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# Lesson 7: The Derivative as a Function

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### Lesson 7: The Derivative as a Function

1. 1. Section 2.2 The Derivative as a Function V63.0121.002.2010Su, Calculus I New York University May 24, 2010 Announcements Homework 1 due Tuesday Quiz 2 Thursday in class on Sections 1.5–2.5 . . . . . .
2. 2. Announcements Homework 1 due Tuesday Quiz 2 Thursday in class on Sections 1.5–2.5 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 2 / 28
3. 3. Objectives 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. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 3 / 28
4. 4. Derivative . . . . . .
5. 5. Recall: the derivative 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. The derivative … …measures the slope of the line through (a, f(a)) tangent to the curve y = f(x); …represents the instantaneous rate of change of f at a …produces the best possible linear approximation to f near a. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 4 / 28
6. 6. Derivative of the reciprocal function Example 1 Suppose f(x) = . Use the x definition of the derivative to find f′ (2). . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 6 / 28
7. 7. Derivative of the reciprocal function Example 1 Suppose f(x) = . Use the x definition of the derivative to x . find f′ (2). Solution . ′ 1/x − 1/2 2−x . f (2) = lim = lim x . x→2 x−2 x→2 2x(x − 2) −1 1 = lim =− x→2 2x 4 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 6 / 28
8. 8. Outline What does f tell you about f′ ? How can a function fail to be differentiable? Other notations The second derivative . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 7 / 28
9. 9. What does f tell you about f′ ? 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′ ? . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 8 / 28
10. 10. What does f tell you about f′ ? 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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 8 / 28
11. 11. Derivative of the reciprocal function Example 1 Suppose f(x) = . Use the x definition of the derivative to x . find f′ (2). Solution . ′ 1/x − 1/2 2−x . f (2) = lim = lim x . x→2 x−2 x→2 2x(x − 2) −1 1 = lim =− x→2 2x 4 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 9 / 28
12. 12. What does f tell you about f′ ? 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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 10 / 28
13. 13. Graphically and numerically y . x2 − 22 x m= x−2 3 5 . . 9 . 2.5 4.5 2.1 4.1 2.01 4.01 . .25 . 6 . limit 4 . .41 . 4 . 1.99 3.99 . .0401 . 4.9601 . 3 . .61 3 4 . .. 1.9 3.9 1.5 3.5 . .25 . 2 . 1 3 . . 1 . . . . ... . . x . 1 1 . .. .1 3 . . .5 .99 .5 . 12 . 2.9 2 2 .01 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 11 / 28
14. 14. What does f tell you about f′ ? 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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 12 / 28
15. 15. What does f tell you about f′ ? 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 still! . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 12 / 28
16. 16. What does f tell you about f′ ? 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 still! Either way, f(x + ∆x) − f(x) f(x + ∆x) − f(x) < 0 =⇒ f′ (x) = lim ≤0 ∆x ∆x→0 ∆x . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 12 / 28
17. 17. Another important derivative fact Fact If the graph of f has a horizontal tangent line at c, then f′ (c) = 0. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 13 / 28
18. 18. Another important derivative fact Fact If the graph of f has a horizontal tangent line at c, then f′ (c) = 0. Proof. The tangent line has slope f′ (c). If the tangent line is horizontal, its slope is zero. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 13 / 28
19. 19. Outline What does f tell you about f′ ? How can a function fail to be differentiable? Other notations The second derivative . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 14 / 28
20. 20. Differentiability is super-continuity Theorem If f is differentiable at a, then f is continuous at a. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 15 / 28
21. 21. Differentiability is super-continuity 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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 15 / 28
22. 22. Differentiability is super-continuity 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. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 15 / 28
23. 23. Differentiability FAIL Kinks f .(x) . x . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 16 / 28
24. 24. Differentiability FAIL Kinks f .(x) .′ (x) f . x . . x . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 16 / 28
25. 25. Differentiability FAIL Kinks f .(x) .′ (x) f . . x . . x . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 16 / 28
26. 26. Differentiability FAIL Cusps f .(x) . x . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 17 / 28
27. 27. Differentiability FAIL Cusps f .(x) .′ (x) f . x . . x . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 17 / 28
28. 28. Differentiability FAIL Cusps f .(x) .′ (x) f . x . . x . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 17 / 28
29. 29. Differentiability FAIL Vertical Tangents f .(x) . x . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 18 / 28
30. 30. Differentiability FAIL Vertical Tangents f .(x) .′ (x) f . x . . x . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 18 / 28
31. 31. Differentiability FAIL Vertical Tangents f .(x) .′ (x) f . x . . x . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 18 / 28
32. 32. Differentiability FAIL Weird, Wild, Stuff f .(x) . x . This function is differentiable at 0. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 19 / 28
33. 33. Differentiability FAIL Weird, Wild, Stuff f .(x) .′ (x) f . x . . x . This function is differentiable at But the derivative is not 0. continuous at 0! . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 19 / 28
34. 34. Outline What does f tell you about f′ ? How can a function fail to be differentiable? Other notations The second derivative . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 20 / 28
35. 35. Notation Newtonian notation f′ (x) y′ (x) y′ Leibnizian notation dy d df f(x) dx dx dx These all mean the same thing. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 21 / 28
36. 36. Link between the notations f(x + ∆x) − f(x) ∆y dy f′ (x) = lim = lim = ∆x→0 ∆x ∆x→0 ∆x dx dy Leibniz thought of as a quotient of “infinitesimals” dx dy We think of as representing a limit of (finite) difference dx quotients, not as an actual fraction itself. The notation suggests things which are true even though they don’t follow from the notation per se . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 22 / 28
37. 37. Meet the Mathematician: Isaac Newton English, 1643–1727 Professor at Cambridge (England) Philosophiae Naturalis Principia Mathematica published 1687 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 23 / 28
38. 38. Meet the Mathematician: Gottfried Leibniz German, 1646–1716 Eminent philosopher as well as mathematician Contemporarily disgraced by the calculus priority dispute . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 24 / 28
39. 39. Outline What does f tell you about f′ ? How can a function fail to be differentiable? Other notations The second derivative . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 25 / 28
40. 40. The second derivative 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! . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 26 / 28
41. 41. The second derivative 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: d2 y d2 d2 f f(x) dx2 dx2 dx2 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 26 / 28
42. 42. function, derivative, second derivative y . .(x) = x2 f .′ (x) = 2x f .′′ (x) = 2 f . x . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 27 / 28
43. 43. Summary A function can be differentiated at every point to find its derivative function. The derivative of a function notices the monotonicity of the function (fincreasing =⇒ f′ ≥ 0) The second derivative of a function measures the rate of the change of the rate of change of that function. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 28 / 28