MAT137 Calculus! Lecture 17

Transcription

1 MAT137 Calculus! Lecture 17 Today: 4.10 Related Rated Local and Global Extrema Next: Mean Value Theorem v official website

2 Arctan This inverse is called the arc tangent function: y = arctan x, x R. Since these functions are inverses to each other, we have tan(arctan x) = x for all x R; ( arctan(tan x) = x for all x π 2, π ). 2

3 Inverse Trigonometric Functions Calculate: 1 tan(arcsin 1 3 ) 2 arctan(tan 2) 3 tan(arctan 2)

4 Inverse Trigonometric Functions Calculate: sec(arctan x)

5 Derivative of Inverse Trigonometric Functions Find d dx arctan x.

6 Inverse Trigonometric Functions Name Notation Definition Maps arcsine y = arcsin x sin y = x [ 1, 1] [ π 2, ] π 2 arccosine y = arccos x cos y = x [ 1, 1] [0, π] arctangent y = arctan x tan y = x (, ) ( π 2, ) π 2 arccosecant y = arccsc x csc y = x (, 1] [1, ) [ π 2, 0) ( ] 0, π 2 arcsecant y = arcsec x sec y = x (, 1] [1, ) [ ) ( 0, π 2 π 2, π] arccotangent y = arccot x cot y = x (, ) (0, π).

7 Derivative of Inverse Trigonometric Functions d 1 dx (arcsin x) = 1 x 2 d 1 dx (arctan x) = x 2 +1 d dx (arcsec x) = 1 x x 2 1 d 1 dx (arccos x) = 1 x 2 d 1 dx (arccot x) = x 2 +1 d 1 dx ( arccsc x) = x. x 2 1

8 Finding Local and Global Extrema Example 1 Find local and global extrema of the function with this graph

9 Extreme Value Theorem Theorem (Extreme Value Theorem) If f is continuous on a closed and bounded interval [a, b], then f attains both a maximum value M and a minimum value m in [a, b].

10 Local Extrema Theorem (Local Extreme Value Theorem) If f has a local maximum or minimum at an interior point c of its domain, then f (c) = 0 or f (c) does not exist

11 Finding Local and Global Extrema Example 2 Find the local and global extrema of the function f (x) = x 2/3 (x 1) 3 on the interval [ 1, 2]

12 Finding Local and Global Extrema Example 2 Find the local and global extrema of the function f (x) = x 2/3 (x 1) 3 on the interval [ 1, 2]

13 Finding Local and Global Extrema Let f be continuous on a closed and bounded interval [a, b]. The Extreme Value Theorem says that f attains a maximum and a minimum. The Local Extreme Value Theorem says the the only places where a function can possibly have an extreme value are 1 interior points where f = 0, 2 interior points where f is undefined, 3 endpoints of the domain of f.

14 Ladder and Slippery floor Trying to reach her calculus textbook, Belle props a 10-foot ladder against the bookcase. Unfortunately, the floor was very slippery because the Beast had just mopped it, and the base of the ladder slides away from the bookcase at a rate of 1 foot per second. How fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6 ft from the wall?

15 Ladder and Slippery floor Trying to reach her calculus textbook, Belle props a 10-foot ladder against the bookcase. Unfortunately, the floor was very slippery because the Beast had just mopped it, and the base of the ladder slides away from the bookcase at a rate of 1 foot per second. According to the model we used, what happens as the top of the ladder approaches the ground? Is the model appropriate for small values of y?

16 Strategy 1 Read the problem carefully. 2 Draw a diagram if possible. 3 Introduce notation. Assign symbols to all quantities that are functions of time. 4 Express the given information and the (unknown) required rate in terms of derivatives. 5 Find an equation that relates the relevant variables. 6 Use the Chain Rule to differentiate both sides of the equation with respect to t. 7 Plug the given information into the resulting equation and solve for the unknown. 8 State the final answer in a coherent form, specifying the units that you are using.

17 UP! Thousands of balloons were tied to Carl s home, as a result the house begins to rise. The house leaves the ground 500 ft away from an observer and rises vertically at a rate of 14 ft/min. At what rate is the observer s viewing angle changing at the instant when the house is exactly 500 ft above the ground?

18 Shrek and Donkey At noon, after a long phone conversation fighting over where to og for lunch, Shrek and Donkey decided to go to different places. Donkey is 4 km west of Shrek s home. Shrek starts walking south at 3 km/h and, Donkey begins walking north at 5 km/h. How fast is the distance between them changing at 1:00 pm?