Charles Robert Darwin (12 February April 1882)

Transcription

1 I attempted mathematics, but it was repugnant to me, chiefly from my not being able to see any meaning in the early steps in algebra. This impatience was very foolish, and in after years I have deeply regretted that I did not proceed far enough at least to understand something of the great leading principles of mathematics; for men thus endowed seem to have an extra sense. Charles Robert Darwin (12 February April 1882)

2 MATH 1131Q - Calculus 1. Álvaro Lozano-Robledo Department of Mathematics University of Connecticut Day 18 Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 2 / 30

3 Clicker Question... (frequency DD) I do not go to your (Álvaro s) office hours because... (A) I don t know when and/or where they are! (B) I find the whole concept of office hours by a professor intimidating. (C) I don t really have questions. (D) I ask for help elsewhere (my TA, Q Center, etc). (E) I can t make it to your office at those times. Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 3 / 30

4 Applications of Derivatives

5 Maximum and Minimum Values of a Function Definition Let c be a number in the domain D of a function f. Then f (c) is the 1 absolute maximum value of f on D if f (c) f (x) for all x in D. 2 absolute minimum value of f on D if f (c) f (x) for all x in D. 3 local maximum value of f if f (c) f (x) when x is near c. 4 local minimum value of f if f (c) f (x) when x is near c.

6 Maximum and Minimum Values of a Function Definition Let c be a number in the domain D of a function f. Then f (c) is the 1 absolute maximum value of f on D if f (c) f (x) for all x in D. 2 absolute minimum value of f on D if f (c) f (x) for all x in D. 3 local maximum value of f if f (c) f (x) when x is near c. 4 local minimum value of f if f (c) f (x) when x is near c. Theorem (Extreme Value Theorem) If f is continuous on a closed interval [a, b], then f attains an absolute maximum value f (c) and an absolute minimum value f (d) at some numbers c and d in [a, b].

7 Maximum and Minimum Values of a Function Definition Let c be a number in the domain D of a function f. Then f (c) is the 1 absolute maximum value of f on D if f (c) f (x) for all x in D. 2 absolute minimum value of f on D if f (c) f (x) for all x in D. 3 local maximum value of f if f (c) f (x) when x is near c. 4 local minimum value of f if f (c) f (x) when x is near c. Theorem (Extreme Value Theorem) If f is continuous on a closed interval [a, b], then f attains an absolute maximum value f (c) and an absolute minimum value f (d) at some numbers c and d in [a, b]. Theorem (Fermat s Theorem) If f has a local maximum or minimum at c, and if f (c) exists, then f (c) = 0.

8 Maximum and Minimum Values of a Function Theorem (Fermat s Theorem) If f (x) has a local maximum or minimum at c, and if f (c) exists, then f (c) = 0. Definition A critical number of a function f (x) is a number c in the domain of f such that (1) f (c) = 0, or (2) f (c) does not exist. Example Find the critical points of f (x) = x 3 3x + 1.

9 Maximum and Minimum Values of a Function Theorem (Fermat s Theorem) If f (x) has a local maximum or minimum at c, and if f (c) exists, then f (c) = 0. Definition A critical number of a function f (x) is a number c in the domain of f such that (1) f (c) = 0, or (2) f (c) does not exist. Example Find the critical points of f (x) = sin x.

10 Maximum and Minimum Values of a Function Example Find the critical points of f (x) = (4 x)x 3/5. Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 8 / 30

11 Maximum and Minimum Values of a Function How to find the absolute maximum and minimum values of a function f (x) on a closed interval [a, b]:

12 Maximum and Minimum Values of a Function How to find the absolute maximum and minimum values of a function f (x) on a closed interval [a, b]: 1 Find the critical values of f in [a, b]. Calculate the values of f at all critical values.

13 Maximum and Minimum Values of a Function How to find the absolute maximum and minimum values of a function f (x) on a closed interval [a, b]: 1 Find the critical values of f in [a, b]. Calculate the values of f at all critical values. 2 Find the values of f at endpoints of the interval and at any point where f is defined but not continuous.

14 Maximum and Minimum Values of a Function How to find the absolute maximum and minimum values of a function f (x) on a closed interval [a, b]: 1 Find the critical values of f in [a, b]. Calculate the values of f at all critical values. 2 Find the values of f at endpoints of the interval and at any point where f is defined but not continuous. 3 The largest of the values from (1) and (2) is the absolute maximum value; the smallest of these values is the absolute minimum value.

15 Maximum and Minimum Values of a Function Example Find the absolute max and min values of f (x) = x 3 3x in the interval [ 1 2, 4]. Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 10 / 30

16 Blue: E(t) = a/(b + ce dt ), Red: E(t) = a + R0 (1 + d) t t, Source

17 Example Find the absolute maximum value of E(t) = a + R0 (1+d) t t in [0, 400], where a = 94, R 0 = 1.077, and d =

18 Example Find the absolute maximum value of E(t) = a + R0 (1+d) t t in [0, 400], where a = 94, R 0 = 1.077, and d =

19 Example Find the absolute maximum value of E(t) = a + R0 (1+d) t t in [0, 400], where a = 94, R 0 = 1.077, and d = Solution: There is a unique critical point in [0, 400] given at The values of the function t = Ln(R 0) 2Ln(1 + d) = E(0) = 95 E(400) E( ) So the absolute maximum occurs at t = 259 and the value is

20 This slide left intentionally blank Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 14 / 30

21 The Mean Value Theorem Theorem (The Mean Value Theorem) Let f be a function that satisfies the following hypotheses: 1 f is continuous on the closed interval [a, b], and 2 f is differentiable on the open interval (a, b). Then, there is a number c in (a, b) such that f (b) f (a) f (c) =, or, equivalently, f (b) f (a) = f (c)(b a). b a

22 The Mean Value Theorem Theorem (The Mean Value Theorem) Let f be a function that is continuous on the closed interval [a, b], and differentiable on the open interval (a, b). Then, there is a number c in (a, b) such that f (b) f (a) f (c) =, or, equivalently, f (b) f (a) = f (c)(b a). b a Example Show that the Mean Value Theorem is true for f (x) = x 2 and the interval [0, 3].

