Section 4.4 Day 1

Antidifferentiation → Indefinite Integral: (the family of functions) Definite Integration → Definite Integral: a number On the calculator: Alpha – Window − 4

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

An Illustration of FTOC

Examples

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.

Day 2 When using u-substitution with a definite integral, it is often convenient to determine the limits of integration for the variable u rather than to convert the antiderivative back to the variable x and evaluate at the original limits.

Change of Variables for Definite Integrals If the function u = g(x) has a continuous derivative on the closed interval [a, b] and f is continuous on the range of g, then

Example 1 The new upper and lower limits of integration: When x = 0, then u = 02 + 1 = 1. (lower limit) When x = 1, then u = 12 + 1 = 2. (upper limit)

Example 2

The new upper and lower limits of integration:

Day 3

Given with the initial condition y(2) = -1. Find y(3) 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).

Method 2: Use the Fundamental Theorem of Calculus

Sometimes there is no antiderivative so we have to use Method 2 and our graphing calculator. 3. f ′(x) = sin(x2) and f (2) = -5. Find f (1).

4. 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 ′

5. 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)

(a) f (0)

(b) f (7)

(c) f (9)

Then sketch the graph of f using the points from a through c 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)

6. 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.

Applications of the First Fundamental Theorem Day 4

Mean Value Theorem for Derivatives If f is continuous on [a, b] and differentiable on (a , b) such that: a c1 c2 b Mean Value Theorem for Derivatives

In Section 4.2, we saw that the area of a region under a curve is greater than the area of an inscribed rectangle and less than the area of a circumscribed rectangle. The Mean Value Theorem for Integrals states that somewhere between the inscribed and circumscribed rectangles there is a rectangle whose area is precisely equal to the area of the region under the curve

Mean Value Theorem for Integrals b b Too small a Too big b a Exact Area b a Area of Rectangle L W c Mean Value Theorem for Integrals

Mean Value Theorem for Integrals If f is continuous on [a, b] and differentiable on (a , b) such that:

Example 1

The value f (c) given in the Mean Value Theorem for Integrals is called the average value of f on [a, b]. Average Value of f

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).

Applications Examples

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

The units for the integral are (dollars/day)(days) = dollars 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. is the change in the amount of water in the reservoir between time t1 and t2.

3. If the rate of growth of a population is dn/dt, is the 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 of a child from 5 years old to 10 years old. The units are 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. 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 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 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. Use a calculator to find how much water was added to the barrel during these 5 minutes?

b. Find the average flow rate for these five minutes.

Functions Defined by Integrals Day 6

The Definite Integral as a Number Constant Constant f is a function of x

The Definite Integral as a Function of x F is a function of x Constant f is a function of t

1. A function g is defined on the interval [0, 6] by where f is the function graphed on the next slide. Fill in the following table of values.

Graph of f x   g(x) 1 2 3 4 5 6 ½ 1 ½ Graph of g ½ 1

2. Let where f is the function graphed on the next slide 2. Let where f is the function graphed on the next slide. Fill in the following table of values.

x   g(x) -2 2 4 5 Graph of f -   − 1  − ½

3. The graph of a function f consists of a quarter circle and line segments. Let g be the function given by Graph of f

b. Find all values of x on the open interval (-1, 5) at which g has a relative maximum. Justify your answer. g′ = f = 0 or undef. f = 0 at x = 2 g has a relative max. at x = 2 because g′ = f which changes from positive to negative at x = 2. Graph of f

c. Find the absolute minimum value of g on [-1, 5] and the value of x at which it occurs. Justify your answer. The candidates for the absolute minimum are at the endpoints and the critical numbers. g(-1) = -1 g(2) =  g(5) =  − 4 The absolute minimum is at x = -1.

Day 7

4. Use the Fundamental Theorem of Calculus to rewrite the following accumulation functions. Then find the derivative of each function.

The Second Fundamental Theorem of Calculus

Second Fundamental Theorem of Calculus (Chain Rule Version)

7. Use the Second Fundamental Theorem to evaluate:

d. Find the x-coordinate of each point of inflection on the graph of g in (-1, 5). Justify your answer. x = 0 and x = 3 are points of inflection because g′ changes from increasing to decreasing (x = 0) and decreasing to increasing (x = 3).