Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Sobolev space wikipedia , lookup
Infinitesimal wikipedia , lookup
Riemann integral wikipedia , lookup
Path integral formulation wikipedia , lookup
Function of several real variables wikipedia , lookup
Itô calculus wikipedia , lookup
History of calculus wikipedia , lookup
Multiple integral wikipedia , lookup
M ATH 1325 – B USINESS C ALCULUS Section 11.4/11.5 The Fundamental Theorem of Calculus/Evaluating Definite Integrals, Part I Establishes a connection between differential calculus and integral calculus. Differentiation and integration are inverse processes. The FTC makes integration easy. The Fundamental Theorem of Calculus: If f is continuous on [a, b], then Z b f (x)dx = F(b) − F(a) where a F is any antiderivative of f . (F 0 = f ) Z b f (x)dx by subtracting F at the endpoints. If we know the antiderivative F of f , we can evaluate a NOTE: BE CAREFUL! There is a big difference between a definite integral and an indefinite integral!! A definite integral is a . An indefinite integral is a Z 3 Ex: Evaluate 3x2 − 4xdx −1 Properties of Integrals: Assume f and g are continuous functions. Then: Z a f (x)dx = 0 1. a Z b 2. f (x)dx = − Z a a f (x)dx b Z b c f (x)dx = c f (x) where c is constant 3. a Z b 4. [ f (x) ± g(x)]dx = a Z b f (x)dx + a f (x)dx ± a Z c 5. Z b Z b g(x)dx a Z b f (x)dx = c f (x)dx for a < c < b a Z 13 Z 9 f (x)dx = 11 and Ex: If it is known that 1 Z 13 f (x)dx = 5, find 1 f (x)dx. 9 . Math 1325 Section 11.4 (with 11.5 – Part I) Continued Ex: Find the area under f (x) = x2 − 6x + 10 from x = −1 to x = 2. Ex: Find Z 25 √ 4 Ex: Find 2 √ x− dx x Z 4 3 2x − 4x2 + 5x − 2 1 x2 dx 2 Math 1325 Section 11.4 (with 11.5 – Part I) Continued Net Change: The net change in a function f over an interval [a, b] is given by f (b) − f (a) = Z b f 0 (x)dx, a provided f 0 is continuous on [a, b]. Ex: A division of Ditton Industries manufactures a deluxe toaster oven. Management has determined that the daily marginal cost function associated with producing these toaster oven is given by C0 (x) = 0.0003x2 − 0.12x + 20 where C0 (x) is measured in dollars per unit and x denotes the number of units produced. Management has also determined that the daily fixed cost incurred in the production is $800. (a) Find the total cost incurred by Ditton in producing the first 300 units of these toaster ovens per day. (b) What is the total cost incurred by Ditton in producing the 201st through 300th units per day? b 1 f (x)dx b−a a Ex: In a report issued by the Economist Intelligence Unit in 2013, the number of commercial vehicle registrations in the United States (in thousands) by year from 2010 through 2015 is approximated by the function N(t) = 1.3926t 3 − 9.2873t 2 + 74.719t + 228.3(0 ≤ t ≤ 5) where t is measured in years with t = 0 corresponding to 2010. If the projection holds up, what would the approximate average commercial vehicle registration per year be in the period from 2010 through 2015? (Round your answer to the nearest thousand.) Z Average Value of a Continuous Function f over [a, b]: 3