Lesson 5-3b Fundamental Theorem of Calculus

Quiz Homework Problem: ( 3e x + 7sec 2 x) dx Reading questions: Fill in the squares below f(x) dx = F(█) – F(█) If F’(x) = f(x) ∫ █ █ b b a a ∫ = 3e x + 7tan x + c

Objectives Understand both forms of the Fundamental Theorem of Calculus Understand the relation between integration and differentiation

Vocabulary Definite Integral – is a number, not a function

∆y ∆x Secant Line ∆y f(b) – f(a) slope = = ∆x b – a ∆y ∆x Tangent Line ∆y slope ≈ ∆x ∆y ∆x area = ∆y ∆x = f(b) (b-a) ∆y ∆x area ≈ ∆y ∆x ∆y lim ---- = m t ∆x ∆x→0 lim ∆y ∆x = area = f(x) dx ∆x→0 Area of rectangle Area under the curve = f’(a) ab Differentiation vs Definite Integration ababab ∫ a b

Fundamental Theorem of Calculus, Part 1 If f is continuous on [a,b], then the function g defined by is continuous on [a,b], differentiable on (a,b) and g’(x) = f(x) Your book has a proof of this if you are interested. We can see g’(x) = f(x) by using derivatives and FTC part 2. g(x) = f(t) dt a ≤ x ≤ b ∫ x a

Fundamental Theorem of Calculus Combining parts 1 and 2 of the FTC, we get F(x) is a function of x and F(a) is just some constant value. Now when we do the following: g(x) = f(t) dt = F(x) – F(a) ∫ x a g(x) = f(t) dt = F(x) – F(a) = ∫ x a d ---- dx g’(x) = = F’(x) – 0 F’(x) = f(x) g’(x) = f(x)

Example Problems Find the derivative of each of the following: 1) t² dt ∫ 1 x ⅓t³ | [⅓x³ ] – [⅓1³ ] d(⅓x³ – ⅓) / dx = x² See any pattern? t=1 t=x t 3/2 2) dt  t² + 17 ∫ 2 x We can’t do what we did in #1! Use the pattern: x 3/  x² + 17

Example Problems with TI-89 Find the derivative of each of the following: Can we use our calculator here? YES!! Hit F3 select derivatives; hit F3 select integration; type in function (t²), integrate with respect to (t), lower limit of integration (1), upper limit of integration (x); close ). Type, and differentiate with respect to x and close ). Should look like this: d( F3 ∫(t^2,t,1,x),x) 1) t² dt ∫ 1 x = x²

Example Problems cont Find the derivative of each of the following: 3) tan²(t) cot (t) dt ∫ x 4 4) t tan (t) dt ∫ x π/4 We can’t do what we did in #1! Use the pattern: -x tan (x) Negative because of location of x in integral We can’t do what we did in #1! Use the pattern: - tan²(x) cot(x) Negative because of location of x in integral

Example Problems cont Find the derivative of each of the following: 5) 5t sin (t) dt ∫ 1 2x We can’t do what we did in #1! Use the pattern: We can’t do what we did in #1! Use the pattern: 10x sin(2x) (2) Remember Chain Rule 6) √2 + sin (t) dt ∫ 2 x²x² √2 + sin (x²) (2x) Remember Chain Rule

Example Problems cont Find the derivative of each of the following: 7) √1 + t 4 dt ∫ x x³x³ 8) t² dt ∫ sin x cos x We can’t do what we did in #1! Use the pattern: cos²x (-sinx) - sin²x(cosx) Remember F(b) – F(a) and Chain Rule We can’t do what we did in #1! Use the pattern: √1 + (x³) 4 (3x 2 ) - √1 + x 4 Remember F(b) – F(a) and chain Rule

Summary & Homework Summary: –Definite Integrals are a number –Evaluated at endpoints of integration –Derivative of the integral returns what we started with (with Chain Rule) Homework: –Day One: pg : 19, 22, 27, 28, –Day Two: pg : 3, 7, 9, 61 (see appendix E)

Second Set Example Problems Find the derivative of each of the following: 1) sin³(t) dt ∫ 1 x² We can’t do what we did in #1! Use the pattern: sin³(x²) (2x) Remember Chain Rule

Second Set Example Problems Find the derivative of each of the following: 2) (1/t²)dt ∫ x 5 We can’t do what we did in #1! Use the pattern: - (1/x²) Negative because of location of x in integral

Second Set Example Problems Find the derivative of each of the following: 3) √1 + t 4 dt ∫ x x²x² We can’t do what we did in #1! Use the pattern: √1 + (x²) 4 (2x) - √1 + x 4 Remember F(b) – F(a) and chain Rule

Last Example Problem Consider F (x) = Find F (x). For what number x does F attain its minimum value? ______ Justify your answer. How many inflection points does the graph of F have? _____ Justify your answer. t – dt for -∞ < x < ∞ t² + 7 ∫ 0 x x – 3 F’(x) = x² + 7 When F’(x) = 0 so x = 3 First Derivative Test: therefore a min When F’’(x) = 0 so x = -1, x = 7