How does a calculator compute 2?
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1 How does a calculator compute 2?
2 y = x
3 y = x and y = 2 + x 2
4 y = x and y = x 4 x 2 8
5 y = x and y = x 6 5x x 3 6
6 y = x and y = x 32 35x x x 4 28
7 Function Value at 2 x 2 = x x 4 x x 6 5x x x 32 35x x x =.5 (error 0.085) 2 =.375 (error 0.039) 8 23 =.4375 (error 0.023) 6 79 = (error 0.05) 28
8 MATH 3Q - Calculus. Álvaro Lozano-Robledo Department of Mathematics University of Connecticut Day 7 Álvaro Lozano-Robledo (UConn) MATH 3Q - Calculus 8 / 34
9 Three Announcements Office hours on Thursday are cancelled. I ll have office hours Friday :30-2:30 instead. Second midterm on Tuesday, November 4th. More details later on. Final Exam on Saturday, December 3th, -3pm. More details later on. Álvaro Lozano-Robledo (UConn) MATH 3Q - Calculus 9 / 34
10 Related Rates
11 Example A trough of water is 8 meters deep and its ends are in the shape of isosceles triangles. The width of the trough is 5 meters and height is 2 meters. If water is being pumped in at a constant rate of 6 m 3 /s. At what rate is the height of the water changing when the water has a height of 20 cm? (Source)
12 Example A trough of water is 8 meters deep and its ends are in the shape of isosceles triangles. The width of the trough is 5 meters and height is 2 meters. If water is being pumped in at a constant rate of 6 m 3 /s. At what rate is the height of the water changing when the water has a height of 20 cm? (Source)
13 Applications of Derivatives
14 Linear Approximations Álvaro Lozano-Robledo (UConn) MATH 3Q - Calculus 3 / 34
15 Linear Approximations The approximation of f (x) by its tangent line is called the linear approximation or tangent line approximation of f. Álvaro Lozano-Robledo (UConn) MATH 3Q - Calculus 3 / 34
16 Linear Approximations Álvaro Lozano-Robledo (UConn) MATH 3Q - Calculus 4 / 34
17 Linear Approximations The approximation of f (x) f (a) + f (a)(x a) is called the linear approximation of f at a. Álvaro Lozano-Robledo (UConn) MATH 3Q - Calculus 4 / 34
18 Linear Approximations Álvaro Lozano-Robledo (UConn) MATH 3Q - Calculus 5 / 34
19 Linear Approximations The function L(x) = f (a) + f (a)(x a) is called the linearization of f at a. Álvaro Lozano-Robledo (UConn) MATH 3Q - Calculus 5 / 34
20 Example Find the linearization of y = sin(x) at a = 0, and use it to find an approximate value of sin(0.5). Álvaro Lozano-Robledo (UConn) MATH 3Q - Calculus 6 / 34
21 Example Find the linearization of y = sin(x) at a = 0, and use it to find an approximate value of sin(0.5) Álvaro Lozano-Robledo (UConn) MATH 3Q - Calculus 6 / 34
22 Example Find the linearization of y = sin(x) at a = 0, and use it to find an approximate value of sin(0.5) = Álvaro Lozano-Robledo (UConn) MATH 3Q - Calculus 7 / 34
23 Example Use a linearization to approximate the value of 0. Álvaro Lozano-Robledo (UConn) MATH 3Q - Calculus 8 / 34
24 Example Use a linearization to approximate the value of 0. 0 = L(0) = 9 6 = error = 0 L(0) = Álvaro Lozano-Robledo (UConn) MATH 3Q - Calculus 8 / 34
25 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 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 The function f (x) = sin x in R has an absolute maximum value of and an absolute minimum value of. These are attained as f (π/2) = and f ( π/2) =.
26 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 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 The function f (x) = sin x in [ 0.5, 0.5] has an absolute maximum value of sin(0.5) and an absolute minimum value of sin( 0.5). These are attained as f (0.5) = sin(0.5) and f ( 0.5) = sin( 0.5).
27 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 local maximum value of f if f (c) f (x) when x is near c. 2 local minimum value of f if f (c) f (x) when x is near c The function f (x) = x 3 3x + has a local maximum value of 3 at c =, and a local minimum value of at c =.
28 Maximum and Minimum Values of a Function 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].
29 Maximum and Minimum Values of a Function 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]. WARNING! Continuity is essential in the extreme value theorem.
30 Maximum and Minimum Values of a Function Theorem (Fermat s Theorem) If f has a local maximum or minimum at c, and if f (c) exists, then f (c) = 0.
31 Maximum and Minimum Values of a Function Theorem (Fermat s Theorem) If f has a local maximum or minimum at c, and if f (c) exists, then f (c) = The function f (x) = x 3 3x + has a local maximum value of 3 at c =, and a local minimum value of at c =
32 Maximum and Minimum Values of a Function Theorem (Fermat s Theorem) If f has a local maximum or minimum at c, and if f (c) exists, then f (c) = WARNING! The converse is not necessarily true! The function f (x) = x 3 satisfies f (0) = 0, but there is no local max or min at 0.
33 Maximum and Minimum Values of a Function Theorem (Fermat s Theorem) If f has a local maximum or minimum at c, and if f (c) exists, then f (c) = WARNING! A local or absolute max/min may occur at places where f is not defined! The derivative of the function f (x) = x is undefined at 0, but there is an absolute minimum at 0.
34 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 () f (c) = 0, or (2) f (c) does not exist. Example Find the critical points of f (x) = x 3 3x +.
35 This slide left intentionally blank Álvaro Lozano-Robledo (UConn) MATH 3Q - Calculus 34 / 34
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