Section 4.4 Day 1
Antidifferentiation → Indefinite Integral: • (the family of functions) • Definite Integration → Definite Integral: • a number
Fundamental Theorem of Calculus • If a function f is continuous on the closed interval [a, b], then • If f (x) = F′(x) or F(x) is the antiderivative of f (x), then FTOC can be written as
x2 – 1 -(x2 – 1)
2. Find the area of the region bounded by the graph of y = 2x2 – 3x + 2, the x-axis, and the vertical lines x = 0 and x = 2.
Thursday HW: p. 291 (1-41 odd) • Friday HW: Practice with FTOC Worksheet, • p. 305 (71, 75 ,77, 79)
Given • with the initial condition y(2) = -1. Find y(3). • Method 1 • Integrate use the initial condition to find C. Then write the particular • solution , and use your particular solution to find y(3).
Sometimes there is no antiderivative so we have to use Method 2 and our graphing calculator. • Ex.f ′(x) = sin(x2) and f (2) = -5. Find f (1).
Ex. The graph of f ′ consists of two line segments and a semicircle as shown on the right. Given that f (-2) = 5, find: • (a) f (0) • (b) f (2) • (c) f (6) Graph of f ′
(a) f (0) Graph of f ′
(b) f (2) Graph of f ′
(c) f (6) Graph of f ′
Ex. The graph of f ′ is shown. Use the figure and the fact that f (3) = 5 to find: • (a) f (0) • (b) f (7) • (c) f (9)
Then sketch the graph of f using the points from a through c. • + area → f is increasing • − area → f is decreasing • Relative maximum at (3, 5) • Relative minimum at (7, -4)
Ex. • A pizza with a temperature of 95°C is put into a 25°C room when t = 0. The pizza’s temperature is decreasing at a rate of r(t) = 6e-0.1t °C per minute. Estimate the pizza’s temperature when t = 5 minutes.
Derivatives Mean Value Theorem • If f is continuous on [a, b] and differentiable on (a , b) such that: a c1 c2 b Mean Value Theorem for Derivatives
Integrals Too big Too small Exact Area c a b a b Area of Rectangle Mean Value Theorem for Integrals L W
The value f (c) given in the Mean Value Theorem for Integrals is called the average value of f on [a, b] and is denoted by
Example 2 • Find the average value of f (x) = 3x2 – 2x on the interval [1, 4].
First Fundamental Theorem of Calculus • The First Fundamental Theorem of Calculus could also be called the “Total Change Theorem”. If you are given the rate of change of a function, f ′(x), you can find the accumulated change in the function f (x). or
1. Suppose that C(t) represents the cost per day to heat your house measured in dollars per day, where t is measured in days and t = 0 corresponds to January 2, 1993. Interpret
Answer: • The units for the integral are • (dollars/day)(days) = dollars. • The integral represents the total cost in dollars to heat your house for the first 90 days of 1993 (January through March). • The second expression is the average cost per day to heat your house for the first 90 days of 1993. The units would be in dollars/day, the same units as C(t).
2. If V(t) is the volume of water in a reservoir at time t, then its derivative V ′(t) is the rate at which water flows into the reservoir at time t. • change in the amount of water in the reservoir between time t1 and t2.
3. If the rate of growth of a population isdn/dt, • increase in population during the time period from t1 to t2.
4. If w′(t) is the rate of growth of a child in pounds per year, what does • It represents the increase in weight from 5 years old to 10 years old. The units is pounds.
5. If oil leaks from a tank at a rate of r(t) gallons per minute at time t, what does • It represents the decrease of oil in the tank over the 1st 2 hours or 1st 120 minutes. The units are gallons.
6. A honey bee population starts with 100 bees and increases at a rate n′(t) bees per week. • n(15) represents the total amount of bees after 15 weeks.
7. A cup of coffee at 90° is put into a 20° room when t = 0. The coffee’s temperature is changing at a rate of r(t) = -7e-0.3t °C per minute with t in minutes. Find the coffee’s temperature when t = 10.
8. The rate at which water is being pumped into a tank is r(t) = 20e0.02t where t is in minutes and r(t) is in gallons per minute. • a. Approximately how many gallons of water have been pumped into the tank in the first five minutes? • b. Find the average rate at which the water is being pumped into the tank during the first five minutes.
9. A faucet was turned on at t = 0, and t minutes later water was flowing into a barrel at a rate of t2 + 4t gallons per minute, 0 ≤ t ≤ 5. • a. How much water was added to the barrel during these 5 minutes? • b. Find the average flow rate for these five minutes.
a. b. HW: p 291 (43, 47); FTOC Worksheet #2