46 Meet the Mathematician: Isaac Newton Notes English, 1643–1727 Professor at Cambridge (England) Philosophiae Naturalis Principia Mathematica published 1687 V63.0121.041, Calculus I (NYU) Section 2.1–2.2 The Derivative September 26, 2010 41 / 46 11
- 12. V63.0121.041, Calculus I Section 2.1–2.2 : The Derivative September 26, 2010 Meet the Mathematician: Gottfried Leibniz Notes German, 1646–1716 Eminent philosopher as well as mathematician Contemporarily disgraced by the calculus priority dispute V63.0121.041, Calculus I (NYU) Section 2.1–2.2 The Derivative September 26, 2010 42 / 46 Outline Notes Rates of Change Tangent Lines Velocity Population growth Marginal costs The derivative, deﬁned Derivatives of (some) power functions What does f tell you about f ? How can a function fail to be diﬀerentiable? Other notations The second derivative V63.0121.041, Calculus I (NYU) Section 2.1–2.2 The Derivative September 26, 2010 43 / 46 The second derivative Notes If f is a function, so is f , and we can seek its derivative. f = (f ) It measures the rate of change of the rate of change! Leibnizian notation: d 2y d2 d 2f f (x) dx 2 dx 2 dx 2 V63.0121.041, Calculus I (NYU) Section 2.1–2.2 The Derivative September 26, 2010 44 / 46 12
- 13. V63.0121.041, Calculus I Section 2.1–2.2 : The Derivative September 26, 2010 function, derivative, second derivative Notes y f (x) = x 2 f (x) = 2x f (x) = 2 x V63.0121.041, Calculus I (NYU) Section 2.1–2.2 The Derivative September 26, 2010 45 / 46 What have we learned today? Notes The derivative measures instantaneous rate of change The derivative has many interpretations: slope of the tangent line, velocity, marginal quantities, etc. The derivative reﬂects the monotonicity (increasing or decreasing) of the graph V63.0121.041, Calculus I (NYU) Section 2.1–2.2 The Derivative September 26, 2010 46 / 46 Notes 13

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