Lesson 7: The Derivative as a Function
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This document contains lecture notes on derivatives from a Calculus I class at New York University. It discusses the derivative as a function, finding the derivative of other functions, and the relationship between a function and its derivative. The notes include examples of finding the derivative of the reciprocal function and state that if a function is decreasing on an interval, its derivative will be nonpositive on that interval, while if it is increasing the derivative will be nonnegative. It also contains proofs and graphs related to derivatives.
Lesson 7: The Derivative as a Function
- 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. 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. 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. Derivative . . . . . .
- 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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