My name is... WHAT?? (as seen on your exams)
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1 My name is... WHAT?? (as seen on your exams) Alvaro Loreno Alvaro Loranzo-Rubledo Alaro Alzano-Robledo Alvareo Lozano-Robledo Alvaro Lozano-Rebledo Dr. Lorenzo-Robledo Alvaro-Loranzo Alvaro Alverez Lonzaro Alavero Robledo Alforno Alvaro Alonzano Alavaro Alzano-Robl Alvaro Lozano-Rebello Alonzano Alvaro-Lozado-Robledo Alvarao Francisco Alvaro Alvarado-Robledo Alozano Alvero Roberzo Alonzo Alvaro Lazano Robero Alvazo Loanzo Alozano Robledo Alavaro-Loranzo Alfonso-Lobledo Alvo Loranzo Alerso Alvarzo Alvaro Lazano Alvaro Lozono Alanzoro
2 My name is... WHAT?? (as seen on your exams) A different one that takes the cake...
3 My name is... WHAT?? (as seen on your exams) A different one that takes the cake... David Dagget
4 My name is... WHAT?? (as seen on your exams) A different one that takes the cake... David Dagget
5 My name is... WHAT?? (as seen on your exams) A different one that takes the cake... David Dagget David Daggett Assistant Professor-in-Residence Department of Molecular and Cell Biology
6 MATH 1131Q - Calculus 1. Álvaro Lozano-Robledo Department of Mathematics University of Connecticut Day 12 Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 3 / 28
7 So... the first midterm exam was... (A) Essentially impossible! So hard : ( (B) Tough. :( (C) Fair. : (D) Kind of easy, actually. :) (E) So easy I could have done it in my sleep! :D Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 4 / 28
8 My grade in this exam... (A) Aced it! I m a Calculus Commander. :D (B) I think I did better than most. :) (C) I did OK. I shall pass. : (D) I am worried. :( (E) (F) I bombed it. : ( Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 5 / 28
9 Derivatives (Rates of Change)
10 Let f (x) and g(x) be differentiable functions (i.e., f (x) and g (x) exist at every point x). Derivatives f (x + h) f (x) f (x) = lim. h 0 h (c) = 0, where c is a constant. (x n ) = nx n 1, where n is real. (a x ) = ka x, where a > 0, k = lim h 0 a h 1 h. (e x ) = e x. (sin x) = cos x. (cos x) = sin x. Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 7 / 28
11 Let f (x) and g(x) be differentiable functions (i.e., f (x) and g (x) exist at every point x). Derivatives f (x + h) f (x) f (x) = lim. h 0 h (c) = 0, where c is a constant. (x n ) = nx n 1, where n is real. (a x ) = ka x, where a > 0, k = lim h 0 a h 1 h. (e x ) = e x. (sin x) = cos x. (cos x) = sin x. Rules (cf ) = cf, where c is a constant. (f + g) = f + g. Product rule: (fg) = f g + fg. Quotient rule: f f g = g fg. g 2 Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 7 / 28
12 The Chain Rule Let y = F(x) be a composition of two (differentiable) functions, i.e., y = f (g(x)). Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 8 / 28
13 The Chain Rule Let y = F(x) be a composition of two (differentiable) functions, i.e., y = f (g(x)). We can write this as y = f (u), where u = g(x). Then: dy dx = dy du du dx Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 8 / 28
14 The Chain Rule Let y = F(x) be a composition of two (differentiable) functions, i.e., y = f (g(x)). We can write this as y = f (u), where u = g(x). Then: or, equivalently, dy dx = dy du du dx (f (g(x))) = f (g(x)) g (x) Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 8 / 28
15 Let f (x) and g(x) be differentiable functions (i.e., f (x) and g (x) exist at every point x). Derivatives f (x + h) f (x) f (x) = lim. h 0 h (c) = 0, where c is a constant. (x n ) = nx n 1, where n is real. (a x ) = ka x, where a > 0, k = lim h 0 a h 1 h. (e x ) = e x. (sin x) = cos x. (cos x) = sin x. Rules (cf ) = cf, where c is a constant. (f + g) = f + g. Product rule: (fg) = f g + fg. Quotient rule: f f g = g fg. g 2 Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 9 / 28
16 Let f (x) and g(x) be differentiable functions (i.e., f (x) and g (x) exist at every point x). Derivatives f (x + h) f (x) f (x) = lim. h 0 h (c) = 0, where c is a constant. (x n ) = nx n 1, where n is real. (a x ) = ka x, where a > 0, k = lim h 0 a h 1 h. (e x ) = e x. (sin x) = cos x. (cos x) = sin x. Rules (cf ) = cf, where c is a constant. (f + g) = f + g. Product rule: (fg) = f g + fg. Quotient rule: f f g = g fg. g 2 Chain rule: (f (g(x))) = f (g(x))g (x). Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 9 / 28
17 Example (f (g(x))) = f (g(x)) g (x) Find the derivative of F(x) = sin e 3x+7.
18 Example Find the tangent line to the circle of radius 1 at the point (3/5, 4/5). c B Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 11 / 28
19 Example Find the tangent line to the circle of radius 1 at the point (3/5, 4/5). c B Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 12 / 28
20 Implicit Differentiation Example Find the tangent line to the folium of Descartes at the point (3, 3). x 3 + y 3 = 6xy Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 13 / 28
21 Implicit Differentiation Example Find the tangent line to the lemniscate at the point (3, 1). 2(x 2 + y 2 ) 2 = 25(x 2 y 2 ) Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 14 / 28
22 Implicit Differentiation: Inverse Functions Example Find the derivative of y = sin 1 (x), for x [ 1, 1], y [ π/2, π/2] Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 15 / 28
23 Implicit Differentiation: Inverse Functions Example Find the derivative of y = sin 1 (x), for x [ 1, 1], y [ π/2, π/2] Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 16 / 28
24 Implicit Differentiation: Inverse Functions Example Find the derivative of y = sin 1 (x), for x [ 1, 1], y [ π/2, π/2] Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 17 / 28
25 Implicit Differentiation: Inverse Functions Example Find the derivative of y = cos 1 (x), for x [ 1, 1], y [0, 2π]. Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 18 / 28
26 Implicit Differentiation: Inverse Functions Example Find the derivative of y = tan 1 (x), for x (, ), y [ π/2, π/2]. Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 19 / 28
27 Implicit Differentiation: Logarithmic Functions Example Find the derivative of y = ln(x) for x (0, ). Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 20 / 28
28 Example Find the derivative of y = ln(sin(x)). Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 21 / 28
29 Implicit Differentiation: Logarithmic Functions Example Find the derivative of y = log a (x) for x (0, ), and a > 0. Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 22 / 28
30 Implicit Differentiation: Logarithmic Functions Example Find the derivative of y = ln( x ) for x (, ). Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 23 / 28
31 Implicit Differentiation: the Power Rule Example Find the derivative of y = x n, for x > 0, where n is a real number. Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 24 / 28
32 Logarithmic Differentiation Example Find the derivative of y = x x, for x > 0. Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 25 / 28
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