23 Example A driver goes through a toll booth at noon, drives 180 miles on the same highway, and exits through another toll 2 hours later. If the speed limit on the highway is 65 miles per hour... 1 Was the driver speeding at some point in the trip? 2 The speed tickets are $100 if the speed at any point in time was between 65 and 75 mi/h, and $100 + x if the speed was 75 + x mi/h at any point. What the largest fine you can charge on the driver?

24 Example A driver goes through a toll booth at noon, drives 180 miles on the same highway, and exits through another toll 2 hours later. If the speed limit on the highway is 65 miles per hour... 1 Was the driver speeding at some point in the trip? 2 The speed tickets are $100 if the speed at any point in time was between 65 and 75 mi/h, and $100 + x if the speed was 75 + x mi/h at any point. What the largest fine you can charge on the driver?

25 Important Consequences of the Mean Value Theorem Theorem (The Mean Value Theorem) Let f be a function that is continuous on the closed interval [a, b], and differentiable on the open interval (a, b). Then, there is a number c in (a, b) such that f (c) = (f (b) f (a))/(b a) or, equivalently, f (b) f (a) = f (c)(b a).

26 Important Consequences of the Mean Value Theorem Theorem (The Mean Value Theorem) Let f be a function that is continuous on the closed interval [a, b], and differentiable on the open interval (a, b). Then, there is a number c in (a, b) such that f (c) = (f (b) f (a))/(b a) or, equivalently, f (b) f (a) = f (c)(b a). Theorem If f (x) = 0 for all x in an interval (a, b), then f is constant on (a, b). Proof.

27 Important Consequences of the Mean Value Theorem Theorem (The Mean Value Theorem) Let f be a function that is continuous on the closed interval [a, b], and differentiable on the open interval (a, b). Then, there is a number c in (a, b) such that f (c) = (f (b) f (a))/(b a) or, equivalently, f (b) f (a) = f (c)(b a). Theorem If f (x) = 0 for all x in an interval (a, b), then f is constant on (a, b). Proof. Let x 1 and x 2 be numbers in (a, b)), with a < x 1 < x 2 < b. Then, f is continuous on [x 1, x 2 ] and differentiable on (x 1, x 2 ), because f exists (and equals 0). Hence, by the MVT there is some c in (x 1, x 2 ) such that f (x 2 ) f (x 1 ) = f (c)(x 2 x 1 ) = 0 (x 2 x 1 ) = 0. Therefore, f (x 2 ) f (x 1 ) = 0, and so f (x 2 ) = f (x 1 ). It follows that f has the same value for any two x 1 and x 2 on (a, b), and so it is constant.

28 Important Consequences of the Mean Value Theorem Theorem If f (x) = 0 for all x in an interval (a, b), then f is constant on (a, b).

29 Important Consequences of the Mean Value Theorem Theorem If f (x) = 0 for all x in an interval (a, b), then f is constant on (a, b). Corollary If f (x) = g (x) for all x in the interval (a, b), then f g is constant on (a, b); that is, f (x) = g(x) + C, where C is a constant. Proof.

30 Important Consequences of the Mean Value Theorem Theorem If f (x) = 0 for all x in an interval (a, b), then f is constant on (a, b). Corollary If f (x) = g (x) for all x in the interval (a, b), then f g is constant on (a, b); that is, f (x) = g(x) + C, where C is a constant. Proof. Consider the function F(x) = f (x) g(x). Then, F (x) = f (x) g (x) = 0 in the interval (a, b). Thus, by the theorem, F(x) = C is constant. Hence, f (x) g(x) = C or, equivalently, f (x) = g(x) + C.

31 Sketching Graphs of Functions

32 Definition We say that a function f (x) is... 1 Increasing in the interval [a, b], if f (c) f (d) for any c < d. 2 Decreasing in the interval [a, b], if f (c) f (d) for any c < d. 3 Concave upward in the interval [a, b], if the graph of f lies above all of its tangent lines for points on the interval. 4 Concave downward in the interval [a, b], if the graph of f lies below all of its tangent lines for points on the interval

33 Using the First Derivative Theorem (Increasing/Decreasing Test) Let f be a differentiable function on an interval (a, b). 1 If f (x) > 0 on (a, b), then f is increasing on (a, b). 2 If f (x) < 0 on (a, b), then f is decreasing on (a, b). Theorem (The First Derivative Test for Critical Points) Suppose that c is a critical number of a continuous function f. Then: 1 If f changes from positive to negative at c, then f has a local maximum at c. 2 If f changes from negative to positive at c, then f has a local minimum at c. 3 If f does not change sign at c, then f has no local maximum or minimum at c. Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 23 / 30

34 Using the First Derivative Example Sketch the graph of the function y = 3x 5 5x Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 24 / 30

35 Using the First Derivative Example Sketch the graph of the function y = 3x 5 5x Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 24 / 30

36 Using the First Derivative Example Sketch the graph of the function y = 3x 5 5x Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 25 / 30