1. Problem of the Day x 0 0.5 1.0 1.5 2.0 f(x) 3 3 5 8 13 A table of values for a continuous function f is shown above. If four equal subintervals of [0, 2] are used, which of the following is the trapezoidal approximation of A) 8 B) 12 C) 16 D) 24 E) 32

2. Problem of the Day x 0 0.5 1.0 1.5 2.0 f(x) 3 3 5 8 13 A) 8 B) 12 C) 16 D) 24 E) 32

3. Graph f(x) = ln x Graph f(x) = e x What do you notice about these 2 functions and their relationship?

4. Graph f(x) = ln x Graph f(x) = e x What do you notice about these 2 functions and their relationship? They appear to be reciprocals of each other and thus inverses.

5. The inverse of the natural logarithmic function f(x) = ln x is called the natural exponential function and is denoted by f -1 (x) = e x i.e. y = e x if and only if x = ln y And ln(e x ) = x and e ln x = x (inverse property)

6. Examples 7 = e x + 1 ln 7 = ln(e x + 1 ) ln 7 = x + 1 ln 7 - 1 = x.946 ≈x natual log both sides and apply inverse property

7. Examples ln(2x - 3) = 5 e ln(2x - 3) = e 5 2 x - 3 = e 5 x = ½(e 5 + 3) x ≈ 75.707 exponential both sides and apply inverse property

8. Caution! cannot do if multiple pieces on each side of = ln(5x + 1) + ln x = ln 4 (5x + 1) + x = 4 Incorrect Correct ln(5x + 1)x = ln 4 (5x + 1)x = 4

9. Operations and Properties e a e b = e a + b ea = e a - b eb 1. Domain (-∞, ∞) 2. Range (0, ∞) 3. continuous, increasing, 1 to 1 4. concave up 5. lim e x = 0 lim e x = ∞ x -∞ x ∞

10. Derivatives (the natural exponential function is the only function besides the zero function that is its own derivative)

11. Examples u = 2x -1 du = 2 u = -3x -1 d u = (-3)(-1)x -2

12. How Fast Does a Flu Spread? The spread of a flu in a certain school is modeled by the equation P(t) = 100 1 + e 3 - t w here P(t) = total number of students infected t days after the flu was first noticed. Many of them may already be well again at time t. Estimate the initial number of students infected with the flu

13. How Fast Does a Flu Spread? The spread of a flu in a certain school is modeled by the equation P(t) = 100 1 + e 3 - t w here P(t) = total number of students infected t days after the flu was first noticed. Many of them may already be well again at time t. Estimate the initial number of students infected with the flu t = 0 P(0) = 100 = 5 1 + e 3 - 0

14. How Fast Does a Flu Spread? The spread of a flu in a certain school is modeled by the equation P(t) = 100 1 + e 3 - t w here P(t) = total number of students infected t days after the flu was first noticed. Many of them may already be well again at time t. How fast is the flu spreading after 3 days (i.e. rate of change)

15. How Fast Does a Flu Spread? at t = 3 100e 0 = 25 (1 + e 0 ) 2

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